Nevanlinna Theory, Normal Families, and Algebraic Differential Equations [1st ed.] 331959799X, 978-3-319-59799-7, 978-3-319-59800-0, 3319598007

Table of contents :
Front Matter ....Pages i-xviii
Selected Topics in Complex Analysis (Norbert Steinmetz)....Pages 1-31
Nevanlinna Theory (Norbert Steinmetz)....Pages 33-73
Selected Applications of Nevanlinna Theory (Norbert Steinmetz)....Pages 75-112
Normal Families (Norbert Steinmetz)....Pages 113-136
Algebraic Differential Equations (Norbert Steinmetz)....Pages 137-172
Higher-Order Algebraic Differential Equations (Norbert Steinmetz)....Pages 173-224
Back Matter ....Pages 225-235

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Norbert Steinmetz

Nevanlinna Theory, Normal Families, and Algebraic Differential Equations

123

Norbert Steinmetz FakultRat fRur Mathematik TU Dortmund Dortmund Germany

ISSN 0172-5939 Universitext ISBN 978-3-319-59799-7 DOI 10.1007/978-3-319-59800-0

ISSN 2191-6675 (electronic) ISBN 978-3-319-59800-0 (eBook)

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Rolf Nevanlinna (1895–1980) Hans Wittich (1911–1984) Photo: Privately owned

Introduction and Preface

Nevanlinna Theory, Normal Families, and Algebraic Differential Equations—how are these topics related to each other? Zalcman’s Re-scaling method set up a way to combine Nevanlinna Theory and Normal Families in both directions—to prove qualitative (‘soft’) results in Nevanlinna Theory by using Normal Family methods, and to prove normality criteria using results from Nevanlinna Theory as a pattern. Part of Schiff’s Normal Families is dedicated to this interesting and fruitful connection. In the present Chap. 4, some old and new results in this direction are presented. Moreover, the connection of Normal Families with Algebraic Differential Equations is discussed on an elementary level. In his seminal paper on the value distribution of meromorphic functions and their derivatives, Hayman opened a new field of application of Nevanlinna Theory. Some of his results on differential polynomials are outlined in his indispensable Meromorphic functions. He initiated a vast field of research in the 1970s and 80s, which to the author’s knowledge has never appeared in book form. Nevanlinna himself was the first to apply his theory to problems of uniqueness of meromorphic functions, known as the Five- and Four-Value Theorems. The enormous progress initiated by Gundersen in the 1980s and early 90s has also never been presented in book form. In combination with introductory applications to Algebraic Differential Equations, these topics constitute Chap. 3. The benefits of applying Nevanlinna Theory to the field of Algebraic Differential Equations were first recognised in the early 1930s, and then systematically since the 50s due to the pioneering work of Wittich. Apart from single chapters and a few remarks in the books [15, 84, 85, 96, 202], just Laine’s monograph Nevanlinna Theory and Complex Differential Equations is dedicated to this field. Since the beginning of the new century much progress has been made in the context of Painlevé Differential Equations.1 For the first time, Re-scaling and Normal Family

1

We quote from the introduction to L.A. Rubel’s Entire and meromorphic functions [146], where the author expressed the need to have more examples of interesting meromorphic functions. One vii

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arguments were used to gain new insight into the nature of the so-called Painlevé transcendents and solutions to other algebraic differential equations. Inspired by Zalcman’s Re-scaling Lemma and older ideas due to Yosida, the so-called Yosida classes entered the stage quite naturally. This material is presented in Chaps. 5 and 6, and in parts also in Chap. 4. The monograph [60] is concerned with Painlevé’s Equations from the complex analytic point of view, but was published ‘too early’ to incorporate the newest developments, methods, and results. This, of course, may happen to every book. Regrettably, the theory of Differential Equations in the Complex Domain is not commonly accepted as a genuine part of Complex Analysis, though it is completely based on complex analytic techniques. Chapters 2, 3, and 4 urgently demonstrate that the theory of Algebraic Differential Equations provides indispensable tools, and their solutions often mark the range of validity of important results in Nevanlinna Theory. Chapter 1 is included for the convenience of the reader. It provides material from classical Complex Analysis, which—apparently or seemingly—does not belong to the generally accepted background. This is particularly true for the topics of Ordinary Differential Equations and Asymptotic Expansions. In Chap. 2 not only the classical Nevanlinna Theory is outlined, but also Cartan’s Theory of Entire Curves and the Selberg–Valiron Theory of Algebroid Functions is briefly presented. This includes generalisations of the Second Main Theorem and various applications to problems in Complex Analysis which then become ‘elementary’, The text is not written redundancy-free. Some of the problems are dealt with at an early stage using the methods at hand, and are picked up later on when new tools are available. Several examples and exercises require extensive computations, which in principle can be realised by hand. It is, however, much more convenient to use some computer algebra system like MAPLE. Non-experts like the author can use MAPLE like a separate sheet of paper to carry out auxiliary computations. The present text was written within the first 2 years of my ultimate sabbatical, but has a long history. It developed from research in Nevanlinna Theory, Analytic Differential Equations, and related subfields of Complex Analysis starting in the late 1970s, with a break in the 90s, which were occupied by other activities. The choice of the material naturally depends on personal preferences, experiences, and skills. Rather than to aim at completeness of the presentation, I intended to explain the main ideas and results exemplarily. I appreciate the support and criticism by friends and colleagues. Of course, the responsibility for mistakes and misinterpretations remains with me. I also profited very much from the survey by A. Eremenko and J. Langley within the English translation of the monograph Value Distribution of Meromorphic Functions by A.A. Gol’dberg and I.V. Ostrovskii, and, in particular, from helpful comments by the

promising source of such examples is the Painlevé transcendents. However, in spite of a growing literature on these functions, the unfortunate fact is that the “proofs” are incomplete and not rigorous [. . . ]—and also in earlier and the earliest papers, one could add.

Introduction and Preface

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referees. It is thanks to them if the present version has become more reader-friendly than the version they commented on. Finally, I gratefully acknowledge the seminal impact the wonderful books Aufgaben und Lehrsätze aus der Analysis I und II (Problems and Theorems in Analysis I and II) by G. Pólya and G. Szegö had on me since my time as a student. They accompanied me for life. I would be pleased if some of the exercises I posed in place of proofs and worked out examples came close to the spirit of Pólya and Szegö. Definitions and theorems form the skeleton of a theory, while it is brought to life by applications, examples, and exercises only. Dortmund, Germany

Norbert Steinmetz

Contents

1

Selected Topics in Complex Analysis . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Algebraic Functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Local Branches.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 Regular and Singular Points . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.3 The Newton Polygon . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.4 Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Normal Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Sequences of Holomorphic Functions . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 Sequences of Meromorphic Functions.. . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Normal Families of Meromorphic Functions .. . . . . . . . . . . . . . . . . 1.3 Ordinary Differential Equations .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Holomorphic Functions of Several Variables.. . . . . . . . . . . . . . . . . 1.3.2 Cauchy’s Existence Theorem . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.3 Linear Differential Equations and Systems .. . . . . . . . . . . . . . . . . . . 1.3.4 Some Algebraic Aspects . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.5 Cauchy’s ‘calcul des limites’ . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.6 The Complex Implicit Function Theorem .. . . . . . . . . . . . . . . . . . . . 1.3.7 Dependence on Parameters and Initial Values .. . . . . . . . . . . . . . . . 1.3.8 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.9 Painlevé’s Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Asymptotic Expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Asymptotic Series . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.2 Asymptotic Integration of Differential Equations .. . . . . . . . . . . . 1.4.3 Asymptotic Integration of Algebraic Differential Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Miscellanea .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.1 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.2 The Phragmén–Lindelöf Principle . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.3 Wiman–Valiron Theory.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5.4 The Schwarzian Derivative .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 2 2 4 5 6 6 6 9 9 10 11 12 13 14 15 16 16 17 19 19 20 22 24 24 25 28 30

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2 Nevanlinna Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 The First Main Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.1 The Poisson–Jensen Formula . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.2 The Nevanlinna Functions.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.3 The Nevanlinna Characteristic . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.4 Valiron’s Lemma .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.5 Cartan’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.6 The Ahlfors–Shimizu Formula .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.7 The Order of Growth . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.8 Canonical Products . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 The Second Main Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 The Lemma on the Logarithmic Derivative . . . . . . . . . . . . . . . . . . . 2.2.2 The Second Main Theorem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.3 The Deficiency Relation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.4 Higher Derivatives .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.5 Generalisations of the Second Main Theorem . . . . . . . . . . . . . . . . 2.2.6 Zeros of Linear Differential Polynomials .. . . . . . . . . . . . . . . . . . . . 2.2.7 Yamanoi’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Applications of Nevanlinna Theory .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Theorems of Hadamard and Borel . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.2 The Tumura–Clunie Theorem .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.3 The Order of the Derivative . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Ramified Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.5 Parametrisation of Simple Algebraic Curves .. . . . . . . . . . . . . . . . . 2.4 Cartan’s Theory of Entire Curves . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 The Characteristic of an Entire Curve . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.2 The Ahlfors–Shimizu Formula .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.3 Cartan’s First and Second Main Theorem .. . . . . . . . . . . . . . . . . . . . 2.4.4 Borel Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 The Selberg–Valiron Theory of Algebroid Functions .. . . . . . . . . . . . . . . . 2.5.1 Algebroid Functions .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.2 Two Equivalent Approaches.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.3 The First Main Theorem for Algebroid Functions . . . . . . . . . . . . 2.5.4 The Characteristics of f and A =Ak . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.5 The Logarithmic Derivative . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.6 The Second Main Theorem for Algebroid Functions . . . . . . . . .

33 33 33 34 36 36 38 39 40 41 42 43 46 47 47 48 49 51 52 52 54 54 55 56 58 58 58 59 61 63 63 64 66 67 68 70

3 Selected Applications of Nevanlinna Theory . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Algebraic Differential Equations .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Valiron’s Lemma Again . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.2 Valiron’s Lemma and Malmquist’s First Theorem . . . . . . . . . . . . 3.1.3 Eremenko’s Lemma and Malmquist’s Second Theorem .. . . . . 3.1.4 Elementary Techniques .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

75 75 75 76 77 78

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3.2 Zeros of Differential Polynomials .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Hayman’s Work on f nC2 C af 0 and f n f 0  a . . . . . . . . . . . . . . . . . . . 3.2.2 Generalisations of Hayman’s Theorem . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 Limit Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.4 The Tumura–Clunie Theorem .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.5 Homogeneous Differential Polynomials .. .. . . . . . . . . . . . . . . . . . . . 3.3 Uniqueness of Meromorphic Functions .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 The Five-Value Theorem . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.2 Examples and Counterexamples . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.3 The Four-Value Theorem .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.4 Variations of the Four-Value Theorem .. . . .. . . . . . . . . . . . . . . . . . . . 3.3.5 Reinders’ Example Rediscovered .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.6 Three Functions Sharing Four Values . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.7 Pair Sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.8 Five Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.9 Gundersen’s Example Rediscovered .. . . . . .. . . . . . . . . . . . . . . . . . . .

81 81 82 84 88 91 94 94 95 97 98 103 106 107 108 110

4 Normal Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Re-scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Zalcman’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Pang’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.3 Functions of Poincaré, Abel, and Zalcman . . . . . . . . . . . . . . . . . . . . 4.1.4 The Theorems of Picard and Montel.. . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.5 Normal Functions .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Applications of the Zalcman–Pang Lemma . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 Examples of Bloch’s Principle . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 From Finite to Infinite Order .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 From Normal Families to Differential Equations . . . . . . . . . . . . . 4.2.4 From Differential Equations to Normal Families . . . . . . . . . . . . . 4.3 The Yosida Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Yosida Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 A Modified Spherical Derivative .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 A Modified Ahlfors–Shimizu Formula . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 The Distribution of Zeros and Poles . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.5 The Class Y0˛;ˇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.6 Yosida Classes and Riccati Differential Equations .. . . . . . . . . . .

113 113 113 114 115 119 120 122 122 125 127 128 129 129 130 131 132 134 135

5 Algebraic Differential Equations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Linear Differential Equations.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.1 The Order of Growth . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.2 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.3 Sub-normal Solutions.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.4 The Phragmén–Lindelöf Indicator . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.5 The Distribution of Zeros . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.6 Exceptional Fundamental Sets . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.7 Fundamental Sets with Zeros Along the Real Axis . . . . . . . . . . .

137 137 138 139 140 141 142 143 146

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5.2 Riccati Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 A Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Value Distribution . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Truncated Solutions.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 Poles Close to a Single Line . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.5 Locally Univalent Meromorphic Functions . . . . . . . . . . . . . . . . . . . 5.2.6 A Problem of Hayman .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 First-Order Algebraic Differential Equations .. . . . .. . . . . . . . . . . . . . . . . . . . 5.3.1 Malmquist’s Second Theorem . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.2 Binomial Differential Equations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.3 Briot–Bouquet Equations.. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.4 Back to Malmquist’s Second Theorem . . . .. . . . . . . . . . . . . . . . . . . . 5.3.5 Equations of Genus Zero . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.6 Some Examples of Genus Zero . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3.7 Equations of Genus One .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Differential Equations and the Yosida Classes . . . .. . . . . . . . . . . . . . . . . . . . 5.4.1 Application to First-Order Differential Equations . . . . . . . . . . . .

147 148 151 152 156 157 157 158 158 160 161 162 163 166 168 169 169

6 Higher-Order Algebraic Differential Equations. . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.1 Painlevé Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.2 The Painlevé Story .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.3 The Painlevé Transcendents .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.4 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1.5 Theorems of Malmquist Type . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 The Painlevé Property.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2.1 The Painlevé Property for System (IV) and Painlevé (IV) . . . . 6.2.2 The Painlevé Property for Painlevé (I) and (II) .. . . . . . . . . . . . . . . 6.3 Algebraic Differential Equations and the Yosida Classes . . . . . . . . . . . . . 6.3.1 Special Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 The Yosida Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.3 Yosida–Zalcman–Pang Re-scaling . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.4 Yosida Re-scaling .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Value Distribution .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.1 The Cluster Set of the Solutions to (IV) . . .. . . . . . . . . . . . . . . . . . . . 6.4.2 Strings and Lattices . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.3 Weber–Hermite Solutions . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.4 Value Distribution of the Painlevé Transcendents .. . . . . . . . . . . . 6.4.5 Airy Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4.6 The Painlevé Hierarchies . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Asymptotic Expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5.1 Pole-Free Sectors and Asymptotic Expansions . . . . . . . . . . . . . . . 6.5.2 Truncated Solutions.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

173 173 174 175 175 177 180 182 183 186 188 188 189 190 195 196 197 198 201 202 204 205 206 206 210

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6.6 Sub-normal Solutions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.1 Sub-normal Second Transcendents . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6.2 The Deficiency of Zero of Second Transcendents . . . . . . . . . . . . 6.6.3 Sub-normal Solutions to System (IV) and Painlevé (IV) . . . . . 6.6.4 The Deficiency of Zero of Fourth Transcendents . . . . . . . . . . . . .

xv

213 214 217 218 222

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 225 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233

Notation

N, N0 , Z Q, R C, b C Re, Im D, H 4ı .p/ P, Q C, L Y0˛;ˇ ; Y˛;ˇ

Set of positive, non-negative, all integers Set of rational, real numbers Complex plane, Riemann sphere C [ f1g Real- and imaginary part Unit disc, upper or right half-plane Local disc jz  pj < ıjpjˇ of ‘radius’ ı Set of poles and zeros (of the function in question) Cluster set, lattice Yosida classes

. st ; $/-string j  j, .; / k  k, k  k1 dist .z; A/ f ] , f ]˛ Sf , ff ; zg W.w1 ; : : : ; wn / } .a; b; c; d/ M.r; f /, m.r; f / N.r; f /, n.r; f / T.r; f /, S.r; f / TC .r; g/, TS .r; f/, TV .r; f/ ı.a; f /, #.a; f / %.f /, %a .f / ord deg R, degx R d˝ , d˝ resp f hf

Sequence satisfying pkC1 D pk C .$ C o.1//pk t Absolute value, chordal metric on b C Euclidean, maximum norm on Rn and Cn Euclidean distance of z to the set A Spherical derivative, modified spherical derivative Schwarzian derivative Wronskian determinant of w1 ; : : : ; wn Weierstraß P-function Cross-ratio of a; b; c; d 2 b C Maximum modulus, Nevanlinna proximity function Integrated, non-integrated counting function of poles Nevanlinna characteristic, remainder term Cartan, Selberg, Valiron characteristic Deficiency, ramification index of a Order of growth, exponent of convergence of a-points Order of Airy and Weber–Hermite solution Degree, degree w.r.t. x of the rational function R Degree, weight of the differential polynomial ˝ Residue of f at p Phragmén–Lindelöf indicator function

s

xvii

xviii

 

Notation

.r/  .r/ , 1=C  j.r/= .r/j  C (r  r0 ), also an  bn , 1=C  jan =bn j  C .n  n0 / .r/ P .r/ , .r/= .r/ ! 1 as r ! 1, also k f .z/  1 kD0 ck z , asymptotic series representing f End of proof, example, exercise, remark

Chapter 1

Selected Topics in Complex Analysis

In this chapter we will discuss several topics in Complex Analysis which usually are not or only incomprehensively considered in lectures and textbooks, but are of particular interest in the field of Analytic and Algebraic Differential Equations. Our standard reference is Ahlfors’ forever young Complex Analysis [2].

1.1 Algebraic Functions The quadratic equation P.x; y/ D xy2  2x2 y C x C 1 D 0 has two solutions y D p 1;2 .x/ D x ˙ 4P .x/=x .4P D x3  x  1/I they are holomorphic except at x D 0, a zero of the leading coefficient, x D 1, which always has to be considered separately, and the zeros x1 ; x2 ; x3 of the discriminant 4P of P with respect to y. At 1 1 x D 0 the solutions have ‘algebraic poles’, 1;2 .x/ D ˙ix 2 ˙ 2i x 2 C    , while the p singularities at x are ‘algebraic’: 1;2 D x ˙ c1 x  x ˙ c2 .x  x / C    . At x D 1, 1 .x/ D 2x  12 x2 C    , say, has a pole and 2 .x/ D 12 x1 C 12 x2 C    has a zero. Analytic continuation along closed curves avoiding the points x and 0 leaves the branches  invariant or permutes them. The aim of this section is to extend these simple results to general algebraic equations1 P.x; y/ D am .x/ym C am1 .x/ym1 C    C a0 .x/ D 0 .am .x/ 6 0/;

(1.1)

where P is an irreducible polynomial; irreducibility just means that P cannot be written non-trivially as P1 P2 . By 4P we denote its discriminant with respect to y.

1

Though the variables are complex, we prefer to write x and y to emphasise their equivalence.

© Springer International Publishing AG 2017 N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, DOI 10.1007/978-3-319-59800-0_1

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1 Selected Topics in Complex Analysis

1.1.1 Local Branches For x fixed, Eq. (1.1) has m mutually distinct finite solutions y D  .x/ provided am .x/4P .x/ ¤ 0. Locally defined holomorphic functions solving some Eq. (1.1) are called algebraic functions, and, more precisely, branches of the Algebraic Function defined by (1.1). Local existence is guaranteed by Theorem 1.1 Suppose P.x0 ; y0 / D 0 and Py .x0 ; y0 / ¤ 0: Then in a neighbourhood of x D x0 there exists a unique holomorphic solution y D .x/ to Eq. (1.1) with .x0 / D y0 . Proof For  > 0 sufficiently small, P.x0 ; y/ ¤ 0 holds on 0 < jy  y0 j  , hence there exists a ı > 0 such that P.x; y/ ¤ 0 holds on jx  x0 j < ı, jy  y0 j D . For x fixed, Eq. (1.1) has n.x/ D

1 2i

Z jyy0 jD

Py .x; y/ dy P.x; y/

solutions on jy  y0 j < . Since n.x/ is integer-valued and depends continuously on x, n.x/ D n.x0 / D 1 follows. Thus  is defined on jx  x0 j < ı. The proof that  is holomorphic is part of the following exercise. Z 1 yPy .x; y/ 2 dy. Exercise 1.1 Prove that .x/ D 2i jyy0 jD P.x; y/ (Hint. Compute the residue of yPy .x; y/=P.x; y/ at y D .x/.)

1.1.2 Regular and Singular Points Finite points x0 such that P.x0 ; y/ D 0 has m finite and mutually distinct solutions are called regular, and singular otherwise. At regular points there exist m holomorphic branches. The set S of finite singular points consists of the zeros of am .x/4P .x/. Theorem 1.2 Any branch admits unrestricted analytic continuation in C n S. Analytic continuation along any closed curve avoiding S permutes the branches. Proof Given x0 … S, let r.x0 / denote the largest radius such that all branches are holomorphic on jx  x0 j < r.x0 /. Then a simple geometric argument shows that jr.x1 /  r.x2 /j  jx1  x2 j holds, and this guarantees that every branch admits unrestricted analytic continuation in C n S. Actually, it turns out that r.x/ D dist .x; S/. The second assertion is obvious.

2

Note the similarity to the Bürmann–Lagrange inversion formula.

1.1 Algebraic Functions

3

Let x0 be any finite singularity. Analytic continuation along the circle jxx0 j D r with sufficiently small radius permutes the branches  , and repeating this q times (q suitably chosen) leaves them invariant. Thus analytic continuation along jtj D r1=q leaves the branches s D ˚ .t/ D  .x0 Ctq / of P.x0 Ctq ; s/ D 0 invariant. They are holomorphic on 0 < jtj < r1=q with an isolated singularity at t D 0; the singularity is not essential, this following from P.x0 Ctq ; ˚ .t// D 0 and the Casorati–Weierstraß Theorem. This proves Theorem 1.3 Finite singularities other than ordinary poles are algebraic singularities or algebraic poles: .x/ D

1 X

cn .x  x0 /n=q

.ck ¤ 0/

nDk

holds on jxx0 j < ı, with cn ¤ 0 for at least one n 6 0 mod q. The series represents q distinct branches according to the chosen branch of .x  x0 /1=q . Remark 1.1 Series of this kind are also called Puiseux series. Algebraic and ordinary poles occur if and only if am .x0 / D 0. In this case one has to consider m P.x; ˛ C 1=/ D P.x; ˛/m C    C am .x/ D 0 .m D degy P; P.x0 ; ˛/ ¤ 0/ at .x; / D .x0 ; 0/. To examine the point x D 1 one has just to discuss the equation  n P.1=; y/ D 0 .n D degx P/ at  D 0. Again x D 1 may be regular, an algebraic singularity or an algebraic or ordinary pole for some or all branches. The reader is encouraged to investigate all cases in detail. Exercise 1.2 Compute (using MAPLE, say) the discriminant w.r.t. y of P.x; y/ D y3  3..Na  1/x  2/xy2  3.2x  a C 1/xy  x3 : p Discuss, in particular, the singularities if a D  12 ˙ 2i 3. p Solution for a D  12 ˙ 2i 3: 4P .x/ D x3 .x  a/3 .x C 1/3 : To sum up we can state the Main Theorem on algebraic functions: Theorem 1.4 The branches defined by Eq. (1.1) admit unrestricted analytic continuation in b C n S, where now S may include x D 1. The singularities are algebraic and algebraic poles.3 Analytic continuation along closed curves in b C n S results in a permutation of the branches.

3

Ordinary poles are not viewed as singularities.

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1 Selected Topics in Complex Analysis

Exercise 1.3 The statement of Theorem 1.4 may be reversed. Prove that if 1 ; : : : ; m are holomorphic functions on some domain D, such that – each  admits unrestricted analytic continuation in b C n S, where S is a finite set; – analytic continuation along any closed curve in b C n S results in a permutation of the branches  ; – the points in S are algebraic singularities or algebraic poles for some or all  . are the branches of some algebraic P.x; y/ D 0. Then 1 ; : : : ; m Q Pm1 equation m  (Hint. Consider m D1 .y   .x// D y C D0 b .x/y and show that the b are well-defined holomorphic functions on b C n S without essential singularities.) We note that irreducibility of P means that every permutation may be achieved by choosing the closed curve appropriately. Otherwise the branches decompose in several independent subsets. Exercise 1.4 Prove that if  and are local branches of Algebraic Functions, then so are  C ;  ; = ;  0 ; and  1 : (Hint. For  , say, consider all products   ; note, however, that the new equation might be reducible. The case of  1 is easier than expected.) Example The equation for D  , where  and satisfy y2  x D 0 and 4 4 3 2 y  x D 0, respectively, obtained in this way is .y  x / D 0.

1.1.3 The Newton Polygon In a neighbourhood of x D 1 there are m solutions  .x/ D c x  C    (c ¤ 0) to Eq. (1.1). To determine the principal terms cx write a .x/ D A x˛ C    (A D ˛ D 0 if a  0) and observe that there must be a k and P an ` such that k C ˛k D ` C ˛`   C ˛ holds for each , since the sum ˛ C Dmax A c ˛k ˛ ˛ ˛k must vanish. This implies   k .k > / and   k .k < /; with equality if  D `, and has a simple geometric interpretation: let C denote the convex hull of the points .0; 0/; .0; ˛0 /; .1; ˛1 /; : : : ; .m; ˛m /; .m; 0/. The upper boundary of C is the graph of a concave polygon, called the Newton–Puiseux polygon (due to Newton, rediscovered by Puiseux), and  is one of its slopes. Exercise 1.5 Suppose that the Newton–Puiseux polygon has vertices .kj ; ˛kj / (0 D kP 0 < k1 <    < ks < ksC1 D m/, and slope  j in Œkj ; kjC1 . Set Pj .c/ D 0 kj , where the sum runs over those indices kj    kjC1 such that the  A c points .; ˛ / belong to the Newton–Puiseux polygon, hence ˛ C  j D max D ˛kj C kj j D ˛kjC1 C kjC1 j holds. Prove that there are exactly kjC1  kj branches  cx j as x ! 1, one for each zero c of Pj (counting multiplicities). Exercise 1.6 Describe the analogue of the Newton–Puiseux polygon at x D 0. (Hint. Consider Q.; y/ D  n P.1=; y/ D 0 at  D 1.)

1.1 Algebraic Functions

5

Exercise 1.7 Prove that the algebraic equation P.x; y/ D 0 in Exercise 1.2 has 2 one regular and q two singular solutions at x D 1: 1 .x/ D 3.Na  1/x C    and x 2;3 .x/ D ˙ 3.1Na/ C    Determine the principal terms of the solutions at x D 0 p and, if a D  12 ˙ 2i 3, also at x D a and x D 1.

1.1.4 Algebraic Curves P Any irreducible equation P.x; y/ D Cd a x y D 0 of algebraic degree d defines a so-called algebraic curve and also a compact Riemann surface, which usually are identified with P. Algebraic curves have a genus g D g.P/. The most descriptive definition is g D 12 .d  1/.d  2/  c.P/; where c.P/ denotes the number of ‘double points’ of P; the term 12 .d  1/.d  2/ is the greatest number of double points an algebraic curve of degree d can have. Two cases are of particular interest. Algebraic curves of – genus zero have a rational parametrisation x D r.t/, y D s.t/, that is, P.r.t/; s.t//  0 holds, and are also called rational curves. – genus one are parametrised by elliptic functions x D r.t/, y D s.t/, and are also called elliptic curves. Remark 1.2 Parametrisations are, of course, not unique. The circle x2 C y2 D 1 1t2 2t has the rational parametrisation x D 1Ct 2 , y D 1Ct2 and the entire parametrisation x D cos t, y D sin t. The Weierstraß functions x D }.tI g2 ; g3 /, y D } 0 .tI g2 ; g3 / (g32  27g23 ¤ 0) parametrise the elliptic curve y2 D 4x3  g2 x  g3 (more on elliptic functions in Sect. 1.5.1). In both cases there exists a parametrisation .r; s/ of smallest algebraic degree resp. elliptic order equal to .degy P; degx P/. Any pair . f ; g/ of non-constant meromorphic functions satisfying P. f .z/; g.z//  0 may be written as . f ; g/ D .r ı h; s ı h/, where h is a non-constant meromorphic resp. entire function (see [10]). Example 1.1 The algebraic curve – in Exercise 1.2 with a D  12 ˙ 2 a/ t.t3/ ; .t1/2

i 2

p

3 has genus zero with parametrisation 2

.t3/  y D s.t/ D  19 .1  a/ t .t1/ . x D r.t/ D 3 3 – x C y C 3xy C  D 0 is reducible if  D 1, has genus zero if  D 0 (and is 2 3t known as the folium of Descartes) with parametrisation x D t3t 3 C1 , y D t3 C1 , and has genus one otherwise. 1 .1 9

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1 Selected Topics in Complex Analysis

1.2 Normal Families 1.2.1 Sequences of Holomorphic Functions Convergence of sequences of holomorphic functions . fn / on some domain D is always measured with respect to the euclidean metric, and is always assumed to be locally uniform. The most important results in this context are the Theorems of Weierstraß and Hurwitz. Weierstraß’ Theorem Suppose fn tends to f , locally uniformly on D. Then f is holomorphic on D, and the sequence . fn0 / tends to f 0 , again locally uniformly on D. Hurwitz’ Theorem Suppose fn tends to f 6 0, locally uniformly on D. Then any zero z0 of f is the limit of some sequence .zn / of zeros of fn (n  n0 ). Any m-fold zero z0 is ‘accompanied’ by zeros of fn on jz  z0 j < n ! 0 of ‘total’ multiplicity m, that P is, fn has zeros zn;j with multiplicities mn;j such that zn;j ! z0 and j mn;j D m, and vice versa. A family F of holomorphic functions on some common domain of definition D is called normal if every sequence in F contains a subsequence that converges either to some holomorphic function f or else to infinity, locally uniformly on D. A sequence . fn / is called normal if the countable family F D f fn W n 2 Ng is normal, that is, if every subsequence . fnk / itself contains a convergent subsequence. Normality is a Local Property The family F is normal on D if and only if it is normal at every z0 2 D, that is, if to every z0 2 D there exists an r.z0 / > 0 such that the family Fz0 of functions f 2 F restricted to jz  z0 j < r.z0 / is normal on this disc. This is a corollary of Cantor’s well-known diagonal argument. Vitali’s Theorem If . fn / is normal on D and fn .z/ converges pointwise on some non-discrete set E  D, then the sequence . fn / converges locally uniformly on D. Apparently, the best-known normality criterion is Montel’s Criterion A family F of holomorphic functions on some domain D is normal if F is locally bounded, in which case all limit functions are holomorphic. Remark 1.3 Montel’s Criterion is a corollary of the Arzelà–Ascoli Theorem and Cauchy’s Integral Formula; the latter implies that the family F 0 of derivatives is locally bounded if F is, hence F is locally equi-continuous.

1.2.2 Sequences of Meromorphic Functions In the context of nonlinear algebraic differential equations it is unavoidable to also consider sequences of meromorphic functions. Here convergence has to be measured with respect to the chordal metric .a; b/ D 12 k 1 .a/  1 .b/kR3

1.2 Normal Families

7

on the Riemann sphere; denotes the stereographic projection S2 ! b C. In terms of a; b 2 b C 1 .a; 1/ D p 1 C jaj2

ja  bj and .a; b/ D p p 1 C jaj2 1 C jbj2

.a; b 2 C/

holds. By definition, is invariant under rotations of the sphere S.z/ D ˛zCˇ N ˛N : In ˇzC the complex plane C, the euclidean and the chordal metric are locally equivalent. Weierstraß’ Theorem holds in parts. Theorem 1.5 Suppose that the sequence of meromorphic functions . fn / tends to f 6 1 with respect to the chordal metric, locally uniformly on some domain D. Then f is meromorphic on D. Proof The limit function is continuous (as a map D ! b C). If f .z0 / is finite, then f is bounded on some disc  W jz  z0 j < ı, and fn tends to f with respect to the euclidean metric. Thus f is holomorphic on this disc. If, however, f .z0 / D 1, the same argument applied to .1=fn / shows that 1=f is holomorphic on , and either z0 is a pole of f or else f  1 holds on . The assertion now follows from the classical connectivity argument: the set where f is meromorphic as well as its complement with respect to D (where f  1) is open. Chordal convergence is delicate and differs in many respect from euclidean convergence. The reason for this is that poles and zeros of fn may ‘collide’, and also that the chordal metric is not compatible with the arithmetic structure of the complex numbers. A simple necessary condition for fn ! f 6 1 is that the poles of fn are uniformly separated from the zeros: to every z0 2 D there exists an r.z0 / > 0 such that jz  z0 j < r.z0 / does not simultaneously contain poles and zeros of any fn . Exercise 1.8 The sequences fn .z/ D .z  1=n/1 and gn .z/ D .z C 1=n/1 converge to z1 , uniformly on the unit disc D, say. Prove, however, that fn C gn does not converge to 2z1 in any neighbourhood of the origin. Exercise 1.9 Suppose that fn is meromorphic on D and tends to f 6 1, and n is holomorphic and tends to  6 1, locally uniformly on D. Prove that fn C n tends to f C , and n fn tends to f , if in the latter case the poles of f are separated from the zeros of . (Hint. It suffices to consider neighbourhoods of poles of f , where jn .z/j  12 j fn .z/j may be assumed in the first case, and 0 < ı < jn .z/j  K in the second.) It is obvious that Hurwitz’ Theorem also remains valid for sequences of meromorphic functions. If fn tends to f on jz  z0 j < r and f has an m-fold pole at z0 , hence is zero-free on jz  z0 j < , then 1=fn .n  n0 / is holomorphic on jz  z0 j < and tends to 1=f also in the euclidean metric. By Hurwitz’ Theorem for holomorphic sequences, 1=fn has m zeros on jzz0 j < ı counted with multiplicities, and fn has m poles there .n  nı ).

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1 Selected Topics in Complex Analysis

Exercise 1.10 Prove that the sequence fn .z/ D .z2  n2 /1 tends to f .z/ D z2 , uniformly on D, say, while fn0 ! f 0 fails. Note that the poles of fn are neither uniformly separated from each other nor from the zero of fn0 at z D 0. The second part of Weierstraß’ Theorem fails to hold for sequences of meromorphic functions. Suppose fn converges to f , locally uniformly on jzj < r, and f has a pole of order m at z D 0, and no other. For fn0 ! f 0 it is necessary that there exists a > 0 such that fn0 (n  n ) has no zeros on jzj < ; and fn has only one pole (of order m) on jzj < , since each pole of fn of order ` is a pole of fn0 of order ` C 1. Exercise 1.11 Prove that the just mentioned first condition implies the second. (Hint. Apply the Argument Principle to fn0 and f 0 , noting that fn00 =fn0 tends to f 00 =f 0 , uniformly on jzj D w.r.t. the euclidean metric.) The second condition is decisive for the validity of Weierstraß’ Theorem. Theorem 1.6 Suppose that fn converges to f , locally uniformly on jzj < r, and f has a pole of order m at z D 0, and no other. Then the above conditions are also sufficient in order that fn0 ! f 0 , locally uniformly on some neighbourhood of z D 0. Proof We have to show that the second condition is sufficient. Assume that fn has an m-fold pole at bn ! 0 and no other on jzj < , that is, fn .z/ D n .z/.z  bn /m holds, where n is holomorphic and tends to D zm f .z/ as n ! 1 w.r.t. the euclidean metric, uniformly on jzj D . Then n ! on jzj < holds by the Maximum Principle. From .0/ ¤ 0 and n0 ! 0 it then follows that fn0 .z/ D

.z  bn / n0 .z/  m .z  bn /mC1

n .z/

!

z

0

.z/  m .z/ D f 0 .z/; zmC1

uniformly on some neighbourhood of z D 0. Remark 1.4 In any case the sequence fn0 tends to f 0 , locally uniformly on the punctured disc 0 < jzj < . If the above conditions are violated, then some of the poles of fn0 collide with zeros of fn0 , and in the limit multiplicities disappear as do zeros of fn0 . If fn D 1=Pn , Pn a polynomial of degree m, the equivalence of both conditions follows from the Gauß–Lucas Theorem (the zeros of P0 are contained in the convex hull of the zeros of P). Example 1.2 In the context of algebraic differential equations it is not always necessary to know that fn ! f implies fn0 ! f 0 ; fn00 ! f 00 , etc. Suppose ˝n Œw is a sequence of polynomials in w; w0 ; : : : ; w.m/ with coefficients that are holomorphic on some domain D, and such that ˝n Œw ! ˝Œw as n ! 1, locally uniformly with respect to the euclidean metric on D CmC1 . Suppose also that ˝n Œ fn .z/  0 and .k/ fn ! f 6 1, locally uniformly on D with respect to the chordal metric. Then fn .k/ tends to f , locally uniformly with respect to the euclidean metric on the domain D n fpoles of f g, and ˝Œ f .z/  0 holds on D.

1.3 Ordinary Differential Equations

9

1.2.3 Normal Families of Meromorphic Functions The definition of normality applies to families F of meromorphic functions wordby-word. In the context of algebraic differential equations the limit function 1 is often excluded for intrinsic reasons. For meromorphic functions the distortion is measured by the spherical derivative j f 0 .z/j . f .z C h/; f .z// D h!0 jhj 1 C j f .z/j2

f ] .z/ D lim

rather than j f 0 j, and the part of Montel’s Criterion is played by Marty’s Criterion A family F of meromorphic functions is normal if and only if the family F ] of spherical derivatives f ] is locally bounded. Exercise 1.12 The family Pd of all polynomials P.zI a/ D ad zd C    C a0 of fixed degree d > 2 and kak D max0d ja j  1 is not normal on 1 < jzj  1 (1 is included). Thus convergence a.n/ ! a is not sufficient for P.zI a.n/ / ! P.zI a/. ] For example, Pn .z/ D 1n zd  zd1 satisfies Pn .n/ D nd2 : Prove, however, that the subfamily of polynomials with jad j D 1 is normal. (Hint. Prove that jP.z/j  12 jzjd and jP0 .z/j  2djzjd1 holds on jzj > 2d.)

1.3 Ordinary Differential Equations As in the theory of ordinary differential equations in the real domain we will consider initial value problems for single differential equations w0 D f .z; w/; w.z0 / D w0 ; systems w0 D f.z; w/;

w.z0 / D w0 ;

(1.2)

with w D .w1 ; : : : ; wn / and f D . f1 ; : : : ; fn /, and higher-order differential equations w.n/ D f .z; w; w0 ;    ; w.n1/ /;

w./ .z0 / D w0 .0   < n/

(1.3)

in the complex domain. Since every n-th order equation may be transformed into a system y0 D f.z; y/, for example into y0 D yC1 .1   < n/ and y0n D f .z; y1 ; : : : ; yn / via y D w.1/ (1    n/, all results obtained for systems (1.2) also hold mutatis mutandis for higher-order problems (1.3). Standard references in the complex domain are the books by Bieberbach [15], Golubew [58], Hille [84, 85], Ince [94], and Laine [102]. Since we expect holomorphic solutions, the components

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of the right-hand side f have to be holomorphic on some domain G  C Cn . It is thus necessary to give a short introduction to the field of holomorphic functions of several variables.

1.3.1 Holomorphic Functions of Several Variables Let G  Cn be any domain. A function f W G ! C is called holomorphic if the partial derivatives f .z C he /  f .z/ h!0 h

fz .z/ D lim

.1    n; z D .z1 ; : : : ; zn / 2 G/

exist; e denotes the -th unit vector and h is complex. The one-dimensional theory then yields (z1 ; : : : ; zn1 fixed) f .z1 ; : : : ; zn1 ; zn / D

1 2i

Z Cn

f .z1 ; : : : ; zn1 ; n / dn n  zn

.jzn  zı j < rn /;

and iteration finally gives Cauchy’s Integral Formula f .z/ D

1 .2i/n

Z

Z  Cn

C1

f .1 ; : : : ; n / d1    dn .1  z1 /    .n  zn /

.jz  zı j < r /I

C denotes the positively oriented circle jz  zı j D r , and f is assumed to be holomorphic on the poly-cylinder fz W jz  zı j < R ; 1    ng with R > r . By a famous Theorem of Hartogs (see [101], for example) the existence of the partial derivatives implies the continuity of f , in contrast to the real case. It is then easy to derive X f .z/ D cp .z  zı /p .jz  zı j < r /; p2Nn0

cp D

1 p 1 .r f /.z0 / D pŠ .2i/n

Z

Z  Cn

C1

.1 

f .1 ; : : : ; n / ı p1 C1 z1 /    .n  zın /pn C1

d1    dn ;

with the usual and useful abbreviations p D . p1 ; : : : ; pn / 2 Nn0 (multi-index), p p pŠ D p1 Šp2 Š    pn Š, and zp D z11 z22    zpnn . In particular, it follows that the partial derivatives are also holomorphic. We note some useful corollaries, which may be proved just as in one complex dimension. – jcp j 

M ı p p with M D max j f .z/j on fz W jz  z j D r ; 1    ng. r1 1    rn n

1.3 Ordinary Differential Equations

11

n X d f .g.z// D fw .g.z//g0 .z/ dz D1 (z 2 D  C, f W G  Cn ! C, and g D .g1 ; : : : ; gn / W D ! G). – Locally f satisfies some Lipschitz condition. – Functions f ; g W G ! C already agree on G if they agree on E1    En , where each set E has an accumulation point zı with .zı1 ; : : : ; zın / 2 G.

– The chain rule, in particular

1.3.2 Cauchy’s Existence Theorem Existence and uniqueness of solutions to the initial value problem (1.2) can be proved just as in the real (Picard–Lindelöf) case. For z0 D 0, w0 D 0, say, the initial value problem is equivalent to the integral equation Z

z

w.z/ D

f.; w.// d:

(1.4)

0

Assuming Z.r; R/ D fz W jzj  rg fw W kwk1  Rg  G and kf.z; w/k1  M on Z.r; R/ we obtain a Lipschitz condition kf.z; u/  f.z; v/k1  Lku  vk1 on Z. ; P/, where < r and P< R are chosen in such a way that M  P holds. The space B of bounded holomorphic functions u W fz W jzj < g ! Cn ;

u.0/ D 0

will be endowed with the norm kuk D supfku.z/k1 e2Ljzj W jzj < g: Then B is a Banach space, since norm convergence and uniform convergence agree. The operator T defined by the right-hand side of (1.4) then turns out to be a contracting self map of the closed ball fu 2 B W kuk  Pg. The inequality k.Tu/.z/k1 e2Ljzj  M  P is obvious, and kTu  Tvk  12 ku  vk follows from Z k.Tu/.z/  .Tv/.z/k1  L

z 0

ku.t/  v.t/k1 e2Ljtj e2Ljtj jdtj  Lku  vk

e2Ljzj  1 : 2L

The unique fixed point then yields the desired unique solution to the integral equation and also to the initial value problem.

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1 Selected Topics in Complex Analysis

1.3.3 Linear Differential Equations and Systems Linear systems may be compactly written as w0 D A.z/w C b.z/; where the entries of the n n-matrix A and the n-vector b are holomorphic on some domain D. Since this time we may work on the cylinder G D D Cn , there is no restriction like M P, hence every solution exists on jz  z0 j < dist .z0 ; @D/. In particular, for D D C the components of the solutions are entire functions. Theorem 1.7 Suppose that the entries of A and b are polynomials. Then the components of every solution are entire functions of finite order, that is, log jw .z/j D O.jzj / holds for some  > 0 that is independent of the solution w. Proof We fix  and r > 0, and set u. / D kw. ei /k1 , b D max kb.z/k1 , and jzjr

a D max kA.z/k (operator norm to the maximum norm on Cn ). Then b  ˇ.1 C rk / jzjr

and a  ˛.1 C r` / hold, and u satisfies Z u. /  u.0/ C br C a



u.t/ dt 0

.0   r/:

Gronwall’s Lemma4 yields u. /  .u.0/ C br/ea . For jzj D D r this gives `C1 /

jw .z/j  .kw.0/k1 C ˇ.jzj C jzjkC1 //e˛.jzjCjzj

:

The solutions to the homogenous problem w0 D A.z/w form a vector space of dimension n. The Wronskian determinant ˇ ˇ w11 ˇ ˇ w12 ˇ W D W.w1 ; w2 ; : : : ; wn / D ˇ : ˇ :: ˇ ˇw 1n

ˇ w21    wn1 ˇˇ w22    wn2 ˇˇ :: :: ˇˇ : : ˇ w  w ˇ 2n

nn

of any n-tuple of solutions w satisfies W 0 D trace A.z/ W, hence W.z/ ¤ 0 if and only if the w form a basis, also called a fundamental system or fundamental set.

Rt Gronwall’s Lemma. Suppose u is continuous and satisfies u.t/  ˛ C ˇ 0 u.s/ ds on Œ0; c/ for Rt some ˇ  0. Then u.t/  ˛eˇt holds on Œ0; c/. To prove this set v.t/ D ˛ C ˇ 0 u.s/ ds. Then d u.t/  v.t/ implies v.t/eˇt D .v 0 .t/  ˇv.t//eˇt D ˇ.u.t/  v.t//eˇt  0; hence v.t/eˇt dt decreases and is  v.0/ D ˛, that is, u.t/  v.t/  ˛eˇt holds. 4

1.3 Ordinary Differential Equations

13

Linear equations will be written as w.n/ C an1 .z/w.n1/ C    C a1 .z/w0 C a0 .z/w D b.z/ (a , b holomorphic on some domain D  C); each linear equation is equivalent to some linear system y0 D yC1 .1   < n/ and y0n D b.z/a0 .z/y1   an1 .z/yn via y D w.1/ : For D D C the solutions are entire functions; they have finite order of growth if the coefficients are polynomials. The Wronskian determinant ˇ ˇ ˇ w1 w2    wn ˇˇ ˇ ˇ w0 w02    w0n ˇˇ 1 ˇ W D W.w1 ; w2 ; : : : ; wn / D ˇ : :: :: ˇ ˇ :: : : ˇˇ ˇ ˇ w.n1/ w.n1/    w.n1/ ˇ n 1 2 of any n-tuple of solutions to the homogeneous equation satisfies W 0 D an1 .z/W. Exercise 1.13 Prove the following rules for Wronskian determinants ( f1 ; : : : ; fn ; f meromorphic, w1 ; : : : wm meromorphic and linearly independent). 1. W. ff1 ; : : : ; ffn / D f n W. f1 ; : : : ; fn /.

W.w1 ; : : : ; wm ; w/ D 0. W.w1 ; : : : ; wm / In other words, L can be rediscovered from w1 ; : : : ; wm . W.w1 ; : : : ; wm ; f1 ; : : : ; fn / D W.LΠf1 ; : : : ; LΠfn /: 3. W.w1 ; : : : ; wm /   4. W. f1 ; : : : ; fn ; f / D .1/nC1 f nC1 W . f1 =f /0 ; : : : ; . fn =f /0 :

2. w1 ; : : : ; wm form a fundamental system to LŒw D

For more useful results on Wronskian and other determinants, see Muir [123].

1.3.4 Some Algebraic Aspects We consider linear differential operators K, L, M, etc., with coefficients in some field F that is closed with respect to differentiation, for example, the field of rational functions. By u1 ; : : : ; uk , v1 ; : : : v` , etc., we denote (local) fundamental sets to KŒu D 0, LŒv D 0, etc. Without proof we quote an amalgam of results from the book of Ince [94] and the papers by Frank and Wittich [50] and Spigler [159]. The latter contains quite general and useful results in the real and complex domain. – There exists some linear operator M over the same field F such that the products w D u v satisfy MŒw D 0. The order of M may be chosen as small as possible, namely equal to the rank of the Wronskian matrix of the functions u v . Mutatis mutandis the same is true for the sums w D u C v . – Suppose K has rational coefficients. Then there exists some operator L with polynomial coefficients that annihilates the functions u .

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1 Selected Topics in Complex Analysis

– Suppose the solutions to KŒu D 0 also satisfy L1 Œu D    D Ln Œu D 0. Then there exist linear operators M such that L Œu D M ŒKŒu holds.

1.3.5 Cauchy’s ‘calcul des limites’ Cauchy’s method to prove existence was quite different from the ‘modern’ Picard– Lindelöf method. For simplicity we will describe it for a single differential equation w0 D f .z; w/ D

1 X

ajk zj wk ; w.0/ D 0:

(1.5)

j;kD0

P1 P p p The power series 1 pD0 Cp z is said to dominate pD0 cp zPif jcp j  Cp holds for P1 p every p. This will be expressed by writing pD0 cp zp 1 pD0 Cp z I the relation ‘ ’ may easily be extended to power series in several variables. The radius of convergence of the first power series is not less than the radius of convergence of the second. If f in (1.5) is bounded by M on jzj < r, jwj < R, then the power series for f is dominated by the geometric series 1 X j;kD0

Ajk zj wk D



rRM .r  z/.R  w/

Our problem (1.5) has a formal solution

P1

 jajk j  Ajk D Mrj Rk :

` `D1 c` z .

The coefficients have the form

c` D P` .ajk W j C k < `/; where P` is a universal polynomial in the variables ajk with non-negative coefficients ( for example, c1 D a00 and 2Šc2 D a10 C a01 a00 ). To prove convergence of the formal power series solution we consider the comparison problem W0 D

rRM ; W.0/ D 0 .r  z/.R  w/

(or any other problem W 0 D F.z; W/, W.0/ D 0 with f F). The solution r   z  p 2Mr log 1  . 1 D 1; log 1 D 0/ W.z/ D R 1  1 C R r   R exists on jzj < D r 1  exp. 2Mr / , and the coefficients C` D P` .Ajk W j C k < `/ P ` of its power series expansion W D 1 `D1 C` z are positive. From jc` j D jP` .ajk W j C k < `/j  P` .jajk j W j C k < `/  P` .Ajk W j C k < `/ D C` it then follows that the solution to (1.5) also exists at least on jzj < :

1.3 Ordinary Differential Equations

15

Exercise 1.14 The solution to w0 D z C w2 , w.0/ D 0 may be expressed in terms of so-called Airy functions and is holomorphic on jzj < r  1:986 (the first pole). P n Prove that the power series ansatz w D 1 c nD1 n z leads to c1 D 0, c2 D 1=2, and Pn2 .n C 1/cnC1 D kD2 ck cnk .n  2/; and deduce cn D jcn j  .1=2/.1C2n/=5. n (Hint. Choose B > 0 and A > 0 such p that jcn j  AB holds for every n  2.) Thus the solution exists on jzj < 5 4  1:32. The Picard–Lindelöf method yields existence on jzj < 1  0:63; 1 is the largest number subject to 1 . 1 C P2 /  P for some P > 0; the inequality jw0 j  2 C jwj2 on jzj < 2 yields existence on jzj < .=2/2=3  1:35, and (iii) Cauchy’s ‘calcul des limites’ yields existence on jzj <  0:25.

1.3.6 The Complex Implicit Function Theorem This theorem, which allows us to ‘solve’ equations f.z; w/ D 0; w.z0 / D w0

.z 2 Cn ; w 2 Cm /

with regular Jacobian matrix A D fw .z0 ; w0 / for w D w.z/, may be proved in the ‘modern’ way and also by using Cauchy’s ‘calcul des limites’. Again it suffices to describe the method for m D n D 1, that is, to consider f .z; w/ D 0 with f .0; 0/ D 0 and A D fw .0; 0/ ¤ 0. It is convenient to rewrite the given problem as a fixed-point equation w D F.z; w/ D w  A1 f .z; w/ D

1 X

asq zs wq

s;qD0

with a00 D F.0; 0/ D 0Pand a01 D Fw .0; 0/ D 0. It is clear that this problem has a p unique formal solution 1 pD1 cp z ; and again cp D Qp .asq W s C q  p/ holds, where Qp is a universal polynomial with non-negative coefficient. To prove convergence on some disc jzj < and to simplify the argument we introduce new variables Rw and rz, if necessary, to achieve that F is holomorphic on jzj  1, jwj  1, and satisfies jF.z; w/j  1. Then F is dominated by 1 z C .1  z/w2 1wD D ˚.z; w/; .1  z/.1  w/ .1  z/.1  w/ and the new problem W D ˚.z; W/; W.0/ D 0, has the unique holomorphic solution p 1 X 1  z  1  10z C 9z2 W.z/ D D Cp zp .jzj < 19 /: 4.1  z/ pD1

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1 Selected Topics in Complex Analysis

The solution to the original problem then also exists on jzj < jw.z/j  W.jzj/ < 14 hold.

1 9

since jcp j  Cp and

1.3.7 Dependence on Parameters and Initial Values We will now consider initial value problems w0 D f.z; w; /; w.z0 / D w0 ; where the right-hand side f W G  ! Cn is holomorphic on some subdomain of C1CnCp . For the moment the initial values are assumed to be independent of . Since the numbers r; R; M, hence also and P, in the proof of Cauchy s Theorem can be determined uniformly with respect to  on every compact subset of , the local solution, denoted w.zI /, exists on jz  z0 j < and is a holomorphic function of . This follows from the fact that the local solutionR is obtained by successive z approximation: w0 .zI /  w0 and wkC1 .zI / D w0 C z0 f.t; wk .t; /; / dt; hence wk is also holomorphic in . In contrast to the real case the differentiability with respect to  is a by-product of uniform convergence wk .zI / ! w.zI /. The case of varying initial values may be reduced to the case just handled by considering the initial value problem Q 0 D f.z C z0 ; w Q C w0 / D Qf.z; w; Q z0 ; w0 /; w.0/ Q w D0 Q for w.z/ D w.z C z0 /  w0 . Obviously the right-hand side Qf is holomorphic in all its variables. We also note that ‘analytic dependence’ may be replaced with ‘continuous dependence’. Remark 1.5 Suppose kf.z; w; /k  M holds on jzj  , kwk  P, kk  L with

M  P. Then the family w.zI / kkL of solutions to w0 D f.z; w/, w.0/ D 0,   is normal on jzj < since the family kw0 .zI /k1 kkL is bounded. Thus the Theorem on analytic resp. continuous dependence on parameters may be viewed as a corollary of Montel’s Criterion and uniqueness of the solutions.

1.3.8 Analytic Continuation Analytic continuation of solutions to some system (1.2) always means analytic continuation as solutions. Example 1.3 Let f be holomorphic on some domain D  C having natural boundary @D, that is, f cannot be continued beyond any boundary point. Then the

1.3 Ordinary Differential Equations

17

problem w0 D 1 C .w  z/f .w/; w.z0 / D z0 , has the unique solution w D z, which, however, exists as a solution only on D. If analytic continuation is possible, then it is possible by solving initial value problems. Denote the solution to the initial value problem (1.2) by w.zI z0 ; w0 /; it exists on jz  z0 j < , where D .z0 ; w0 / is chosen as large as possible. For 0 < jz1  z0 j < .z0 ; w0 / the solution to w0 D f.z; w/; w.z1 / D w.z1 I z0 ; w0 / D w1 exists at least on jzz1 j < .z0 ; w0 /jz1 z0 j, hence .z1 ; w1 /  .z0 ; w0 /jz1 z0 j holds. Analytic continuation beyond jz  z0 j D r.z0 ; w0 / is obtained if and only if

.z1 ; w1 / > .z0 ; w0 /  jz1  z0 j: The solutions to the linear problem w0 D A.z/w C b.z/, with A, b holomorphic on some domain D, exist on jz  z0 j < dist .z0 ; @D/. Hence we have Theorem 1.8 The solutions to w0 D A.z/w C b.z/ admit unrestricted analytic continuation on D. If D is simply connected, every solution is holomorphic on D.

1.3.9 Painlevé’s Theorem Any solution to x0 D f.t; x/ in the real domain has a maximal interval .˛; ˇ/ of existence. As t ! ˛C and t ! ˇ, the graph f.t; x.t// W ˛ < t < ˇg leaves every compact subset of the domain of definition of the right-hand side f, and never comes back. In the complex case a maximal domain of existence does not necessarily exist. The analogous result is known as Painlevé’s Theorem Suppose that the solution w to (1.2) admits analytic continuation along every sub-arc  jŒ0;t , 0 < t < 1, of the arc  W Œ0; 1 ! C, and let the continuation along  jŒ0;t at the end point z D .t/ be denoted by !.t/. Then w admits analytic continuation along  if and only if there exists some sequence tk ! 1 such that !.tk / ! !  with .z ; !  / 2 G .z D .1//. In other words, if w cannot be continued along  , then as t ! 1; the curve ..t/; !.t//jŒ0;1/ leaves every compact subset of G and never comes back – it accumulates on @G. Proof For .z ; !  / 2 G, the solution w.zI z ; !  / exists on jz  z j <  . For Q 0 / with jQz0  z j <  and kw Q0   > 0 sufficiently small, the solution w.zI zQ0 ; w   Q0 D w k <  exists on jz  zQ0 j < =2, say. But this is true for zQ0 D .tk / and w !.tk / if k is sufficiently large, and w.zI z ; !  / provides the analytic continuation of w.zI .tk /; !.tk // into the end-point of  . Remark 1.6 It is obvious that analytic continuation respects holomorphic and continuous dependence on parameters. If, for  D 0 , the solution w.zI 0 / to the initial value problem w0 D f.z; w; 0 /, w.z0 / D w0 , admits analytic continuation along , then this is also true for w.zI / if  is sufficiently close to 0 . p p Example w0 D  C w2 , w.0/ D 0, with w.zI / D  tan. z/.

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1 Selected Topics in Complex Analysis

We will give two applications of Painlevé’s Theorem. Example 1.4 The first application concerns Riccati equations w0 D a.z/ C b.z/w C c.z/w2 ;

(1.6)

where a; b; c are holomorphic on some domain D  C and c.z/ 6 0. Then every solution to Eq. (1.6) admits unrestricted analytic continuation in D, the only singularities are poles. The case when (with the notation used in the proof of Painlevé’s Theorem) !.tk / tends to some finite limit !  is settled by Painlevé’s Theorem. If, however, !.t/ tends to 1 as t ! 1, Painlevé’s Theorem applies to the equation y0 D c.z/  b.z/y  a.z/y2 for y D 1=w, and z is a zero of y and a pole of w; y 6 0 follows from c.z/ 6 0. Remark 1.7 We note that even for differential equations w0 D P.w/, P any polynomial of degree deg P > 2, singularities other than poles may occur. A simple example is w0 D w3 ; the non-trivial solutions w D .2z C c/1=2 have an algebraic pole at z D 12 c D z0 C 12 w2 0 . In the present case the term ‘moving singularity’ is self-explanatory. Exercise 1.15 A different proof for Riccati equations is available. Prove that there are (infinitely many) linear systems u0 D a11 .z/u C a12 .z/v; v 0 D a21 .z/u C a22 .z/v with coefficients related to a; b; c, such that w D u=v solves Eq. (1.6), and every solution is obtained this way. Example 1.5 The next application concerns the implicit initial value problem w02 D P.w/; w.z0 / D w0

.P.w0 / ¤ 0/;

(1.7)

where P is any polynomial of degree at most 4. The problem is equivalent to w0 D v; v 0 D 12 P0 .w/; w.z0 / D w0 ; v.z0 / D w00 ; if only initial values with w02 0 D P.w0 / are admitted. We claim that every solution to Eq. (1.7) is meromorphic in the plane. To prove this we again have to discuss two possibilities: if !.tk / (notation as above) is bounded, then so is P.!.tk //. Assuming !.tk / ! !  , Painlevé’s Theorem immediately applies to some subsequence such that .P.!.tkj ///1=2 converges. If, however, !.t/ ! 1, the transformation w D 1=y leads to y02 D Q.z; y/ D y4 P.z; 1=y/ .degy Q  4/; and Painlevé’s Theorem yields a zero of y and a pole of the original solution. The theory of elliptic functions may be built upon Eq. (1.7). For further applications of Painlevé’s Theorem, in particular to implicit first-order equations P.z; w; w0 / D 0 and higher-order equations and systems, the reader is referred to the textbooks quoted in the beginning of this section, and, of course, to Chaps. 5 and 6.

1.4 Asymptotic Expansions

19

1.4 Asymptotic Expansions P1 =q In this section we will consider infinite series which, whether Dm a z convergent or not, in some sense represent holomorphic functions on open sectors j arg z  0 j < ; q and m are integers.

1.4.1 Asymptotic Series Although q > 1 and m < 0 are unavoidable in the asymptotic theory of algebraic differential equations, it suffices to consider holomorphic functions f and series P1  a z on sectors S W j arg zj < . Then f is said to have an asymptotic  D0 expansion f .z/ 

1 X

a z on S

D0

if for every n 2 N0 and every proper sub-sector Sı W j arg zj <   ı, f .z/ D

n X

a z C O.jzjn1 / as z ! 1 on Sı

D0

holds. We note some elementary facts: – the coefficients a are uniquely determined: a0 D limz!1 f .z/, a1 D limz!1 z. f .z/  a0 /, a2 D limz!1 z2 . f .z/  a0 P a1 z1 /, etc.; 1 0 1 – the derivative I D0 a z P1 of fhas the asymptotic expansion P1 f .z/a  C1 – f .z/  D2 a z has a primitive F.z/  D2 C1 z ; – if g also has an asymptotic expansion on S, then so have f C g, fg, and g=f (provided a0 ¤ 0). The coefficients are obtained in an obvious way; for P1 n example, the product fg has the asymptotic expansion with cn D nD0 cn z Pn a b :  n D0 The second assertion requires a proof. Given ı > 0, Cauchy’s Theorem applies to P fn ./ D f ./  nD0 a   on the disc j  zj  jzj sin ı (z 2 S2ı ) and yields 0

f .z/ D

n X D0

1

a z

1 C 2i

Z jzjDjzj sin ı

fn ./ d: .  z/2

n1 n1 / as z ! 1, uniformly on j  zj D jzj sin ı, Since fn ./ D O.jj Pn / D O.jzj 0 1 we obtain f .z/ D D0 a z C O.jzjn2 / as z ! 1 on S2ı :

20

1 Selected Topics in Complex Analysis

Example 1.6 Asymptotic series representing holomorphic functions need not be convergent. In fact, convergence at z0 implies convergence on jzj > jz0 j. On the other hand, every asymptotic series represents some holomorphic function on some sector. One of the most famous asymptotic expansions is known as Stirling’s formula, log  .z/  12 log 2  .z  12 / log z C z 

1 X nD1

B2n z2nC1 I 2n.2n  1/

Bn is the n-th Bernoulli number. The series converges nowhere. P1 k Exercise 1.16 Suppose f .z/  holds on j arg zj < , and let ˚ be kD0 ck z holomorphic on some neighbourhood of c0 . Prove that ˚ ı f has an asymptotic expansion on j arg zj <   ı, jzj > rı , ı > 0 arbitrary.

1.4.2 Asymptotic Integration of Differential Equations We quote from Wasow [197] two jewels in the asymptotic theory of differential equations, and start with linear systems zq y0 D A.z/y

(1.8)

of rank q C 1  1; A is a holomorphic n n-matrix that has an asymptotic expansion A.z/ 

1 X

Ar zr

.A0 ¤ O/

rD0

on some sector S with central angle

 qC1 .

Theorem 1.9 (Wasow [197], Theorem 12.3 and 19.1) There exists a holomorphic fundamental matrix G O Y.z/ D Y.z/z diag .eQ1 .z/ ; : : : ; eQn .z/ /

such that

P O k=p holds on S with YO 0 ¤ O; p 2 f1; : : : ; ng is some integer; O – Y.z/  1 kD0 Yk z – G is a constant matrix; – Q1 .z/; : : : ; Qn .z/ are polynomials in z1=p . If, in addition, the eigenvalues 1 ; : : : ; n of A0 are mutually distinct, then p D 1,  qC1 det YO 0 ¤ 0, G is a diagonal matrix, and Q has leading term qC1 z .

1.4 Asymptotic Expansions

21

Remark 1.8 Any linear differential equation w.n/ C an1 .z/w.n1/ C    C a1 .z/w0 C a0 .z/w D b.z/

(1.9)

with analytic coefficients having asymptotic expansions on some sector may be transformed into some linear system (1.8). In particular, this holds if the coefficients a and b are polynomials. The procedure, however, is not unique and does not always reveal the rank. In most applications it suffices to have the analogue of Theorem 1.9 for linear equations in the much older form going back to Sternberg [187], see Theorem 5.3. The next theorem deals with nonlinear systems of differential equations zq y0 D Ay C z1 f.z; y/;

(1.10)

and will be stated in a form that is adapted to our purposes. Again q is some non negative integer, S is any sector with central angle  qC1 , A is a regular n n-matrix, and f is holomorphic on S fy W kyk < ıg with asymptotic expansion f.z; y/ 

1 X

fr .y/zr

.fr holomorphic on kyk < ı/;

rD0

uniformly with respect to y on every closed sub-sector of S. Theorem 1.10 (Wasow [197], Theorem 12.1 and 14.1) Under the assumptions stated above, the system (1.10) has a solution with asymptotic expansion y

1 X

ck zk

on S:

kD1

Remark 1.9 The asymptotic series may represent several solutions. The coefficients ck are uniquely determined: from Ay D zq y0  z1 f.z; y/ it follows that Ac1 D  lim f.z; 0/ and AckC1 D Pk .cj W j  k/; z!1

where Pk is a vector-valued polynomial in the variables cj , with coefficients arising from the coefficients in the asymptotic expansion of f. In particular, the asymptotic series coincides with the unique formal series solution. Example 1.7 Theorem 1.10 does not immediately apply to the differential equation w0 D z2 Cw2 . It is, however, obvious that the principal terms of asymptotic solutions, if any, are ˙iz. We set w D y ˙ iz to obtain the differential equation z1 y0 D ˙2iy C z1 .y2 i/;

22

1 Selected Topics in Complex Analysis

to which Theorem 1.10 applies. Given any quadrant S there exists a solution w˙ having an asymptotic expansion on S with principal term ˙iz. It is not hard to show that w˙ .z/  ˙iz C 12 z1 ˙ 38 z3  34 z5 ˙    holds. Exercise 1.17 Theorem 1.11 also does not apply to w0 D z  w2 . It is plausible that p the first approximation is w  ˙ z. The change of variables z D t2 , w.z/ D tv.t/ 2 2 3 leads to t vP D 22v t v D 2.1v/.1Cv/t3 v: Theorem 1.11 applies to the differential equation for y D v 1. Compute the first few terms of the asymptotic 1 5 5=2 4 expansion for y to obtain the asymptotic series ˙z 2  14 z1 32 z  15 C 64 z in the z-plane. To every sector with central angle 2=3 there exist some solution w˙ having this asymptotic expansion.

1.4.3 Asymptotic Integration of Algebraic Differential Equations In contrast to Theorems 1.9 and 1.10, the next theorem on the existence of asymptotic expansions applies to specific solutions. The equation we will consider is QŒw D P.z; w/I

(1.11)

– P.z; w/ is a polynomial in w and rational in z; – QŒw is a finite sum of terms aM .z/w`M MŒw with aM .z/ D AM z˛M .1 C o.1// as z ! 1; AM ¤ 0; ˛M 2 Z; ` MŒw D w0 `1    w.m/ m ; ` D ` .M/; and dM D 2`1 C    C .m C 1/`m  ˛M C 2 for every M. Theorem 1.11 Let w be any solution to the algebraic differential equation (1.11) satisfying w.z/ ! c0 as z ! 1 on some sector S, and assume that P.z; c0 / ! 0 and Pw .z; c0 / ! c ¤ 0 as z ! 1 on S. Then w has an asymptotic expansion w

1 X

ck zk

on S:

kD0

Proof We will start with w.z/ D

n X

c z C o.jzjn / D

n .z/

C o.jzjn /;

(1.12)

D0

which is true for n D 0. To proceed we need w`M MŒw D

`M n MΠn

C o.jzjdM nC1 /

.z ! 1/I

(1.13)

1.4 Asymptotic Expansions

23

the proof will be given below. Using (1.13) we obtain QŒw D QŒ

n

C o.jzjmaxM .˛M dM /nC1 / D QŒ

n

C o.jzjn1 /;

hence w satisfies P.z; w/ D QŒ

n .z/

C o.jzjn1 /:

(1.14)

It follows from P.z; c0 / ! 0 and Pw .z; c0 / ! c ¤ 0 that for every rational function R that tends to zero as z ! 1, the algebraic equation P.z; y/ D R.z/ has a unique solution that tends to c0 as z ! 1. In particular, the equation P.z; y/ D QŒ has a unique solution yn .z/ D from (1.14), (1.15), and

P1

Œn  D0 a z

Z P.z; w.z//  P.z; yn .z// D

w.z/

yn .z/

n .z/

(1.15) Œn

about z D 1 with a0 D c0 , and

P .z; / d D .c C o.1//.w.z/  yn .z//

as z ! 1 (we integrate along the line segment in j  c0 j < ı from yn .z/ to w.z/) it follows that w.z/  yn .z/ D o.jzjn1 /; hence (1.12) holds with n C 1 and PnC1 Œn  Œk in place of n and n , respectively; we note that a D c nC1 .z/ D D0 a z Œn holds for 0    k  n, while the new coefficient is cnC1 D anC1 : To prove (1.13) we first consider the case of n .z/  c0 , hence MŒ n D 0. Then .k/

w.k/ .z/ D n .z/ C o.jzjnk / D o.jzjnk /; `M w MŒw D o.jzj`1 .n1/CClm .nm/ / D o.jzjdM nC1 / is true since  holds with MŒ

Pm

P D dM .n1/ m kD1 `k  dM nC1: Thus (1.13) n  0. Now let c (  1) be the first non-zero coefficient. Then kD1 .nCk/`k

.k/ n .z/ .k/

D O.jzjk /; .k/ w .z/ D n .z/ C o.jzjnk / D O.jzjk /; .k/ .w.k/ .z//`k D . n .z//`k C O.jzj.Ck/.`k 1/ /o.jzjnk / .k/ D . n .z//`k C o.jzjk`k n.`k 1/ /; and MŒw D MŒ Rn D

n

C Rn holds, with remainder term

XY . ` >0

D

.k/ `k n .z//

o.jzjj`j n.`j 1/ /

k¤j j   P X o jzj k¤j `k .kC/j`j n.`j 1/ D o.jzjdM nC1 /: `j >0

24

1 Selected Topics in Complex Analysis

This is true since now   w`M MŒw D D D

Pm

kD1 `k .

`M n MΠn `M n MΠn `M n MΠn

 1/  1, and thus it follows that

n C o.jzj /MΠ

C o jzj C o.jzj

n C o.jzj  P n m kD1 .kC/`k

dM nC1

dM nC1

C o.jzj

/

dM nC1

/

/;

this finishing the proof of Theorem 1.11.

1.5 Miscellanea 1.5.1 Elliptic Functions Non-constant meromorphic functions with R-linearly independent primitive periods #1 and #2 (Im ##21 ¤ 0) are called elliptic (also doubly periodic). The periods span the lattice L D #1 Z ˚ #2 Z, and the basis may be chosen to satisfy j#1 j  j#2 j  j#1 ˙ #2 j. For every basis the parallelogram P D fs#1 C t#2 W 0  s; t < 1g has the same area jIm .# 1 #2 /j. The basic properties of elliptic functions are known as Liouville’s Theorems Let f be an elliptic function and .#1 ; #2 / any basis of the corresponding lattice. Then – fP has (infinitely many) poles; – z2P resz f D 0; – in every parallelogram c C P, f assumes every value in b C equally often. This number is called the elliptic order; it is also the mapping degree of f W P ! b C. Every elliptic function f may be written as f D R.}/ C S.}/} 0; where R and S are uniquely determined rational functions, and } denotes the fundamental Weierstraß P-function Given any lattice L , the Weierstraß P-function is defined by the Mittag-Leffler series }.z/ D

1 C z2

X #2L nf0g

1 1  I .z  #/2 # 2 X

(1.16)

2 and }.z/ D .z  #/3 #2L }.z/: The P-function has elliptic order two and is the negative derivative of the X 1 z 1 1 C C 2; – Weierstraß Zeta-function .z/ D C z z# # # its periodicity is proved via the periodicity of } 0 .z/ D 

#2L nf0g

1.5 Miscellanea

25

which itself is the logarithmic derivative of the Y  z  #z C z22 e 2# : 1 – Weierstraß Sigma-function .z/ D z # #2L nf0g

The series P (on C n L ) and the product converge absolutely and locally uniformly since #2L nf0g j#js converges for s > 2. The P-function satisfies } 02 D 4} 3  g2 }  g3 D 4.}  e1 /.}  e2 /.}  e3 / with g2 D 60

P #2L nf0g

# 4 , g3 D 140

P #2L nf0g

# 6 and g32  27g23 ¤ 0; and

e1 D }. 21 #1 /; e2 D }. 21 #2 /; e3 D }. 21 .#1 C #2 // .e1 C e2 C e3 D 0/: The critical values e1 ; e2 ; e3 , and e4 D 1 are always assumed with multiplicity two. Jacobi’s Elliptic Functions The theory of elliptic functions can also be built on Jacobi’s functions sn (‘sinus amplitudinis’), cn (‘cosinus amplitudinis’), and dn (‘delta amplitudinis’), which may be defined by the initial value problem x01 D x2 x3 ; x1 .0/ D 0; 0 x2 D x1 x3 ; x2 .0/ D 1; x03 D  2 x1 x2 ; x3 .0/ D 1

. 2 ¤ 0; 1/:

Exercise 1.18 Deduce x21 C x22 D  2 x21 C x23 D 1 and 2 2 2 x02 1 D .1  x1 /.1   x1 /; 2 2 2 2 x02 2 D .1  x2 /.1   C  x2 /; 02 2 2 2 x3 D .1  x3 /.  1 C x3 /:

(Hint. Compute x1 x01 C x2 x02 etc.) For  2 ¤ 0; 1 the solutions are elliptic functions. Note, however, that they have different period lattices, in classical notation spanned by .4K; 2iK 0 /, .4K; 2K C 2iK 0 /, and .2K; 4iK 0 /, respectively (Fig. 1.1). For  2 D 0, sn and cn degenerate to the ordinary sine and cosine. What happens if  2 D 1?

1.5.2 The Phragmén–Lindelöf Principle Let D be any domain and assume that the boundary of D is divided into two disjoint parts ˛ and ˇ. The unique bounded harmonic function on D with boundary values N is called 1 on ˛ n ˇN and 0 on ˇ n ˛N (one cannot expect boundary values on ˛N \ ˇ) the harmonic measure of ˛ and denoted !.z; ˛; D/. We will not discuss conditions

26

1 Selected Topics in Complex Analysis

Fig. 1.1 The distribution of zeros of sn (K; K 0 > 0); d sn  D cn  D dn  D 1; dz d sn ı D 1, dz cn ı D dn ı D 1; d sn  D dn  D 1, dz cn  D 1; d sn ˘ D cn ˘ D 1, dz dn ˘ D 1

that ensure the existence of !, but work only in cases where ! is explicitly known or may be estimated appropriately. Exercise 1.19 Prove that – !.z; ˛; D/  !..z/; .˛/; D0 / holds on D, where  W D ! D0 is holomorphic and has a continuous extension to @D such that .˛/  @D0 . – !.z; ˛; D/  !.z; ˛; D0 / holds on D if D  D0 and ˛  @D \ @D0 . Determine the harmonic measure in the following cases: – D D fz W 0 < arg z < g, ˛ D .0; 1/. – D D fz 2 D W Im z > 0g, ˛ D .1; 1/: (Hint. Thales’ Theorem.) Exercise 1.20 Let f be holomorphic on some domain D with boundary values lim supz!˛ j f .z/j  M and lim supz!ˇ j f .z/j  1: Prove the Two-Constants Theorem

j f .z/j  M !.z;˛;D/

.z 2 D/:

(Hint. log j f j may be viewed as a subharmonic function on D or a harmonic function on D n fzeros of f g.) The Phragmén–Lindelöf Principle is an immediate corollary of the TwoConstants Theorem when applied to holomorphic functions f on sectors ˙. We assume that M˙ .r; f / D supfj f .z/j W jzj D r; z 2 ˙g

.r > 0/

is finite. In its simplest (and classical) form it says the following: Theorem 1.12 Let f be holomorphic on the upper half-plane H with boundary values lim supz!x j f .z/j  1 .x 2 R/: Then either MH .r; f /  1 .r > 0/ or else lim infr!1 1r log MH .r; f / > 0 holds. Proof The harmonic measure of the semi-circle ˛ W jzj D r; Im z > 0 measured in the semi-disc D W jzj < r; Im z > 0 is !.z; ˛; D/ D

2 2rIm z 4rjzj arctan 2  2  r  jzj .r2  jzj2 /

1.5 Miscellanea

27

(see Exercise 1.19), and the Two-Constants yields log j f .z/j 

4rjzj 4r2 jzj log MH .r; f / log MH .r; f / D : 2 2 .r  jzj / .r2  jzj2 / r

Exercise 1.21 Prove that – the alternative in Theorem 1.12 remains true if f is holomorphic only on the domain H n fz W jzj  r0 g and satisfies j f .z/j  1 on jzj D r0 , Im z > 0. (Hint. !.z; ˛r ; D \ Dr /  !.z; ˛r ; Dr /:) – replacing the upper half-plane H in Theorem 1.12 with any sector ˙ with central 1 angle ı turns the alternative into M˙ .r; f /  1 or lim infr!1 r=ı log M˙ .r; f / > ı= 0: (Hint. Consider f .z / on some appropriate half-plane.) The version of the Phragmén–Lindelöf Principle we will use here is Theorem 1.13 Let f be holomorphic on the upper half-plane H and continuous on H [ R, and assume logC j f .z/j D o.jzj/ as z ! 1 on H. Then f is bounded on H if it is bounded on R, in which case the following is true: – If f tends to a as z D x ! C1 along the positive real axis, then f tends to a as z ! 1 on every sector 0  arg z     < . – If, in addition, f has a limit as z D x ! 1, then f tends to a as z ! 1 on H. Proof We assume that f is bounded on R, hence also on H by Theorem 1.12. To prove the first assertion we may assume that a D 0 and j f .z/j  1 on H (otherwise consider . f .z/  a/=M in place of f , M sufficiently large). Given  > 0 there exists some x0 > 0 such that j f .x/j <  holds on .x0 ; 1/, hence 1

j f .z/j <  !.z;.x0 ;1/;H/ D  .1  arg.zx0 //   ı=2 holds on 0  arg.z  x0 /    ı=2 by the Two-Constants Theorem, and also on 0  arg z   ı, jzj > x0 (by the ‘inscribed angle Theorem’, the rays arg z D  ı and arg.z  x0 / D   ı=2 intersect at x0 ei.ı/ ). To prove the second assertion one has just to combine the first one with its counterpart on the negative real axis, noting that limy!C1 f .iy/ D a: Exercise 1.22 Prove that Theorem 1.13 mutatis mutandis remains true for functions f satisfying logC j f .z/j D o.jzj=ı / as z ! 1 on ˙ W j arg zj < ı=2, jzj > r0 , and having finite limits as z ! 1 on the rays ˙ W arg z D ˙ı=2. As a corollary we obtain: Corollary 1.1 Let f be holomorphic on j arg zj < ; jzj  r0 of finite order of growth .logC j f .z/j D O.jzjk / as z ! 1/, and assume that  f .z/ D

Ae˛z z .1 C o.1// .z D reiı ! 1/ Beˇz z .1 C o.1// .z D reiı ! 1/

28

1 Selected Topics in Complex Analysis

.A; B; ˛; ˇ; ;  2 C; AB ¤ 0/ holds for every 0 < ı < . Then – Re ˛ D Re ˇ and Im ˛  Im ˇ, and – ˛ D ˇ implies f .z/ D Ae˛z z .1 C o.1// as z ! 1 on j arg zj  =2, say; in particular, f has only finitely many zeros on j arg zj  =2. Proof To prove the first assertion we assume Re ˛ ¤ Re ˇ and even Re ˛ > Re ˇ N ˛). (otherwise consider f .Nz/ instead of f to replace ˛; ˇ with ˇ; N Then h.z/ D e˛z z f .z/

(1.17)

has finite order of growth on j arg zj <  and satisfies h.reiı / ! A ¤ 0 and j f .reiı /j D O.rRe ./ / eRe .ˇ˛/r cos ıCIm .ˇ˛/r sin ı

(1.18)

as r ! 1. The right-hand side of (1.18) tends to zero if ı > 0 is chosen sufficiently small. This, however, contradicts Theorem 1.13 in the form given in Exercise 1.22. In the same way we obtain a contradiction if we assume Re ˛ D Re ˇ and Im ˛ > Im ˇ; then h tends to A as z D reiı ! 1, while (1.18) for Re ˛ D Re ˇ again implies f .reiı / ! 0 as r ! 1. This proves Im ˛  Im ˇ. To prove the second assertion we assume (˛ D ˇ and) Re  > Re . Then h, by (1.17),   again defined tends to A ¤ 0 as z D reiı ! 1, while jh.reiı /j D O rRe ./ tends to zero as r ! 1: This proves Re   Re , and in the same way we obtain Re   Re  (replace f with f .Nz/), hence Re  D Re  D , and h is bounded and tends to A as z D reiı ! 1. By the Phragmén–Lindelöf Principle, h.z/ ! A as z ! 1 is true on 0  arg z  ı; in particular, f .x/ D Ae˛x xCiIm  .1 C o.1// holds as x ! C1 (x real). In the same way we obtain f .x/ D Be˛x xCiIm  .1 C o.1//, hence  D , A D B, and f .z/ D Ae˛z z .1 C o.1// holds on j arg zj < . Exercise 1.23 Let f be an entire function of finite order (logC j f .z/j D O.jzjk /), 2 2 and assume that f .z/ D e˛z Ca1 zCa0 z .1 C o.1// and f .z/ D eˇz Cb1 zCb0 z .1 C o.1// holds as z ! 1 on ı < arg z <   ı and  C ı < arg z < ı, respectively .0 < ı < =2 arbitrary). Prove ˛ D ˇ and Im a1  Im b1 . Prove also that f .z/ D 2 e˛z Ca1 z P.z/ holds with some polynomial P if a1 D b1 . p p p p (Hint. Consider f . z/ with Re z > 0 and f . z/ with Re z > 0 on j arg zj <  and j arg.z/j < 2 , respectively.) 2

1.5.3 Wiman–Valiron Theory P n n Let g.z/ D 1 nD0 an z be a transcendental entire function. Since jan jr tends to zero n as n ! 1, the sequence .jan jr / has a maximum called the maximum term, .r/ D max jan jrn : 0n 0/: By the same reasoning (and its definition), the central index is also increasing, hence is piecewise constant with discontinuities (jumps) at 0  r0 < r1 < r2 <    . From 0 .r/=.r/ D .r/=r on each interval rk < r < rkC1 it follows that 0

Z

log .r/  log .r / D

r r0

.t/ dt t

.0 < r0 < r/;

hence the order of growth may be defined threefold: %.g/ D lim sup r!1

logC logC M.r; g/ logC logC .r/ logC .r/ D lim sup D lim sup : log r log r log r r!1 r!1

Exercise 1.25 To determine  D .r/ asymptotically it often suffices to use the inequalities jan jrn  ja jr and jam jrm < ja jr for the largest index n <  with an ¤ 0 and the smallest index mp>  with am ¤ 0, respectively. Prove p in this manner .r/ D Œr and .r/  r=2 for g.z/ D ez and g.z/ D cos z, respectively. The central result of Wiman–Valiron theory is the relation between .r/ and g0 .z/=g.z/ at points z D zr on jzj D r such that jg.zr /j D M.r; g/. Theorem 1.14 For every transcendental entire function and every n there exists R some set Fn  .0; 1/ of finite logarithmic measure Fn dtt such that g.n/ .zr /  .r/ n  g.zr / zr

.r ! 1; r … Fn /

holds at points zr with jzr j D r and jg.zr /j D M.r; g/. g.zr C

r / .r/

! e as r ! 1 outside F, locally g.zr / uniformly with respect to . The reader is referred to [12, 79, 193, 200]. It is almost obvious how this applies to determine the possible orders of growth of the transcendental solutions to linear differential equations An equivalent statement is that

w.n/ C an1 .z/w.n1/ C    C a1 .z/w0 C a0 .z/w D b.z/

30

1 Selected Topics in Complex Analysis

with polynomial coefficients. Dividing by w.zr / yields 

.r/=zr

n

n1  C an1 .zr /.1 C o.1// .r/=zr C    C a0 .zr /.1 C o.1// D o.1/

as r ! 1 outside F (of finite logarithmic measure). Although this is not exactly an algebraic equation, it suffices to consider the simplified algebraic equation yn C

n1 X

Aj x˛j Cnj yj D 0 .x D zr ; y D .r//

jD0

(assuming aj .z/ D Aj z˛j C   ) to determine y D .r/ approximately. In combination with the Newton–Puiseux method it turns out that the possible orders of growth coincide up to sign with the negative slopes of the Newton–Puiseux polygon. The exceptional set F can be neglected since .r/ is increasing. Exercise 1.26 Compute the order of growth of the transcendental solutions to the linear differential equation w000  4z2 w0  12zw D 0. Solution. .r/  2r2 and zr is approximately real. Why? Example 1.8 The results are not always satisfactorily. In case of the nonlinear differential equation ww00  w02  ww0 D 0 we obtain .1 C 2 .r/  .1 C 1 .r//2 /.r/2  .1 C 1 .r//zr .r/ D 0: Since 1C2 .r/.1C1 .r//2 D 2 .r/21 .r/1 .r/2 cannot be controlled, nothing z can be said about .r/. The solution ee has infinite order.

1.5.4 The Schwarzian Derivative The expression Sf D

 f 00 0 f0



1  f 00 2 f 000 3  f 00 2 D  2 f0 f0 2 f0

(1.21)

is called the Schwarzian derivative of f , classically also written f f ; zg. Exercise 1.27 Prove the chain-rule Sf ıg D .Sf ı g/g02 C Sg . Exercise 1.28 Suppose that f is regular at z0 and f 0 .z0 / ¤ 0. Prove that w2 D 1 . f 0 / 2 (any branch on jzz0 j < ı), and also w1 D fw2 satisfies the linear differential equation w00 C 12 Sf .z/w D 0; hence f D w1 =w2 holds. (Hint. Start with w02 =w2 D  12 f 00 =f 0 .)

(1.22)

1.5 Miscellanea

31

Exercise 1.29 Deduce that – Sf  0 if and only if f is a Möbius transformation; – SMıf D Sf for every Möbius transformation M; – Sf is regular except at the critical points of f (zeros of f 0 , multiple poles of f ); 1 .1  p/2 c1 C    if z0 is a . p  1/-fold critical point. C – Sf .z/ D 2 .z  z0 /2 .z  z0 / Remark 1.10 The Schwarzian derivative occurs in a natural way in several subfields of Complex Analysis. We mention – – – – – – –

conformal mapping of circular domains, one of its origins; univalence criteria of Nehari type; Nevanlinna Theory (equality in the Second Main Theorem); Teichmüller space; quasiconformal mappings; iteration of transcendental functions (and of real functions on real intervals); normal family criteria.

Exercise 1.30 Suppose f is holomorphic at z D 0 with f 0 .0/ ¤ 0. Prove that for mutually distinct constants a; b; c; d, . f .az/; f .bz/; f .cz/; f .dz// D 1 C 16 .a  b/.c  d/Sf .0/z2 C O.jzj3 / .a; b; c; d/ holds, where .a; b; c; d/ D

.ac/.bd/ .ad/.bc/

.z ! 0/

denotes the cross-ratio.

Exercise 1.31 Prove that the cross-ratio .w1 ; w2 ; w3 ; w4 / of any four distinct solutions w to the Riccati equation w0 D a.z/ C b.z/w C c.z/w2 is constant. (Hint. Compute the logarithmic derivative of the cross-ratio and make use of w0 w0 w w

D b.z/ C c.z/.w C w /.)

Chapter 2

Nevanlinna Theory

In this chapter Nevanlinna Theory, Cartan’s Theory of Entire Curves, and the Selberg–Valiron Theory of Algebroid Functions will be outlined, particularly with regard to the applications in the subsequent chapters and including recent developments in the context of the Second Main Theorem. Nevanlinna Theory provides the most effective tools in the modern theory of meromorphic functions, and even simplifies the older theory of entire functions considerably.1

2.1 The First Main Theorem 2.1.1 The Poisson–Jensen Formula Nevanlinna Theory ([124, 126, 128]) is based on the Poisson–Jensen formula for meromorphic functions; ‘meromorphic’ always means meromorphic in the plane. By P.z; rei / D Re

r2  jzj2 rei C z D rei  z jrei  zj2

ˇ r.z  c/ ˇ ˇ ˇ and g.z; c/ D  log ˇ 2 ˇ r  cN z

we denote the Poisson kernel and Green’s function with pole at c of the disc jzj < r, respectively.

1 H. Weyl (Meromorphic functions and analytic curves. Princeton University Press 1943) called it “one of the few great mathematical events of the [XX] century”. W.K. Hayman dedicated his monograph [78] to Rolf Nevanlinna, “the creator of the modern theory of meromorphic functions”, and A.A. Gol’dberg and I.V. Ostrovskii wrote in the preface to [57]: “After his [Nevanlinna’s] work, the value distribution theory acquired, in some way, a complete form. The main classical results of the theory of entire functions have been included in Nevanlinna’s Theory in a natural way.”

© Springer International Publishing AG 2017 N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, DOI 10.1007/978-3-319-59800-0_2

33

34

2 Nevanlinna Theory

Poisson–Jensen Formula Let f be meromorphic with zeros and poles a and b ; written down according to multiplicities. Then log j f .z/j D

1 2

Z

2 0

P.z; rei / log j f .rei /j d X X  g.z; a / C g.z; b / ja j 2, respectively, with k  1 in both cases. To proceed we will first prove a rather general theorem on the zeros of  D f n PΠf  C QΠf ;

(3.11)

which covers the relevant results in [77, 83, 158]; P and Q denote not identically vanishing differential polynomials in f . Theorem 3.6 ([32, 162]) Let f be any transcendental meromorphic function and  be given by (3.11). Then  1 D S.r; f / N r; 

(3.12)

implies n  dQ C 2. Moreover, for dQ D n  2, m.r; f / C m.r; 1=f / C N1 .r; f / C N1 .r; 1=f / D S.r; f / holds, where N1 .r; f / D N.r; f /  N.r; f / and N1 .r; 1=f / ‘counts’ the multiple poles and zeros, respectively. The function f is a solution to the differential equation wPŒwQ0 Œw C nwn PŒw  .nw0 PŒw  n C wP0 Œw/QŒw D 0;

(3.13)

3.2 Zeros of Differential Polynomials

83

where  6 0 is a small meromorphic function with respect to f , and P0 Œw and Q0 Œw d denote the total derivatives dzd PŒw and dz QŒw. Remark 3.2 In case of  D f n f 0  a and  D f nC2 C af 0  c (c 2 C) as well as  D f n . f 0 /n1    . f .k/ /nk  a and  D . f nCk /.k/  a (always a ¤ 0) the inequality n  2 is necessary for (3.12), hence the above mentioned results are corollaries of Theorem 3.6. Proof We assume n  dQ C2 and (3.12) to prove that  does not vanish identically. Otherwise Clunie’s Lemma (Exercise 3.4) applies to f n PŒ f  D QŒ f  and yields m.r; PŒ f / D S.r; f /: At any pole z0 of f of order p which is not a pole of any of the coefficients, QŒ f  has a pole of order at most pdQ  p.n  2/, while f n has a pole of order pn. Thus PŒ f  vanishes at z0 , and this implies N.r; PŒ f / D S.r; f /: Finally, it follows from f n D  PŒ1f  QŒ f  and T.r; PŒ f / D S.r; f / that nT.r; f /  T.r; QŒ f / C S.r; f /  dQ T.r; f / C S.r; f /  .n  2/T.r; f / C S.r; f /; which clearly is impossible. Differentiating (3.11) we obtain f n1 n D

0 QΠf   Q0 Πf  

(3.14)

and n D nf 0 PΠf  C fP0 Πf  

0 fPΠf : 

(3.15)

Again Exercise 3.4, applied to (3.14), yields m.r; / D S.r; f /: We note that Clunie’s Lemma requires that the coefficients are small functions with respect to f . The ‘coefficient’  0 = , however, is not necessarily small. Analysing the proof of Clunie’s Lemma shows that it suffices to know that m.r;  0 = / D S.r; f / holds. We also note that  cannot vanish identically, since otherwise   cQŒ f  and f n PŒ f  C .1  c/QŒ f   0 holds for some c ¤ 0. This, however, was excluded in the first part of the proof. Like there, it follows that T.r; / D S.r; f /

84

3 Selected Applications of Nevanlinna Theory

holds since  is regular at almost every pole2 of f . Clunie’s Lemma applies to f n2 f D

 1 0 QΠf   Q0 Πf  n 

(3.16)

and yields m.r; f / D S.r; f /, and in combination with  1f0 1 0 1 D PΠf  C P0 Πf   PΠf  f  f n n

(3.17)

we obtain m.r; 1=f / D S.r; f /: Since  vanishes at almost every multiple zero and pole of f , it also follows that N1 .r; 1=f / C N1 .r; f / D S.r; f /: Finally, the differential equation (3.13) is a combination of (3.14) and (3.15). Since for n > dQ C 2,  vanishes at almost every pole of f , but does not vanish identically, n  dQ C 2 follows.

3.2.3 Limit Cases We will now discuss the following differential polynomials which form various limit cases n D dQ C 2 with respect to Theorem 3.6. 1:  2:  3:  4: 

D D D D

f 2 . f 0 /n1    . f .k/ /nk  1with nk  1; f 2 f 0  1 as a special case; . f kC2 /.k/  1 with k  1; and f 4 C f 0  1:

(3.18)

The common idea will be to derive a Riccati differential equation from Eq. (3.13) and afterwards express the derivatives f . j/ in terms of f , hence rewrite  as an ordinary polynomial in f with small coefficients. To this end we will frequently make use of the following exercises. Exercise 3.10 Let f be meromorphic with m.r; f / C N1 .r; f / D S.r; f / such that f .z/ D

.z0 / C .z0 / C O.jz  z0 j/ .z ! z0 / z  z0

(3.19)

2 Unless otherwise stated, ‘almost every pole, zero, etc’. means ‘up to a sequence of poles, zeros, etc. with counting function S.r; f /’.

3.2 Zeros of Differential Polynomials

85

holds at almost every simple pole, where 6 0 and  are small meromorphic functions w.r.t. f . Prove that w D f .z/ satisfies w0 D a0 .z/ C a1 .z/w C a2 .z/w2

(3.20)

with a2 D 1= , a1 D .2  0 /= , and T.r; a0 / D S.r; f /. (Hint. It is convenient first to consider f = instead of f , hence to assume  1. In this case show that f 0 C f 2  2f is regular at almost every pole of f .) Exercise 3.11 Assuming (3.12) for  D f 2 . f 0 /n1    . f .k/ /nk  1, employ Eq. (3.13) qC2 0 to verify that (3.19) holds with coefficients D qC2 2 ;  D  qC4  ; and q D 2n1 C    C .k C 1/nk  k C 1: Exercise 3.12 Do the same for  D . f kC2 /.k/  1 with D (Hint. Do not try to compute PΠf  D . f

kC2 .k/

2

2kC2 k

and  D 0.

/ =f explicitly!)

Exercise 3.13 Let f be meromorphic with m.r; 1=f / C N1 .r; 1=f / D S.r; f /, and assume that, similar to Exercise 3.10, f 0 .z0 / D .z0 / and f 00 .z0 / D 2.z0 / holds at almost every simple zero of f . Prove that w D f .z/ satisfies Eq. (3.20) with a0 D , a1 D .2 C 0 /= and T.r; a2 / D S.r; w/. Exercise 3.14 Assume that  D f 4 C f 0  1 satisfies N.r;  / D S.r; f /, and let z0 be any simple zero of f which is neither a zero of  nor a zero or pole of . From (3.15) and (3.14) deduce that – f 0 .z0 / D .z0 / and f 00 .z0 /  0 .z0 / D 14 .z0 / .z0 / with D  0 = ; – f 00 .z0 / D .z0 /..z0 /  1/, hence f 00 .z0 /.1  34 .z0 // D 0 .z0 /..z0 /  1/; – some Riccati equation (3.20) for w D f .z/ if .z/ 6 43 . Theorem 3.7 Let f be transcendental meromorphic. Then each of the differential polynomials (3.18) 1:, 2: and 3: has infinitely many zeros in the stricter sense that lim sup r!1

N.r; 1= / > 0: T.r; f /

Remark 3.3 The third case is due to Hennekemper [83]; Wang [195] settled the case . f m /.`/ for m  3; m > `, which in view of Theorem 3.7 is relevant if m D `C1  3. Bergweiler and Eremenko [14] finished the case m D ` C 1  2. Proof We assume (3.12) for  D f 2 . f 0 /n1    . f .k/ /nk  1 (first case), and take from Exercise 3.10 and 3.11 that w D f .z/ solves w0 D a0 .z/ C a1 .z/w C a2 .z/w2 D P2 .z; w/

86

3 Selected Applications of Nevanlinna Theory 0

q  with a1 D  qC4  , a2 D

2 qC2 ,

and T.r; a0 / D S.r; f /. This implies

w00 D P3 .z; w/ D 2Ša2 .z/2 w3 C    ; w000 D P4 .z; w/ D 3Ša2 .z/3 w4 C    etc:; hence  .z/ D H.z; f .z// with H.z; w/ D w2 P2 .z; w/n1    PkC1 .z; w/nk  1 D AqC2 .z/wqC2 C    C A2 .z/w2  1 qm

and AqC2 D 2Šn2 3Šn3    kŠnk a2 , q D dP D 2n1 C    C .k C 1/nk , and m D dP D n1 C    C nk : Equation (3.14) with n D 2 and QŒ f  D 1 takes the form 2.z/wH.z; w/ C Hz .z; w/ C Hw .z; w/P2 .z; w/ D 0

(3.21)

for w D f .z/: Obviously this is an identity in .z; w/. Now every (algebroid) solution y D .z/ to H.z; y/ D 0 solves y0 D P2 .z; y/;

(3.22)

KΠy D y2 . y0 /n1    . y.k/ /nk  1 D 0:

(3.23)

hence also

For a proof, write H.z; w/ D .w  .z//` .z; w/ with .z; .z// 6 0 and employ Hz .z; w/ D `.w  .z//`1 0 .z/.z; .z// C O.jw  .z/j` /; and Hw .z; w/ D `.w  .z//`1 .z; .z// C O.jw  .z/j` / as w ! .z/ to obtain  0 .z/ C P2 .z; .z// D O.jw  .z/j/ from (3.21). Now (3.22) admits at most two solutions that are algebroid over the field of small functions w.r.t. f , since the cross-ratio . f ; 1 ; 2 ; 3 / of any four solutions to (3.22) is constant (see Exercise 1.31). Also the solutions to (3.23) cannot be single-valued. It follows that H.z; w/ D AqC2 .z/.w  1 .z// 1 .w  2 .z// 2 . 1 C 2 D q C 2/; .1/qC2 AqC2 1 1 2 2 D 1 and 1 2 C 2 1  0 since H contains no linear term in w. The branches 1 and 2 are permuted under analytic continuation, and this yields 1 D 2 D and 21 D 22 D g; where p g is meromorphic, but g is not; g is even an entire function, since the solutions p p to KŒy D 0 cannot have poles. Now . g/0 D a0 C a1 g C a2 g is equivalent to g0 g

0

2q

 qC4 2q  D 2a1 D  qC4 .ı1 ¤ 0/ on one  and a0 C a2 g D 0. This implies g D ı1  qm



hand, while AqC2 g D AqC2 . 1 2 / D 1 and AqC2 D const  a2 D const  qm qm

1 yield g D ı2  on the other. From this and 2 D qC2 we obtain m D 2 C1 < 2, hence m D 1, D 3, and q D 4. Now m D 1 means KŒy D y2 y.k/  1, so that

3.2 Zeros of Differential Polynomials

87

p KΠg cannot vanish identically. This settles the first and, of course, also the second case. The proof of the third assertion runs along the same lines. This time w D f .z/ solves w0 D P2 .z; w/ D a0 .z/ 

k .z/w2 2kC2



 T.r; a0 / D S.r; f / ;

and again  .z/ D H.z; f .z// holds, where H.z; w/ is a polynomial with respect to w. The solutions y D .z/ to H.z; y/ D 0 now solve y0 D P2 .z; y/ and . ykC2 /.k/ D 1; hence .z/pkC2 is a polynomial ˚.z/ D zk =kŠ C    of degree k. It is obvious that

.z/ D kC2 ˚.z/ cannot solve 0 D P2 .z; / D a0 .z/ C a2 .z/ 2 , which is equivalent 1 2 1 ˚.z/ kC2 1 ˚ 0 .z/ D a0 .z/ C a2 .z/˚.z/ kC2 . This settles the third case. to kC2 While the cases 1., 2., and 3. may be regarded as generalisations of Hayman’s (H2), the next theorem returns to case (H1) with n D 2. Theorem 3.8 For every transcendental meromorphic function, f 0 C f 4 has infinitely many zeros. If f 0 C f 4 omits the value c D 1, say, then f satisfies f 0 D 2 C 2f 2 , hence f .z/ D tan.2z C z0 / coincides with the second example in Exercise 3.9. Proof The first part (due to Mues [119]) is a corollary of Theorem 3.7: If f 0 C f 4 has only finitely many zeros, then so has ˚ D g4 . f 0 C f 4 / D g2 g0  1 (set f D 1=g), since g and ˚ have no common zeros. To prove (a slightly stronger version of) the second assertion (see [162]) we set  D f 0 Cf 4 1 and assume N.r; 1= / D S.r; f /. According to Theorem 3.7 and Exercise 3.14, we have to consider two cases. a) .z/ 6 4=3. Then w D f .z/ satisfies w0 D P2 .z; w/ D a0 .z/ C a1 .z/w C a2 .z/w2 by Exercise 3.14, hence  .z/ D H.z; f .z// holds with H.z; w/ D w4 C a2 .z/w2 C a1 .z/w C a0 .z/  1: Again every solution y D .z/ to H.z; y/ D 0 satisfies y0 D P2 .z; y/ and also y4 C y0  1 D 0. The first equation can have only p two solutions 1;2 of this kind. The second equation has four constant solutions 4 1, while the non-constant solutions have algebraic singularities. In this (non-constant) case we have 1 D  2 D since 1 and 2 are interchanged under analytic continuation, which, however, contradicts 4 ˙ 0  1 D 0. All in all we have to discuss the case H.z; w/ D .w2  2 /2 D .w2 C1/2 (H.z; w/D .w  1 /4 and H.z; w/ D .w  1 /3 .w  2 / cannot occur since H contains no term a3 w3 ). This leads to the assertion w0 D 2.w2 C 1/ for w D f .z/. b) .z/  4=3. Here a non-trivial problem arises, namely to decide whether the 4 4 0 0 differential equation ww00 C 16 3 w  4.w  3 /.w  1/ D 0 has a transcendental 4 0 meromorphic solution such that w .z0 / D 3 holds at every zero of w. The equation

88

3 Selected Applications of Nevanlinna Theory

passes the Painlevé test for zeros as well as for poles, that is, to every z0 there exist p formal solutions 43 .zz0 /Cc2 .zz0 /2 C   and ˙ 46 .zz0 /1 Cc0 Cc1 .zz0 /C   . We will thus pursue a different track. Exercise 3.15 From (3.15) with .z/  4=3 deduce that f0 0 . f 4 = /0 16 D 12  3 D3 4 f f  f = and prove that  is zero-free, hence 4=f D 3g0 =g holds, where g is an entire function; g and g0 have simple zeros, they coincide with the zeros and poles of f , respectively, and  D C=g04 holds for some C ¤ 0. Exercise 3.16 (Continued) Deduce that a) 256g4 C 27g04  108gg02g00  81C D 0; b) 256g2  54g002  27g0 g000 D 0; c) 512gg0  135g00 g000  27g0 g.4/ D 0; and prove that .r/  43 r (central index) holds, hence %.g/ D 1: To finish the proof of Theorem 3.8 we note that g0 .z0 / D 0 implies g000 .z0 / D 0, hence  D g000 =g0 is entire and has Nevanlinna characteristic T.r; / D o.log r/. Thus  D 2 is non-zero constant and g.z/ D c0 C c1 e z C c2 e z holds. This, however, is incompatible with b).

3.2.4 The Tumura–Clunie Theorem The significance of the weight of a differential polynomial is restricted to the poles of the function f under consideration. If f is entire or ‘almost’ entire, that is, if N.r; f / D S.r; f / holds, the role of the weight is taken by the degree. For example, T.r; PŒ f /  dP T.r; f / C S.r; f / holds in place of T.r; PŒ f /  dP T.r; f / C S.r; f /. It is self-evident that Theorem 3.6 has an analogue as follows. Theorem 3.9 Let f be an almost entire transcendental function and suppose that  D f n PŒ f  C QŒ f  with PŒ f QŒ f  6 0 satisfies N.r; 1= / D S.r; f /. Then dQ  n  1 holds. In the limit case dQ D n  1, f satisfies the differential equation (3.13) and m.r; 1=f / C N1 .r; 1=f / D S.r; f /. Proof We assume dQ  n  1 rather than dQ  n  2 in the proof of Theorem 3.6. Then (3.14) and (3.15) still hold and  and D  0 = are small functions. To falsify the hypothesis dQ < n  1 we remark that in this case m.r; f / D S.r; f / would follow from (3.16) and Clunie’s Lemma, in contrast to N.r; f / D S.r; f /. Again m.r; 1=f / D S.r; f / is obtained from (3.17), while N1 .r; 1=f /  N.r; 1=/ D S.r; f / follows from the fact that  has zeros of order at least . p  1/ at p-fold zeros of f . The differential equation (3.13) is just a combination of (3.14) and (3.15).

3.2 Zeros of Differential Polynomials

89

The classical Tumura–Clunie Theorem ([28], also [78], p. 69) deals with (almost) entire functions and differential polynomials  D f n C QŒ f  .QŒ f .z/ 6 0; dQ  n  1/ N.r; 1= / D S.r; f /;

(3.24)

and may be stated as follows. Theorem 3.10 Suppose f is (almost) entire and satisfies (3.24). Then  n  .z/ D f .z/  a.z/ holds; a, , and

 and

1 n 1 n

f 0 .z/ D .z/ C a0 .z/ D .z/ C

.z/f .z/ .z/a.z/

D  0 = are small functions with respect to f .

We will now return to meromorphic functions and the zeros of ordinary polynomials  .z/ D P.z; f .z//. Without loss of generality we may assume that P.z; w/ D wn C an2 .z/wn2 C    C a0 .z/

.n  2/

is square-free. Theorem 2.6 yields N.r; 1= /  .n  2/T.r; f / C S.r; f /. The Tumura–Clunie Theorem referring to N.r; 1= / rather than N.r; 1= / reads as follows. Theorem 3.11 ([122]) Under the hypotheses stated just now, N.r; 1= / D S.r; f / implies P.z; w/ D w2 C a0 .z/: Moreover, w D f .z/ satisfies the Riccati equation w0 D a0 .z/c.z/ C

a00 .z/ 2a0 .z/ w

C c.z/w2

.c 6 0 small/:

p Remark 3.4 Since both branchespof a0 .z/ also p satisfy the above Riccati equa2 tion, f .z/ C a0 .z/ D . f .z/  a0 .z//. f .z/ C a0 .z// has no zeros except possibly at poles of a00 =a0 and c. Proof Theorem 3.6 applies to  .z/ D f .z/n C Q.z; f .z// with Q.z; w/ D an2 .z/wn2 C    C a0 .z/: In particular, degw Q D n2 and an2 6 0 holds, and the differential equation (3.13) for w D f .z/ has the form wQz .z; w/ C n.z/wn C n.z/Q.z; w/ C .wQw .z; w/  nQ.z; w//w0 D 0:

(3.25)

90

3 Selected Applications of Nevanlinna Theory

Since wQw .z; w/nQ.z; w/ D 2an2 wn2 C   does not vanish identically, Malmquist’s First Theorem yields w0 D b0 .z/ C b1 .z/w C b2 .z/w2

.b small/:

(3.26)

Replacing w0 in (3.25) with b0 .z/ C b1 .z/w C b2 .z/w2 we obtain an identity in .z; w/, which in terms of P may be written as wPz .z; w/ C n.z/P.z; w/ C .wPw .z; w/  nP.z; w//.b0 .z/ C b1 .z/w C b2 .z/w2 / D 0: As in the proof of Theorem 3.7 it is shown that the solutions w D .z/ to the algebraic equation P.z; w/ D 0 also solve (3.26), and from degw P D n  2 it follows that there are two solutions 1 and 2 . This implies n D 2 and P.z; w/ D a0 w2 C a0 .z/; while b1 D 2a00 and b0 D a0 b2 D a0 c follow from (3.26) for w D p ˙ a0 .z/. Example 3.2 Pang and Ye [138] considered  D f n CQŒ f , where f is meromorphic and QŒw D w.k/ C a1 w.k1/ C    C ak w C akC1 has constant coefficients, such that QŒ f .z/ 6 0. Assuming N.r; 1= / D S.r; f / and n D k C 3, the authors derived a Riccati differential equation f 0 D c. f  ˛/. f  ˇ/ with constant coefficients (Theorem 1 in [138]) by a process similar to ours, but much more elaborate. We note that ˛ and ˇ are Picard values of f , and n  k C 3 is a necessary condition for N.r; 1= / D S.r; f /. Thus  D f n C bn2 f n2 C    C b0 is an ordinary polynomial in f , which necessarily has the form  D . f  ˛/n1 . f  ˇ/n2 with n1 C n2 D n and n1 ˛ C n2 ˇ D 0. Exercise 3.17 Prove that, given n1 ; n2 , ˛, ˇ with n1 ˛ C n2 ˇ D 0, ˛ˇ ¤ 0, n D n1 C n2 , and k D n  3  1, there exist c ¤ 0 and a1 ; : : : ; akC1 such that  D f n C f .k/ C a1 f .k1/ C    C ak f C akC1 D . f  ˛/n1 . f  ˇ/n2 holds for any non-constant solution to f 0 D c. f  ˛/. f  ˇ/. (Hint. Prove that f ./ D c f C1 C    D c PC1 . f / holds forp D 1; 2; : : :.) As an example we quote  D f 6 C f 000  1c f 0 C 1 with c D 3 1=2, f 0 D c.1 C f 2 /, hence f .z/ D tan.cz/ and  .z/  .1 C f .z/2 /3 ¤ 0. Remark 3.5 Theorems on the zeros of differential polynomials may have a different shape. They may be purely qualitative: ‘ has (infinitely many) zeros’, ‘semi’quantitative: ‘lim sup N.r; 1= /=T.r; f / > 0’ like in the cases being discussed here, or quantitative like in Weissenborn [198], where the inequality T.r; f /  N.r; 1= / C N.r; f / C S.r; f / in the Tumura–Clunie context is proved. Perhaps the most prominent example is

3.2 Zeros of Differential Polynomials

91

Hayman’s alternative, which is based on the inequality (see [77])   1 1  C .2` C 2/N r; .`/ C S.r; f / .`  1/; `T.r; f /  .2` C 1/N r; f f 1 and says that f or f .`/  1 (possibly both) have infinitely many zeros, with counting function a numerical portion of T.r; f /. Further examples of this kind are   .`  1/T.r; f /  N r; 1= C o.T.r; f //

. D f ` . f .k/ /n  1; k; `; n  1/

(Jiang and Huang [97]) and .n  2/T.r; f /  N.r; 1= / C S.r; f /

. D f n C an1 .z/f n1 C    C a0 .z//:

An example for the qualitative case is Theorem 5 in [139], which says that if f is entire with zeros of multiplicity at least k, then f n f .k/ (n; k  1) assumes every value a ¤ 0 infinitely often.

3.2.5 Homogeneous Differential Polynomials Hayman [77] also considered homogeneous differential polynomials, thereby opening a second field of research concerning the zeros of differential polynomials. He proved that f .z/ D eazCb .a ¤ 0/ is the only transcendental meromorphic function of finite order such that ff 0 f 00 has no zeros, and asked whether this will remain true if the additional hypotheses ‘f 0 ¤ 0’ and ‘f has finite order’ are dropped. This is the case for entire functions. Clunie [28] proved that ff .k/ ¤ 0 for some k  2 and f entire implies f .z/ D eazCb : For transcendental meromorphic functions and k  3 this was proved by Frank [46], and by Mues [119] for k D 2 if f has finite (lower) order. Only much later and with quite different methods, Langley [104] was able to settle this problem in full generality. The corresponding result for ff 0 ¤ 0 is not true: z for example, f .z/ D ee (entire of infinite order) and also the non-constant solutions to w0 D w.w C 1/ (meromorphic of finite order) have this property. We will now describe a far-reaching method due to Frank [46] which applies to homogeneous differential polynomials fLŒ f , where L is any differential operator with polynomial coefficients. It suffices to consider the canonical form LŒy D y.k/ C ak2 .z/y.k2/ C    C a0 .z/y

.k  3/:

(3.27)

Without any hypothesis on the zeros of fLΠf , Frank and Hellerstein [47] proved the a priori estimate   T.r; f 0 =f / D O N.r; 1=f / C N.r; 1=LΠf / C r%max C S.r; f 0 =f /I

92

3 Selected Applications of Nevanlinna Theory

%max denotes the maximal order of growth of the solutions to LŒy D 0, and r%max may be replaced with log r if LŒy D y.k/ . In particular, f 0 =f has finite order of growth if fLŒ f  is zero-free. Assuming this, there exists an entire function satisfying gk D LŒ f =f . The zeros of g are simple and coincide with the poles of f (which may have arbitrary multiplicities), hence h D g. f 0 =f / is also an entire function. Both functions g and h have order of growth at most %max , and even are polynomials if LŒy D y.k/ . The main idea now is to consider the functions  0 w D gy0  hy D g f y =f

.1    k/;

where . y1 ; : : : ; yk / denotes any fundamental set to the equation LŒy D 0, normalised by W. y1 ; : : : ; yk / D 1. From   f D gk LŒ f  D gk W. y1 ; : : : ; yk ; f / D gk f kC1 W y1 =f ; : : : ; yk =f ; 1       0 0 D .1/kC1 fW fg y1 =f ; : : : ; fg yk =f D .1/kC1 f W.w1 ; : : : ; wk / it follows that W.w1 ; : : : ; wk / D .1/kC1 , hence these functions also annihilate some linear differential operator MŒw D w.k/ C bk2 .z/w.k2/ C    C b0 .z/wI the coefficients are entire functions satisfying T.r; b / D m.r; b / D S.r; f 0 =f /, hence are polynomials, and even constants if L has constant coefficients. The fact that MŒgy0  hy D 0 is equivalent to LŒy D 0 then leads to the following relations: .ka0 g0 C a00 g/ D .b0  a0 /h C

k X

b h./ ;

D1

.b1  a1 

a0 /g

! j g. jC1/ C .b  ka /g C bj   1 jDC1 0

k X

! j . j/ h D .b  a /h C bj  jDC1 k X

.1    k  2/;

! k 00 g C .bk2  ak2 /g D kh0 : 2

(3.28) By an elimination process for linear operators we obtain some differential equation KŒv D v .s/ C cs1 .z/v .s1/ C    C c0 .z/v D 0 for v D g.z/, again with polynomial or even constant coefficients, and g, h, and f 0 =f D h=g can be computed with considerable effort.

3.2 Zeros of Differential Polynomials

93

1. The case of LŒy D y.k/ (k  3) leads to w1 D h and w D .  1/z2 g  z1 h .2    k/; and after intricate computations to f 0 =f D h=g D a and f .z/ D eazCb (Frank [46]). 2. If L has constant coefficients so have M and K. Then g and h are exponential polynomials, and tedious computations lead to three cases as follows [166]. a. f .z/ D exp.az C b C eczCd /; b. f .z/ D eazCb =.z  z0 /n ; c. f .z/ D eazCb =.eczCd  1/n . Apart from f .z/ D eazCb (which just requires ak C ak2 ak2 C    C a0 D 0), every case is realised by a unique operator L in the case of a. and b., and by k C 1 uniquely determined operators L in the case of c. 3. Combining the arguments in [47] and [166] with the theory of asymptotic integration, Brüggemann [20] was able to prove the conjecture of Frank and Langley (see [17]) that if fLŒ f  is zero-free (L with polynomial coefficients) and f has infinitely many poles, then 1

f D .H 0 / 2 .k1/ H n ; n 2 N and H 00 =H 0 D p

(3.29)

holds for some non-constant polynomial p. We note that this occurs in 3.c. in a trivial way, and non-trivially for 000

2

0

LŒy D y  .z C 2/y  zy

and 1=f .z/ D e

z2 =2

Z

z

et

2 =2

dtI

0

2

f has infinitely many poles and fLΠf .z/ D 6e3z f .z/5 is zero-free. Obvi2 ously (3.29) holds with n D 1, k D 3, and H 0 .z/ D ez =2 . 4. The case k D 2 resisted efforts for a long time and was finally settled by Langley [104], who also solved several related problems, see, for example, [105]. We note that for k D 2 the information in (3.28) is almost worthless, and the arguments differ substantially from those in the case of k  3. The following exercises will show that the results just stated are all sharp. Exercise 3.18 Set F.z/ D ez =.ez 1/ and f .z/ D eazCb F.z/k . Prove that F 0 D FF 2 and f ./ =f D Q .F/ hold, where Q is a polynomial of degree . Then LΠf  D f .k/ C ak1 f .k1/ C    C a0 f

(3.30)

is zero-free if and only if Q.w/ D Qk .w/ C ak1 Qk1 .w/ C    C a1 Q1 .w/ C a0 has the form const  wm .w  1/km for some m 2 f0; 1; : : : ; kg; for each m this is possible in exactly one way.

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3 Selected Applications of Nevanlinna Theory

Exercise 3.19 Set f .z/ D eazCe and prove f ./ =f D Q .ez /; where Q is a polynomial of degree . Then (3.30) is zero-free if and only if Q.w/ D const  wk ; this uniquely determines L. Similarly, for f .z/ D eazCb zn (n  1) construct L in a unique way such that LΠf  has no zeros. z

Exercise 3.20 ([103]) Suppose that p is a non-constant polynomial, H is entire and satisfies H 00 =H 0 D p, f (of finite order with infinitely many poles) and g (entire of infinite order) are given by f D .H 0 /k H n and g D .H 0 /k eH , respectively, and L D .D C p.z//.D C 2p.z//    .D C kp.z// .D D

d /: dz

Prove that LŒ f  D cH nk for some c ¤ 0 and LŒg D eH hold. (Hint. Prove f 0 C kpf D n.H 0 /.k1/ H .nC1/ and g0 C kpg D .H 0 /.k1/ eH .)

3.3 Uniqueness of Meromorphic Functions The very first application of Nevanlinna Theory was dedicated to the question of how many pre-images f 1 .fag/ determine a meromorphic function. Distinct meromorphic functions f and g are said to share the value a if f 1 .fag/ D g1 .fag/. Moreover, if f  a and g  a (resp. 1=f and 1=g if a D 1) have the same zeros counting multiplicities (the same divisor) we will speak of sharing counting multiplicities.

3.3.1 The Five-Value Theorem Nevanlinna’s first uniqueness result in [125] is known as the Five-Value Theorem; it initiated extensive research on sometimes obscure ‘value sharing problems’. The serious investigations after Nevanlinna started in the late seventies and early eighties with the work of Gundersen [65, 66, 68]. Shortly afterwards Mues [120] introduced the ‘method of auxiliary functions’ with sustainable effect. The underlying idea is as follows: if f is meromorphic and  is a small function (w.r.t. f ), which vanishes on a sequence of poles, zeros, etc. of f , then either  vanishes identically or else the sequence in question has counting function S.r; f /. Though the idea is simple, the main problem consists in constructing relevant auxiliary functions and requires a lot of experience. We set T.r/ D maxfT.r; f /; T.r; g/g and denote by S.r/ the common error term. Five-Value Theorem ([125]) Distinct meromorphic functions f and g cannot share five values. In other words, five pre-images f 1 .fag/ determine f uniquely.

3.3 Uniqueness of Meromorphic Functions

95

Proof We assume that f and g share the finite values a (1    q) and denote by N.rI a/ the (integrated) counting function of common a-points, each point being counted once, ignoring multiplicities. The Second Main Theorem then yields .q  2/T.r/ 

q X

 N.rI a / C S.r/  N r;

D1

1  C S.r/ f g

(3.31)

 T.r; f / C T.r; g/ C S.r/  2T.r/ C S.r/; which obviously is a contradiction if q  5.

3.3.2 Examples and Counterexamples We will discuss several (counter-)examples to show that the Five-Value Theorem is the best possible and has no complete analogue for four values. To start with, the functions f .z/ D ez and g.z/ D ez share four values: 1 and 1 counting multiplicities, and the Picard exceptional values 0 and 1 (of course, also counting multiplicities). At the other end of the scale we have Gundersen’s example [65]: Exercise 3.21 Prove that the functions f .z/ D

ez C 1 .ez  1/2

and g.z/ D

.ez C 1/2 8.ez  1/

share the values 1; 0; 1 and 1=8 in the following manner: f has only simple zeros and 1-points, and only double poles and .1=8/-points, while g has only simple poles and .1=8/-points, and only double zeros and 1-points at the very same places. Prove also that x D f .z/ and y D g.z/ parametrise the rational curve 6xy C y C x2  8xy2 D 0 and satisfy the first-order differential equations .x0 C x C 18 /2 D 2.x C 18 /.x  14 /2

and . y0 C y2  y/2 D y. y  1/. y C 12 /2 :

Gundersen’s example has one more interesting property, namely f .z/ D  12 implies g.z/ D 14 , and vice versa. Reinders [142, 144] characterised this example three-fold: Theorem 3.12 Assume that F and G share four mutually distinct finite values a , and one of the following conditions holds: – there exist values a; b ¤ a .1    4/ such that F.z0 / D a implies G.z0 / D bI – for every  the zeros of .F  a /.G  a / have multiplicity three; – for every  either F  a or else G  a has only double zeros.

96

3 Selected Applications of Nevanlinna Theory

Then F and G are either Möbius transformations of each other (this is possible in the first case) or else have the form F D M ı f ı h and G D M ı g ı h, where f and g are the functions being discussed in Exercise 3.21; M is a Möbius transformation and h a non-constant entire function. Exercise 3.22 (Reinders [143]) The elliptic functions defined by the differential equation u0 2 D 12 u.u C 1/.u C 4/ have elliptic order two. Prove that the elliptic functions 1 u.z/u0 .z/ fO .z/ D p 8 3 u.z/ C 1

1 .u.z/ C 4/u0 .z/ and gO .z/ D p 8 3 .u.z/ C 1/2

of elliptic order four share the values 1; 0; 1; 1 in the following manner: each of these values is assumed in an alternating way with multiplicity 1 by one of these functions, and with multiplicity 3 by the other, see Fig. 3.1. Prove also that w D fO .z/ solves p w04  4 3.w2 C 1/w03 C 3888w2 .w2  1/2 D 0; and fO and gO parametrise the algebraic curve .x  y/4 D 16xy.x2  1/. y2  1/: (Hint. From u02 D 12u.u C1/.u C4/ and u00 D 18u2 C60u C24 deduce .u C1/fO 2 D P.u/ and .u C 1/fO 0 D Q.u/ with polynomials P and Q. To derive the differential equation for fO compute the resultant R.z; w; w1 / of the polynomials .u C 1/w2  P.u/ and .u C 1/w1  Q.u/ w.r.t. u. The algebraic curve is simply obtained by computing the resultant R.x; y/ of yu.u C1/x.u C4/ and 16xy.u C1/2 u2 .u C4/2 (corresponding to y=x D gO =fO and xy D fO gO , respectively), again w.r.t. u; u, x, y, w, and w1 are just regarded as complex variables.) The number five in the Five-Value Theorem is the best possible, as follows from our examples, which are of a quite different character otherwise. We note, however, that the proof of the Five-Value Theorem reveals important information common to all pairs of meromorphic functions f and g that share four values. This information has its origin in the equality sign in the Second Main Theorem. Fig. 3.1 The distribution of poles  (three-fold for f , simple for g, and two-fold for u) and ı (three-fold for g and simple for f ), and zeros  (three-fold for f and simple for g) and ˘ (three-fold for g and simple for f )

◦ ∗

 •

 •

◦ ∗

◦ ∗

 •

 •

◦ ∗

◦ ∗

 •

 •

◦ ∗

3.3 Uniqueness of Meromorphic Functions

97

Exercise 3.23 Suppose that f and g share four finite values a . Use (3.31) for q D 4 to prove – T.r; f / D T.r/ C S.r/ and T.r; g/ D T.r/ C S.r/; 4 P – N.rI a / D 2T.r/ C S.r/; D1  1  D 2T.r/ C S.r/; – N r; f g    1 1  C N r; D 2T.r/ C S.r/ .b ¤ a /, and also – N r; f b gb N.r; f / C N.r; g/ D 2T.r/ C S.r/. Remark 3.6 We remind the reader of the special meaning of the term ‘almost’: by definition, almost every point in a given sequence (of poles, zeros, etc.) has some property P if the subsequence of points not having this property has counting function S.r/. For example, it follows from Exercise 3.23 that almost every zero of f  g is simple and corresponds to some common a -point of f and g, hence also almost every a -point is simple for at least one of the functions f and g. Also almost all critical points of f and g correspond to the a -points. Nevanlinna Theory tolerates sequences with counting function S.r/.

3.3.3 The Four-Value Theorem The next result, known as the Four-Value Theorem, is also due to Nevanlinna [125]. It states that the first example f .z/ D ez , g.z/ D ez is unique if the pairs . f ; g/ and .M ıf ıh; M ıgıh/ are identified (M any Möbius transformation, h any non-constant entire function). Four-Value Theorem Suppose f and g share four values a counting multiplicities. Then relabelling the values if necessary, the cross-ratios of a1 ; a2 ; a3 ; a4 and f ; g; a3 ; a4 satisfy .a1 ; a2 ; a3 ; a4 / D . f ; g; a3 ; a4 / D 1I a1 and a2 are Picard exceptional values of f and g, and f D M ı g holds with some Möbius transformation M that fixes a3 and a4 and permutes a1 and a2 . Proof (Mues [120]) It follows from Exercise 3.23 and the hypothesis ‘counting multiplicities’ that almost every a -point is simple for f and g, f 0 and g0 have almost no zeros, and at least two of the values a satisfy N.rI a / ¤ S.r/I we may assume that a4 D 1 and a3 D 0 have this property. The auxiliary function D

f 00 g00  0 f0 g

98

3 Selected Applications of Nevanlinna Theory

satisfies m.r; / D S.r/, and from the previous discussion of the critical points it follows that N.r; / D N.r; / D S.r/ holds. Moreover,  vanishes at almost every common pole of f and g. This yields a contradiction if  6 0, namely N.rI 1/ D S.r/. If, however,  vanishes identically, f D c1 g C c0 follows at once by twofold integration, and N.rI 0/ ¤ S.r/ yields c0 D 0, while c1 ¤ 1 is part of the hypothesis. Since also the values a ( D 1; 2) are shared by f and g, while a D c1 a is impossible, it follows that these values are Picard exceptional values for f and g, and from f D c1 g, a1 D c1 a2 and a2 D c1 a1 it follows that c1 D 1, a2 D a1 , f D g, and .a1 ; a2 ; 0; 1/ D . f ; g; 0; 1/ D 1: Remark 3.7 Nevanlinna’s proof was quite different and based on Theorem 2.10. Suppose that distinct meromorphic functions f and g share the values 0; 1; c, and f 1 f c 1 in the strong sense, namely assume that h1 D gf ; h2 D g1 ; and h3 D gc are zero-free entire functions. Then the Borel identity .1  c/h1 C ch2  h3  h1 h2 C ch1 h3 C .1  c/h2 h3  0 holds, and by Theorem 2.10 at least two of the functions h1 ; h2 ; h3 ; h1 h2 ; h1 h3 ; h2 h3 are linearly dependent. No matter which are, this leads to the conclusion that g is a Möbius transformation of f . Since Möbius transformations may have only two fixed points, two of the shared values, 0 and 1, say, must be Picard values of f and g, and this implies g D a=f . Then f .z0 / D g.z0 / D 1 implies a D 1 (1 D a=1) and c D 1=c, thus g D 1=f , f D e , g D e , and c D 1. Exercise 3.24 Discuss exemplarily the case h1 D h2 h3 , that is, g f D

: . f  1/. f  c/ .g  1/.g  c/ Mues’ idea to introduce auxiliary functions not only simplified the proof but pointed the way to further developments. For example, it follows from the proof that the hypothesis ‘counting multiplicities’ may be replaced with ‘almost every a point is simple for f and g’. This is not just a trivial generalisation but will turn out to be crucial in what follows.

3.3.4 Variations of the Four-Value Theorem The papers on ‘value sharing problems’ are legion. We will restrict ourselves to the fundamental question of whether and how far the hypothesis ‘counting multiplicities’ in the Four-Value Theorem may be relaxed. To this end we will prove

3.3 Uniqueness of Meromorphic Functions

99

two theorems due to Gundersen [65, 66, 68]. The proofs are based on suitably chosen auxiliary functions, the most important one, D

f 0 g0 . f  g/2 P. f /P.g/

4 Y   P.x/ D .x  a /

(3.32)

D1

was introduced by Mues in the early 1980s, and independently at almost the same time by Czubiak and Gundersen; if a4 D 1 the factor x  a4 has to be omitted. Our general assumption is that f and g share four values a . If appropriate, three of these values may be chosen at will. We note that  carries the whole information on the shared values, but unfortunately is incapable of taking into account multiplicities. Exercise 3.25 Prove that  is a small entire function, that is, m.r;  / D T.r;  / D f 2 f 0 g0 ff 0 gg0 f 0 g2 g0 S.r/ holds. (Hint.  D 2 C .) P. f / P.g/ P. f / P.g/ P. f / P.g/ We are now prepared to prove Theorem 3.14 below. Although this theorem is stronger than Theorem 3.13, it will be instructive to prove this result first. The proof, due to Rudolph [147], is ‘auxiliary function-based’. Theorem 3.13 Suppose that f and g share four values a , three of them counting multiplicities. Then the fourth value is also shared counting multiplicities. Proof Suppose f and g share the finite values a (1    3) counting multiplicities, and a4 D 1 without further hypothesis. Then g0 f0  D P. f / P.g/

3 Y   nowP.x/ D .x  a / D1

satisfies T.r; / D S.r/ and vanishes at the common poles of f and g. Thus we have either  6 0 and N.rI 1/ D S.r/ by the First Main Theorem, or else  vanishes identically. In this case it follows from P. f / D P.g/ for some constant  ¤ 0 that a4 D 1 is also shared counting multiplicities. On the other hand it is not hard to show that the hypotheses ‘N.rI 1/ D S.r/’ and ‘a4 D 1 is shared counting multiplicities’ are equally strong in the sense that in combination with the hypotheses on the other values a they lead to the same conclusion. The proof of Gundersen’s Theorem 3.14 given here is due to Mues [120], see also [181]. It is even more ‘auxiliary function-based’ and requires repeated separation into cases. In part it will be organised in the form of exercises. Theorem 3.14 Suppose that f and g share four values a , two of them counting multiplicities. Then all values a are shared counting multiplicities. Proof We may assume that f and g share the values a3 D 0 and a4 D 1 counting multiplicities and a1 a2 D 1 holds; this may be achieved by considering cf and cg with c2 a1 a2 D 1 in place of f and g, and will simplify matters significantly.

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3 Selected Applications of Nevanlinna Theory

Exercise 3.26 Prove that under this hypothesis the auxiliary function 2

D

2

f0 X f0 g00 g0 X g0 f 00 C2   0 2 C 0 f f f  a g g g  a D1 D1

(3.33)

is regular at every zero and pole that is simple for f and g, and also at every a -point ( D 1; 2), despite multiplicities. Prove also that T.r; / D S.r/ and .z1 /2 D .a1 C a2 /2  .z1 / holds at every common simple pole z1 . (Hint. Many computations are easily realised by using some computer algebra system like MAPLE, which is most suitable when operating with ‘rational expressions’ and finite Laurent series.) One consequence is that either N.rI 1/ D S.r/ or else  2  .a1 C a2 /2  holds. This argument may be repeated with f and g replaced by F D 1=f and G D 1=g, which share the very same values a2 ; a1 ; 1; 0. Exercise 3.27 Prove that the pair .F; G/ D .1=f ; 1=g/ has the same Mues function (3.32), and the auxiliary function 2

˚D

2

F0 X F0 G00 G 0 X G0 F 00  C C 2   2 F0 F D1 F  a G0 G G  a D1

corresponding to (3.33) has the form 2

˚D

2

f0 X f0 g00 g0 X g0 f 00 2   0 C2 C 0 f f f  a g g g  a D1 D1

when written in terms of f and g. This was the reason to insist on a2 D 1=a1 ! Again either N.rI 0/ D S.r/ or else ˚ 2  .a1 C a2 /2  holds. We thus have to discuss four cases as follows: 1. N.rI 1/ C N.rI 0/ D S.r/, 2. ˚  , 3. ˚  , and 4. N.rI 1/ D S.r/ and ˚ 2  .a1 C a2 /2  as one of two equivalent cases. Exercise 3.28 In the first case prove that N.rI a / D T.r/ C S.r/ ( D 1; 2) holds (Exercise 3.23, second assertion). This is in some sense weaker than saying that f and g also share a1 and a2 counting multiplicities, but strong enough to ensure that the proof of the Four-Value Theorem works. Exercise 3.29 Prove that in the second case g D cf holds for some constant c ¤ 0, this implying that a1 and a2 are Picard exceptional values of both functions, and f and g share all values counting multiplicities. In both cases the proof of Theorem 3.14 is finished.

3.3 Uniqueness of Meromorphic Functions

101

Exercise 3.30 In the third case integrate ˚   twice to obtain g0 f0 D . f  a1 /. f  a2 / .g  a1 /.g  a2 /

and

 g  a  f  a1 1 DC f  a2 g  a2

for some integer  and some constant C ¤ 0. From T.r; f / D T.r; g/CS.r/ conclude that  D ˙1. Discuss the cases  D 1 and  D 1 separately to reduce the proof of Theorem 3.14 either to the second case or else to the Four-Value Theorem. In the fourth case we first assume a1 C a2 ¤ 0 in addition, and consider the auxiliary function

1 D ˚  .a1 C a2 /

f 0 . f  g/ : f .g  a1 /. f  a2 /

Exercise 3.31 Prove  N.r; 1 /  N r;

 1  1   N.rI a1 / and m.r; 1 /  m r; C S.r/; g  a1 g  a1

hence T.r; 1 /  T.r/  N.rI a1 / C S.r/, and similar estimates for 1 D ˚  .a1 C a2 /

g0 . f  g/ g. f  a1 /.g  a2 /

and the functions 2 and 2 that are obtained by permuting a1 and a2 in the definition 0 f 0 . f g/ 1 f 0 . f a1 / of 1 and 1 . (Hint. f .ga D  f . f fa2 / C ga :) 1 /. f a2 / 1 f . f a2 / Exercise 3.32 Prove that each of the functions 1 ; 1 ; 2 ; 2 vanishes at common simple zeros z0 of f and g. (Hint. Use MAPLE and 1=a1 C 1=a2 D a1 C a2 .) To proceed we have to discuss two subcases as follows. a)  6 0 or  6 0 for  D 1; 2;

b) 2 D 2  0, say.

Exercise 3.33 Prove that in case a) N.rI 0/  T.r/  N.rI a / ( D 1; 2) holds, and in combination with N.rI 1/ D S.r/ and Exercise 3.23 also 2N.rI 0/  2T.r/  N.rI a1 /  N.rI a2 / C S.r/ D N.rI 0/ C N.rI 1/ C S.r/ D N.rI 0/ C S.r/; thus N.rI 1/ C N.rI 0/ D S.r/. This means that we are back in the first case. Prove also that in case b) . f  a2 /f 0 .g  a2 /g0 D f . f  a1 / g.g  a1 / holds, hence f and g share four values counting multiplicities.

102

3 Selected Applications of Nevanlinna Theory

It remains to consider the fourth case with a1 C a2 D 0 and a1 a2 D 1, hence a1 D i, a2 D i, and ˚  0. Integration yields f 2. f

g0 f0 D 2  i/. f C i/ g .g  i/.g C i/

Exercise 3.34 Integrate (3.34) to obtain

. ¤ 0/:

(3.34)

. f C i/.g  i/ 1 1  log D  C c. 2i . f  i/.g C i/ f g

Since at least one of the values ˙i is not a Picard exceptional value for f and g (otherwise we were already in the case of Theorem 3.13),  (or 1=) is a positive integer, hence f assumes the values ˙i ‘always’ with multiplicity , while g has ‘only’ simple ˙i-points (up to a sequence of points with counting function S.r/). Now  D 1 means that f and g also share the values ˙i counting multiplicities, and we are done. From   2 and Exercise 3.34, however, it follows that . f C i/.g  i/ D eH . f  i/.g C i/

.H D 2i.1=f  =g C c/ an entire(!) function/

holds. At every common ˙i-point, H assumes the value ˙2.1  / C 2ic, thus H is non-constant and N.rI i/ C N.rI i/  2T.r; H/ C O.1/ follows. To obtain a contradiction and to finishes the proof we refer to the following exercise. Exercise 3.35 Use N.rI ˙i/  1 T.r/ C O.1/ , N.rI 1/ D S.r/ and Exercise 3.23 to deduce N.rI 0/ D T.r/ C S.r/ and T.r; H/ D m.r; H/  m.r; 1=f / C m.r; 1=g/ C O.1/ D S.r/: Remark 3.8 The diploma thesis [147] of Eva Rudolph was never published, so it is no wonder that several researchers [68, 93, 157, 190, 194, 196] independently rediscovered her results or results that may be deduced from [147]. To state these results we need to define the quantity (introduced by Mues [120]) .a / D lim inf

r!1 .r…E/

N s .rI a / N.rI a /

if N.rI a / ¤ S.r/; and .a / D 1 otherwise;

here N s .rI a / denotes the counting function of those a -points which are simultaneously simple for f and g, and E is the exceptional set for S.r/; we note that .a / D 1 holds in particular if a is shared counting multiplicities, and also if a is a Picard value. Theorem 3.15 Suppose f and g share four values. Then these values are shared counting multiplicities if one of the following conditions is fulfilled in addition: – one value satisfies .a / D 1 and some other .a / > 23 ; for .a1 ; a2 ; a3 ; a4 / D 1 the constant 23 may be replaced with 12 (Mues [120]). This is the strongest result known so far.

3.3 Uniqueness of Meromorphic Functions

103

– one value is shared counting multiplicities, while the others satisfy .a / > 12 (Wang [196], Huang [93]). – one of the values is shared counting multiplicities and satisfies N.rI a /  ı T.r/ for some ı > 45 on a set of infinite measure (Gundersen [68]). – two of the values satisfy .a / > 45 ; for .a1 ; a2 ; a3 ; a4 / D 1 the constant 45 may be replaced with 23 (Wang [194], Huang [93]). – .a / > 34 holds for three of the values (Song and Chang [157]). – .a / > 23 holds for 1    4 (Wang [196], conjectured in [157]). Resume The reader will have noticed that the main problem in the context of the Four-Value Theorem was left aside. It is still an open question whether or not there exists some pair . f ; g/ sharing four values, exactly one of them counting multiplicities. There is some evidence that the answer to this question is ‘No’, but a proof is not in sight. Our examples indicate that meromorphic functions sharing four values (without any additional assumption) might be algebraically dependent and separately satisfy implicit first-order differential equations. So a step in the right direction could be to determine all pairs . f ; g/ of transcendental meromorphic functions that share four values and are algebraically dependent.

3.3.5 Reinders’ Example Rediscovered We will now show that the pair .fO ; gO / in Reinder’s Example 3.22 may be characterised as follows. Theorem 3.16 ([143]) Suppose f and g share the values a .1    4/ such that . f  a /.g  a / resp. 1=.fg/ if a D 1 has only zeros of order at least four. Then there exists some Möbius transformation M and some non-constant entire function h such that f D M ı fO ı h and G D M ı gO ı h holds. Proof Our goal is to derive an algebraic curve that is parametrised by . f ; g/. Choosing a1 D 0, a2 D 1, and a4 D 1 it will turn out that this curve has genus > 1 except when a3 2 f1; 12 ; 2g. In case of a3 D 1, say, the algebraic curve in question is . y  x/4 D 16xy.x2  1/. y2  1/

(3.35)

and has genus 1. We may assume that f and g are elliptic functions (actually f D f1 ı h, g D g1 ı h holds, where f1 and g1 are elliptic functions and h is non-constant entire). Substituting y D tx, x2 D s transforms (3.35) into the algebraic curve s.t  1/4  16t.s  1/.st2  1/ D 0

104

3 Selected Applications of Nevanlinna Theory

of genus 0. The parametrisation s D f .z/2 D

u3 .uC4/ 16.uC1/ ,

u.z/3 .u.z/ C 4/ 16.u.z/ C 1/

tD

uC4 u.uC1/

and g.z/2 D

leads to

u.z/.u.z/ C 4/3 ; 16.u.z/ C 1/3

where now u is meromorphic and assumes the values 0; 1; 4; and 1 always with even multiplicities. Since u4 D 16f 3 =g is an elliptic function, the just mentioned multiplicities are always equal to 2 and u02 D  2 u.u C 1/.u C 4/ holds. Choosing p 4 D 8 3 just means a linear change of the independent variable. Assuming a1 D 0; a2 D 1, and a4 D 1, we will now go into details to derive (3.35) as one of three equivalent equations. It is quite plausible that in ‘almost’ all cases f and g assume each value a either in a .3W1/ or .1W3/ manner. To give a precise statement, let N . pWq/ .rI a / denote the counting functions of those a -points which are p-fold for f and q-fold for g. Then p C q  4 and minfp; qg D 1 holds for almost all a -points. a) In the first step we will prove N.rI a / D 12 T.r/ C S.r/ and N .3W1/ .rI a / D 14 T.r/ C S.r/ and N .1W3/ .rI a / D 14 T.r/ C S.r/: The first assertion follows from Exercise 3.23 and N.rI a /  12 T.r/ C O.1/. To P prove the second for a4 D 1, say, observe that p;q>1 N . pWq/ .rI 1/ D S.r/ and P N.r; f / C N.r; g/  3 . C 1/.N .W1/ .rI 1/ C N .1W/ .rI 1// C S.r/ P  4N.rI 1/ C 4 .  3/.N .W1/ .rI 1/ C N .1W/ .rI 1// C S.r/ P  12 T.r/ C 4 .N .W1/ .rI 1/ C N .1W/ .rI 1// C S.r/; hence P

4 .N .W1/ .rI 1/

C N .1W/ .rI 1// D S.r/; N.r; f / C N.r; g/ D 2T.r/ C S.r/; 1 T.r/ D N.rI 1/ C S.r/ D N .3W1/ .rI 1/ C N .1W3/ .rI 1/ C S.r/; 2 T.r/ D N.r; f / C S.r/  3N .3W1/ .rI 1/ C N .1W3/ .rI 1/ C S.r/ D 2N .3W1/ .rI 1/ C N.rI 1/ C S.r/; and T.r/ D N.r; g/ C S.r/  2N .1W3/ .rI 1/ C N.rI 1/ C S.r/

hold. b) In the second step we will prove that the auxiliary functions ˝D3



3 X g00 f0 g0 f 00 f0 3 0 C6  4 C 10 0 f f  a f g g gf D1

u02 is a zero-free polynomial! u.u C 1/.u C 4/

3.3 Uniqueness of Meromorphic Functions

105

Q which is obtained from ˝ by permuting f and g, vanish identically. To and ˝, prove that ˝ is a small function w.r.t. f and g we observe that ˝ is regular at every a -point of type .1W3/ and even vanishes at a -points of type .3W1/.3 In f0 particular, ˝ has counting function of poles N.r; ˝/ D S.r/: From 10 f g C 0

0

0

0

g g f 6 gf D 6 f f g C 4 f g we obtain

m.r; ˝/  m

 f0  C S.r/: f g

To estimate the first term on the right-hand side we refer to Exercise 3.25, which states that the auxiliary function  defined in (3.32) is small. From 

f0 f 02 g0 . f  g/ D f g . f  a1 /. f  a2 /. f  a3 /.g  a1 /.g  a2 /.g  a3 /

m.r; ˝/ D S.r/ and thus T.r; ˝/ D S.r/ follows. Since, however, ˝ vanishes at every a -point of type .3W1/ (1    4) with counting function 14 T.r/ C S.r/, Q it must vanish identically. The same, of course, is true for ˝. Q is the logarithmic derivative of c) To proceed we note that 14 .˝ C ˝/

.z/ D

. f .z/  g.z//4 P. f .z//P.g.z//

.P.x/ D

3 Y

.x  a //;

D1

which thus is a non-zero constant, and f and g parametrise the algebraic curve .x  y/4 D P.x/P. y/: For a1 D 0, a2 D 1, a3 D c ¤ 0; 1, intricate computations involving poles of type .3W1/ and .1W3/ show that 2c3 3c2 3cC2 D 0 (with solutions c D 1; 2; 12 ) 48 and D c2 cC1 holds, hence (3.35) in the case of c D 1; the interested reader is referred to [143]. We will give a ‘semi-rigorous MAPLE-aided proof’: When asked for the genus of the algebraic curve C ;c W .xy/4 D xy.x1/. y1/.x c/. y  c/; MAPLE responds ‘genus 5’ if neither nor c is specified, ‘genus 3’ if

D 16 but c is not specified, and ‘genus 1’ if . ; c/ D .16; 1/ (or .16; 2/ or .64; 12 /). The ‘proof’ relies on the fact that the genus is > 1 if . ; c/ ¤ .16; 1/, .16; 2/, and .64; 12 /.

3

˝ was constructed for that purpose; the verification is left to MAPLE.

106

3 Selected Applications of Nevanlinna Theory

3.3.6 Three Functions Sharing Four Values The question whether there might be more than two meromorphic functions that share four values was answered in the negative by Cartan [21]. His argument, however, contained a gap which could not be bridged. Persistently pursuing his ideas instead leads to the following counterexample, and even more to a characterisation of the function triples sharing four values. Example 3.3 Given any a ¤ 0; 1, the non-constant solutions to the differential equation u02 D p 4u.u C 1/.u  a/ are elliptic functions of elliptic order two. For a D 12 .1 ˙ i 3/ the algebraic equation P.x; y/ D y3  3..Na  1/x2  2x/y2  3.2x2  .a  1/x/y  x3 D 0

(3.36)

has singular points .x; y/ D .0; 0/; .a; 1/; .1; a/; .1; 1/, and no others. At each of these points two of the algebraic functions y D  .x/ defined by (3.36) have a square-root p singularity, while the other behaves regularly (for example, 1;2 .x/ D ˙c1 x C    and 3 .x/ D c2 x2 C    hold at x D 0). The branches f .z/ D  .u.z// admit unrestricted analytic continuation in the complex plane, hence are meromorphic functions that share the values 0; 1; a; 1; each value is assumed in turn with multiplicities .1; 1; 4/, .1; 4; 1/, and .4; 1; 1/, see Fig. 3.2. The triple . f1 ; f2 ; f3 / not only represents a positive example but also the normalised solution to the above mentioned problem. Lack of space prevents us from presenting the elaborate and involved proof, which can be found in [168]. Theorem 3.17 Any triple F1 ; F2 ; F3 of meromorphic functions sharing four values is given by Fj D M ı fj ı h, where h is non-constant entire, M is some Möbius transformation, and the functions fj are defined in Example 3.3. Remark 3.9 The functions fj have elliptic order six, hence they satisfy some Briot– Bouquet equation Q.w; w0 / D 0 of degree six with respect to w0 . The problem of how to determine Q may be solved as follows: consider the polynomials P.x; y/ and P1 .x; y; y1 / D 4x.x C 1/.x  a/Px .x; y/2  Py .x; y/2 y21 , which reflect the fact that P.u.z/; f .z//  0; dzd P.u.z/; f .z//  0, and u02 D 4u.u C 1/.u  a/, and compute the resultant R. y; y1 / of P and P1 with respect to the variable x to obtain w06  12 .1  aN /.3w  1 C aN /.w  1 C aN /.w  a/w04  256w3 .w  1/3 .w C a/3 D 0: In the same manner one can show that any two of these functions parametrise one and the same algebraic curve of genus one. Exercise 3.36 (Continued) Prove that the algebraic curve P.x; y/ D 0 has the parap t.1Ct/2 p t2 .1Ct/ 1 metrisation x D r.t/ D 12 .9  3 3i/ .1C3t/ 3i/ .1C3t/ : 2 ; y D s.t/ D 2 .9  3 Determine meromorphic functions hj (1  j  3/ such that u.z/ D r.hj .z// and fj .z/ D s.hj .z// holds.

3.3 Uniqueness of Meromorphic Functions

107

◦

Fig. 3.2 u has double poles  ı  and periods #; # 0 ; f1 has simple poles  ı and four-fold poles , and periods 3#; # C # 0 . The four-fold poles of f2 .z/ D f1 .z C #/ and f3 .z/ D f1 .z C 2#/ are  and ı, respectively

•

• ∗

◦ •

∗ ◦

• ∗

◦ •

∗ ◦

• ∗

◦ •

∗ ◦

• ∗

◦ •

∗ ◦

◦ •

3.3.7 Pair Sharing A new direction of research was opened by Czubiak and Gundersen [31], who considered meromorphic functions f and g that ‘share pairs’ .a; b/, this meaning f 1 .fag/ D g1 .fbg/, counting or disregarding multiplicities. For example, the functions f and g in Exercise 3.21 share five pairs .0; 0/, .1; 1/; . 18 ;  18 /; .1; 1/, and . 12 ; 14 /, the latter counting multiplicities. Czubiak and Gundersen introduced the useful technique of ‘algebraic auxiliary functions’ P. f .z/; g.z// to prove that the number ‘five’ is the best possible. Theorem 3.18 Meromorphic functions f and g that share more than five pairs are Möbius transformations of each other. Proof We will give a straightforward proof based on the main idea in [31]. We assume that f and g are not Möbius transformations of each other and share six finite pairs .a ; b /. By N.rI a ; b / we denote the counting function of common .a ; b /points of . f ; g/, not counting multiplicities, and set T.r/ D maxfT.r; f /; T.r; g/g with remainder term S.r/. Let L and L denote the Möbius transformations that map b onto a for 1    3 and 4    6, respectively. Then Nevanlinna’s Second Main Theorem yields 4T.r/ 

6 X

N.rI a ; b / C S.r/

D1

   1 1 C N r; C S.r/  f  L.g/ f  L .g/  2T.r; f / C T.r; L.g// C T.r; L .g// C S.r/ D 2T.r; g/ C 2T.r; g/ C S.r/:   N r;

This implies T.r; f / D T.r/ C S.r/, T.r; g/ D T.r/ C S.r/, and 6 X D1

N.rI a ; b / D 4T.r/ C S.r/:

(3.37)

108

3 Selected Applications of Nevanlinna Theory

We now choose c D .c1 ; : : : ; c6 / 2 C6 non-trivially such that F6 .x; yI c/ D c1 xy2 C c2 xy C c3 y2 C c4 y C c5 x C c6 vanishes at .x; y/ D .a ; b /, 1    5. With any such c consider 6 .z/ D F6 . f .z/; g.z/I c/; and first assume 6 6 0. Then T.r; 6 /  T.r; f / C 2T.r; g/  3T.r/ C S.r/ holds, and 6 vanishes whenever . f .z/; g.z// D .a ; b /, 1    5. This implies 5 X

N.rI a ; b /  T.r; 6 /  3T.r/ C S.r/

D1

hence N.rI a6 ; b6 / D T.r/ C S.r/ follows from (3.37). Since, however, any pair .a ; b / can take the role of .a6 ; b6 /, we obtain N.rI a ; b / D T.r/ C S.r/ in contrast to (3.37). This contradiction shows that 6  0 (representative for the other functions  ) and f .z/ D 

c6 C c4 g.z/ C c3 g.z/2 D M.g.z// c5 C c2 g.z/ C c1 g.z/2

holds. Since T.r; f / D T.r; g/ C S.r/, the representation of M cannot be in lowest terms, hence f is a Möbius transformation of g in contrast to our hypothesis. Remark 3.10 It is obvious that the same proof allows us to replace the pairs of constants with pairs of small functions, if Möbius transformations over the field generated by the functions a1 ; : : : ; a6 ; b1 ; : : : ; b6 rather than C are considered.

3.3.8 Five Pairs We will first collect in the form of exercises some basic facts taken from [72, 92]4 about meromorphic functions f and g that share five pairs .a ; b /. It is assumed that f and g are not Möbius transformations of each other and that the values a and b are finite. If necessary, three of the pairs may be chosen at will. We remind the reader of the special meaning of the word ‘almost’: almost every zero, pole, etc. means ‘up to a subsequence of zeros, poles, etc. with counting function S.r/’. Exercise 3.37 Construct in a non-trivial manner F.x; yI c/ D c1 x2 y C c2 xy C c3 x2 C c4 x C c5 y C c6 4

Cited in [72]. I had no access to [92].

3.3 Uniqueness of Meromorphic Functions

109

such that F.a ; b I c/ D 0 (1    5) holds. If F.z/ D F. f .z/; g.z/I c/ 6 0 prove 3T.r; f / 

5 X

N.rI a ; b / C S.r/  2T.r; f / C T.r; g/ C S.r/:

D1

Exercise 3.38 (Continued) F.z/  0 implies g D M. f /, where M is rational and deg M  2. Prove that M is a Möbius transformation. (Hint. Prove that M 1 .fb g/ D fa g and deg M D 2 is impossible for three pairs .a ; b /.) Q yI cQ/ D cQ 1 y2 xC cQ 2 xyC cQ 3 y2 C cQ 4 yC cQ 5 xC cQ 6 can be constructed. In the same way, F.x; Q Q f .z/; g.z/I cQ/ can vanish From Exercise 3.38 it follows that neither F nor F.z/ D F. identically. This will henceforth be assumed; it implies T.r; f / D T.r; g/ C S.r/ D T.r/ C S.r/ and 5 X

N.rI a ; b / D 3T.r/ C S.r/:

(3.38)

D1

In particular, equality holds in the Second Main Theorem, and every value a ¤ a resp. b ¤ b is ‘normal’ for f resp. g (for example, N.r; f /CN.r; g/ D 2T.r/CS.r/). Exercise 3.39 (Continued) Prove m.r; 1=F/ D S.r/ and N.r; 1=F/ D N.r; 1=F/ C S.r/ D

5 X

N.rI a ; b / C S.r/;

D1

Q Thus almost every zero of F and hence T.r; F/ D 3T.r/ C S.r/, and the same for F. FQ is simple and is also a zero of f  a and g  b for some , and vice versa. Exercise 3.40 (Continued) Given any three pairs .a ; b /, .a ; b /, and .a ; b /, choose .d1 ; d2 ; d3 ; d4 / non-trivially such that H  .x; y/ D d1 xy C d2 x C d2 y C d4 vanishes whenever .x; y/ equals one of the distinguished pairs; H  . f .z/; g.z//  0 is excluded since f and g are not Möbius transformations of each other. Prove that H.z/ D

H234 .x; y/H235 .x; y/H245 .x; y/ ˇˇ ˇ Q y/ xDf .z/;yDg.z/ F.x; y/F.x;

vanishes at almost every zero of f  a2 and g  b2 and has poles of order at most two at almost every zero of f  a1 and g  b1 . Prove also that there are almost no other poles since almost all poles of the numerator and zeros of the denominator, with the exception of the zeros of . f  a1 /.g  b1 /, are cancelled by the poles of the denominator and the zeros of the numerator, respectively. Also H satisfies m.r; H/ D S.r/ by Exercise 3.40, hence T.r; H/ D N.r; H/CS.r/  2N.rI a1 ; b1 /C S.r/:

110

3 Selected Applications of Nevanlinna Theory

Since  D 1 and  D 2 are only place markers, N.rI a ; b /  2N.rI a ; b / C S.r/ and N.rI a ; b /  13 T.r/ C S.r/

(3.39)

follows from (3.38). In particular, none of the values a and b can be exceptional.

3.3.9 Gundersen’s Example Rediscovered The paper [183] contains yet another characterisation of Gundersen’s example (Exercise 3.21). This characterisation uses Reinders’ Theorem 3.12. Theorem 3.19 Suppose meromorphic functions f and g share four pairs .a ; b /, and a fifth pair .a5 ; b5 / counting multiplicities such that     m r; 1=. f  a5 / C m r; 1=.g  b5 / D S.r/

(3.40)

holds. Then there exists some Möbius transformation M such that either f D M ı g holds or else f and M ı g share four values and one pair. Proof The proof is based on the technique developed in [31, 72]. We assume that f and g are not Möbius transformations of each other, and note that (3.40) is automatically fulfilled if a D b , 1    4. Three of the pairs .a ; b / may be prescribed, we will assume .a1 ; b1 / D .0; 0/, .a2 ; b2 / D .2; 4/, and, in particular, a5 D b5 D 1. Then f and g have the same poles counting multiplicities, and (3.40) takes the form m.r; f / C m.r; g/ D S.r/:

(3.41)

We note that almost all poles of f and g are simple since f and g have almost no critical points in common. Exercise 3.41 Prove that there are at least two linearly independent vectors c 2 C5 such that P.x; yI c/ D c1 x2 C c2 xy C c3 y2 C c4 x C c5 y satisfies P.a ; b I c/ D 0

.1    4/:

(3.42)

Assuming P.zI c/ D P. f .z/; g.z/I c/ 6 0, deduce that 4 X

N.rI a ; b / D 2T.r/ C S.r/

and T.r/ D N.rI 1/ C S.r/:

D1

(Hint. Apply the Second Main Theorem to . f ; a1 ; : : : ; a4 / and .g; b1 ; : : : ; b4 /.)

3.3 Uniqueness of Meromorphic Functions

111

Still assuming P 6 0 it follows from equality in the Second Main Theorem that N.r; 1=P/ D N.r; 1=P/ C S.r/ and m.r; 1=P/ D S.r/ holds. In particular, the ratio .z/ D P.zI cQ/=P.zI c/ has Nevanlinna characteristic T.r; / D S.r/ and f and g parametrise the algebraic curve F.x; yI z/ D 1 x2 C 2 xy C 3 y2 C 4 x C 5 y D 0

.k D ck  cQ k /

(3.43)

over the field C./. This is also true if P.zI c/ or P.zI cQ/ vanishes identically. Exercise 3.42 Prove that 1 3 6 0. (Hint. For 3  0, say, g would be a Möbius transformation or a rational function of f of degree two over the field C./, the latter contradicting T.r; g/ D T.r; f / C S.r/. And the former?) The algebraic curve (3.43) has the rational parametrisation (set x D ty) xD

t.4 t C 5 / 4 t C 5 p.z; t/ q.z; t/ D 2 D 2 ; yD s.z; t/ 1 t C 2 t C 3 s.z; t/ 1 t C 2 t C 3

. D  .z//

over C./: In terms of f and g this yields t.z/.4 .z/t.z/ C 5 .z// p.z; t.z// D s.z; t.z// 1 .z/t.z/2 C 2 .z/t.z/ C 3 .z/ 4 .z/t.z/ C 5 .z/ q.z; t.z// D g.z/ D s.z; t.z// 1 .z/t.z/2 C 2 .z/t.z/ C 3 .z/ f .z/ D

with

t.z/ D

f .z/ : g.z/

From (3.39) with a1 D b1 D 0 it follows that f and g have ‘many’ common zeros. There are three possibilities to be discussed: the common zeros correspond to the – poles of t, in which case 4  0 and almost all zeros of f are simple, while the zeros of g have order two. Moreover, t has almost no zeros (N.r; 1=t/ D S.r/). – zeros of t, in which case 5  0 and almost all zeros of g are simple, while the zeros of f have order two. Moreover, t has almost no poles (N.r; t/ D S.r/). – zeros of 4 .z/t.z/ C 5 .z/ with 4 5 6 0. Then almost all zeros of f and g are simple, and t has almost no zeros and poles (N.r; 1=t/ C N.r; t/ D S.r/). Taking all pairs .a ; b / (1    4) into account, the following holds: for every  there exist  ;  ; ˛ ; ˇ ; ˇQ 2 C./k such that p.z; t/  a s.z; t/ D  .t  ˛ /.t  ˇ / q.z; t/  b s.z; t/ D  .t  ˛ /.t  ˇQ /

.ˇ ¤ ˇQ /

At first glance one would expect that ˛ ; ˇ ; ˇQ are algebraic over C./. But this is not the case, since an analytic continuation which permutes ˛ and ˇ would also permute ˛ and ˇQ , in contrast to ˇ 6 ˇQ .

k

112

3 Selected Applications of Nevanlinna Theory

holds; occasionally the factor .t  ˇ / and .t  ˇQ / corresponding to ˇ  1 and ˇQ  1, respectively, might be missing. The functions ˛ are mutually distinct, and the same is true for ˇ and ˇQ . Also ˇ and ˇQ are exceptional for t, unless one of them coincides with ˛ . Since t has at most two exceptional functions, we obtain the following picture: for  D 1 and  D 4, say, we have ˇ  ˛ , that is, the pairs .a ; b / are attained by . f ; g/ in a .2 W 1/ manner, while for  D 2 and  D 3 this happens the other way .1 W 2/. This means that, in addition to (3.43), we also have Fy .a ; b I z/  0

. D 1; 4/ and Fx .a ; b I z/  0 . D 2; 3/:

(3.44)

Exercise 3.43 Assume 3  1 (this is possible since 3 6 0 is already known). From (3.44), that is, Fy .0; 0/ D Fx .2; 4/ D Fx .a3 ; b3 / D Fy .a4 ; b4 / D 0 deduce 1 D

b4 .b3 C 4/ 2b4 4b4 .b3 C 2a3 / ; 2 D  ; 5 D 0: ; 3 D 1; 4 D a4 .a3  2/ a4 a4 .2  a3 /

(3.45)

In particular, the functions k are constant, and f and g are ordinary rational functions of the meromorphic function t D f =g. Having determined the coefficients (3.45) we will now solve the nonlinear system 4b4 .b3 C 2a3 / b4 .b3 C 4/ 2 2b4 a  b D 0 a b C b2 C a4 .a3  2/ a4 a4 .2  a3 /

.2    4/

(3.46)

obtained from F.a ; b I z/ D 0 with a2 D 2, b2 D 4. The first solution b3 D 2a3 , b4 D 2a4 leads to g D 2f against our hypothesis, while the second solution, again obtained with the help of MAPLE, may be expressed in terms of a4 as follows: a1 D b1 D 0; a2 D 2; b2 D 4; a3 D a4  2; b3 D 2a4  4; b4 D 2a4  8: 2

4 2/ (cross-ratio), there Since .0; 2; a4  2; a4 / D .0; 4; 2a4  4; 2a4  8/ D a.a4 .a 4 4/ exists some Möbius transformation M such that M.b / D a .1    4/, hence f and M ı g share four values a1 ; a2 ; a3 ; a4 and the pair .1; M.1//, and this finishes the proof of Theorem 3.19.

Remark 3.11 It is open whether or not—and how—the hypothesis (3.40) may be relaxed. Is it sufficient to assume that the pair .a5 ; b5 / is shared counting multiplicities? Is it even true that functions sharing five pairs are either Möbius transformations of each other or else are conjugate to the functions in Gundersen’s Example? Recently [75] it was shown that there cannot exist meromorphic functions that share five pairs, two of them counting multiplicities, unless the functions are Möbius transformations of each other. Again it could be useful to consider algebraically dependent pairs . f ; g/ sharing five pairs .a ; b /.

Chapter 4

Normal Families

In this chapter the theory of Normal Families will be deepened and enlarged, and applied to various problems in the fields of entire and meromorphic functions, distribution of zeros of differential polynomials, ordinary differential equations and functional equations. The Yosida classes, which play an outstanding part in the theory of algebraic differential equations, will be introduced and discussed in the final part of the chapter.

4.1 Re-scaling Re-scaling means any transformation fn ! fQn .z/ D an fn .bn C cn z/ that converts a given non-normal sequence . fn / into a normal sequence . fQn /. As an example, we consider the sequence fn .z/ D zn , which is not normal at any point z0 on the unit circle. Replacing z with z0 C 1n z and multiplying by zn 0 yields a new sequence    z=z0 n z z D 1 C f C ; which converges to fQ .z/ D ez=z0 with spherical fQn .z/ D zn 0 n 0 n n derivative fQ ] .z/  fQ ] .0/.

4.1.1 Zalcman’s Lemma What happened in our simple example happens mutatis mutandis in every family F of meromorphic functions that is not normal at some point. Written down formally this leads to the famous Zalcman Re-scaling Lemma [212, 213], which had and still has an enormous impact on many parts of Complex Analysis. Theorem 4.1 Suppose that the family F of meromorphic functions is not normal at some point z0 of the common domain of definition. Then there exist sequences

© Springer International Publishing AG 2017 N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, DOI 10.1007/978-3-319-59800-0_4

113

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4 Normal Families ]

fk 2 F , zk ! z0 , and k D 1=fk .zk / ! 0 such that the re-scaled sequence gk .z/ D fk .zk C k z/ tends to some non-constant meromorphic function satisfying g] .z/  g] .0/ D 1: Proof To simplify matters we assume that z0 D 0 and all functions under consideration are meromorphic on jzj  1. Then F contains functions fn satisfying ] supjzjm is normal on jzj < m; say, and the well-known Cantor diagonal process yields a subsequence .gk / D .hnk / which converges, uniformly on every disc jzj < R. The limit function ] g satisfies g] .z/  lim supn!1 hn .z/  1 D g] .0/.

4.1.2 Pang’s Lemma Zalcman’s Lemma was generalised by Pang [136, 137] in a way which makes it more flexible, and, in particular, applicable to algebraic differential equations. Theorem 4.2 The statement of Theorem 4.1 remains valid if the sequence gk .z/ D fk .zk C k z/ is replaced with gk .z/ D k˛ fk .zk C k z/

.1 < ˛ < 1 arbitrary/:

We note several important supplements to Theorem 4.2. 1. If the zeros and poles of the functions f 2 F have order at least m and n, respectively, then ˛ may vary in m < ˛ < n. 2. In particular, if f has no zeros [m D 1] resp. poles Œn D 1, then every ˛ < n resp. ˛ > m is admitted.

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3. If all zeros have multiplicity at least m, and j f .m/ ./j  K holds for every f 2 F and every zero of f , then even ˛ D m is admitted. 4. ˛ D n is admitted if limz! j f .z/.z  /n j  1=K holds at every pole of every f 2 F. Remark 4.1 The refinements 3. and 4. are due to Chen and Gu [23]. In the proof of Theorem 4.2 the expression .1  jzj/1C˛ j f 0 .z/j .1  jzj/2˛ C j f .z/j2 takes the part of .1  jzj/f ] .z/. The closely related expression f ]˛ .z/ D

jzj˛ j f 0 .z/j jzj2˛ C j f .z/j2

will play a crucial role in the context of Yosida functions, see Sect. 4.3. Example 4.1 The solutions to Painlevé’s first equation w00 D z C 6w2 have double poles with principal part .z  /2 , hence 1 < ˛  2 is admitted. It will turn out that the right choice is ˛ D 2 in the following sense: suppose zk ! 1 and w.zk C z/ 1

is not normal at z D 0. Then some subsequence of k2 w.zk C k z/ with k D zk 4 (note that k need not be positive) tends to some limit function w which satisfies 1

]1

1

w00 D 1 C 6w2 . Choosing k D zk 4 is admitted since w 2 .z/ D O.jzj 4 / holds (a non-trivial fact). More on this kind of re-scaling in Sect. 4.3, Chaps. 5 and 6 .

4.1.3 Functions of Poincaré, Abel, and Zalcman Nothing can be said about the ratio jzk z0 j=jk j in Zalcman’s Lemma. The following examples deal with the extremal cases zk D z0 on one hand, and k D o.jzk  z0 j/ on the other. Poincaré Functions Suppose R is a rational function of degree d > 1 with repelling fixed-point at z D z0 , that is, we assume that R.z/ D z0 C .z  z0 / C a2 .z  z0 /2 C   

.jj > 1/

holds on some neighbourhood of z0 . Then the sequence of iterates Rn D R ı Rn1 is not normal at z0 since Rn .z/ D z0 C n .z  z0 / C    . By Zalcman’s Lemma there exist sequences k ! 0, zk ! z0 , and nk " 1 such that Rnk .zk C k z/ tends to some non-constant meromorphic function f satisfying f ] .z/  f ] .0/ D 1. If we dispense with the latter condition, it is possible to choose nk D k; zk D z0 , and k D k .

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Assuming this for the moment we obtain f .z/ D R. f .z// with f .z/ D z0 C z C c2 z2 C    at z D 0

(4.1)

from RnC1 .z0 C .nC1/ .z// D R.Rn .z0 C n z//I f is called a Poincaré function and solves Schröder’s functional equation (4.1). Assuming that f exists on jzj < r, it may be analytically continued into the discs jzj < jjr; jzj < jj2 r; etc., hence into the whole plane by applying f .z/ D R. f .z=// successively. Moreover, f .z/ D Rn . f .z=n //  Rn .z0 C n z/

and f .z/ D lim Rn .z0 C n z/ n!1

holds, locally uniformly on C. To prove local existence it is convenient to consider the local inverse ˚.z/ D R1 .z/ D 1 z C    (we assume z0 D 0), solve F.˚.z// D 1 F.z/ with

F.z/ D z C b2 z2 C    at z D 0;

and set f D F 1 locally. Exercise 4.1 The local existence of F can be proved as follows ( D 1 ). – Choose D D fz W jzj < rg such that j˚.z/j  jj2=3 jzj holds on D (why is this 2 possible?) and prove that j˚.z/  zj  jzj2 . r (Hint. Apply the Maximum Principle to .˚.z/  z/=z2 :) – Prove that the iterates ˚ n are well-defined and satisfy j˚ n .z/j  jj2n=3 jzj on D; 2 j˚ n .z/j2 2r jjn=3 .Fn D ˚ n =n /:  and deduce jFnC1 .z/  Fn .z/j  r jjnC1 jj – Prove that Fn ! F, uniformly on D, and conclude that F ı ˚ D F. Theorem 4.3 ([41, 177]) The Poincaré functions are meromorphic on the plane with Nevanlinna characteristic T.r; f /  r%

 log deg R  : %D log jj

Proof Set T.r/ D T.r; f /, q D jj, and d D deg R, and observe that Valiron’s Lemma yields T.qr/ D dT.r/ C .r/, where  is bounded. Iterating gives T.qn r/ D d n T.r/ C d n1 .r/ C d n2 .qr/ C    C .qn1 r/ D d n .T.r/ C O.1// as n ! 1, uniformly with respect to r. Set s D qn r and restrict r to r0 < r  qr0 to obtain 0 < C1  T.s/s%  C2 if r0 is chosen sufficiently large.

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Exercise 4.2 Determine the Poincaré function in the following cases 1. R.z/ D z2 , z0 D 1,  D 2. 2. R.z/ D 2z2  1 (the second Chebychev polynomial), z0 D 1,  D 4. 2z 3. R.z/ D , z0 D 0,  D 2. 1  z2 p Solution 1. ez , 2. cos 2z , 3. tan z . Example 4.2 Let f be any Poincaré function and suppose that a 2 C has dn mutually distinct pre-images a under Rn . Then f .n z/  a D Rn . f .z//  a implies  m jjn r;

1  X  1  D C O.1/ m r; f a f  a D1 dn

 2T.r; f / C O.log r/ D 2dn T.jjn r; f / C O.log r/

by the Second Main Theorem. If this is true for every n, then  m r;

1  D o.T.r; f // f a

(4.2)

holds (note that the O.log r/-term depends on n). We note without proof that (4.2) even holds if R1 .fag/ ¤ fag (see [41, 177]). On the other hand, a is a Picard value for f if R1 .fag/ D fag. In particular, f is entire if R is a polynomial. Example 4.3 The Weierstraß P-function has the duplication formula }.2z/ D

.6} 2 .z/  g2 =2/2  2}.z/ D R.}.z//I 4.4} 3 .z/  g2 }.z/  g3 /

R has a repelling fixed-point at z D 1 with ‘multiplier’  D 4 (this is defined by ‘conjugation’: R1 ./ D 1=R.1=/ has a repelling fixed-point at  D 0 with  D R01 .0/ D 4). The corresponding Poincaré function, this time normalised by p f .z/ D 1=z C    , is f .z/ D }. z/: p p p f .4z/ D }. 4z/ D }.2 z/ D R.}. z// D R. f .z//: Abel Functions The polynomial P.z/ D z C zd of degree d  2 has a parabolic fixed-point at the origin (P.0/ D 0, P0 .0/ D 1), and the sequence Pn is not normal at z D 0 since Pn .z/ D z C nzd C    . We assume Pnk .zk C k z/ ! f .z/ 6 const, zk ! 0; k ! 0; and zk k ¤ 0; and claim that k D o.jzk jd /: Otherwise we may assume zdk D bk k with bk ! b 2 C. Then P.zk C k z/ D zk C k .z C b C o.1// and P. f .z//  P.Pnk .zk C k z// D Pnk .P.zk C k z//  Pnk .zk C k .z C b//  f .z C b/

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holds, uniformly on every disc jzj < R, and f satisfies f .zCb/ D P. f .z//. Obviously b D 0 is impossible by Valiron’s Lemma, hence we may assume b D 1, that is, f solves Abel’s functional equation f .z C 1/ D P. f .z//I

(4.3)

f has order of growth at most 1 since f is entire with bounded spherical derivative f ] , see Theorem 4.8 below. This, however, is impossible, since any non-constant solution to Abel’s equation has infinite order of growth (see Yanagihara [207] and the exercise below). This example also shows how sensitively the limit function may depend on the sequence .zk /: replacing zk with zk C zdk while k remains unaltered leads to the limit function P ı f : Example 4.4 Let f be any non-constant solution to Abel’s equation (4.3), where P is any nonlinear rational function with parabolic fixed-point at z D 0; global existence of f is assumed. From Cartan’s Identity and Valiron’s Lemma it easily follows that T.r C 1; f /  T.r; P ı f / C O.1/ D dT.r; f / C O.1/, hence T.r C n; f /  dn ŒT.r; f / C O.1/ .n ! 1/ holds, uniformly w.r.t. r; this yields T.r; f /  cd r (r  r0 ) for some c > 0. Zalcman Functions A rich class of meromorphic functions is obtained by a procedure similar to the construction of the Poincaré and Abel functions. Let R be any rational function of degree d > 1 and denote by .Rn / the sequence of its iterates. If .Rn / is not normal at z0 , that is, if z0 is any point in the Julia set of R, Zalcman’s Lemma yields meromorphic functions f .z/ D lim Rnk .zk C k z/ k!1

.zk ! z0 ; k ! 0/;

(4.4)

which may be viewed as generalisations of the Poincaré and Abel functions; the term Zalcman function was coined in [177]. To describe the most interesting facts concerning the value distribution of these functions we need two concepts from iteration theory. – A critical point is a zero of R0 or a multiple pole of R. The critical points generate the critical orbit CC D fRn ./ W  critical point; n 2 Ng: – The exceptional set E D E.R/ of R is the largest finite set such that R1 .E/ E; it consists of at most two points and is ‘usually’ empty: card E D 1: R is conjugate to some polynomial; card E D 2: R is conjugate to z 7! zd or z 7! zd .

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Theorem 4.4 Zalcman functions for any rational function R and any z0 in the Julia set of R have the following properties.  1  D o.T.r; f // if a … E. – m r; f a – a 2 E is a Picard value of f . – #.a; numberP if a 2 CC n E, and #.a; f / D 0 otherwise. P f / is a positive rational P – .a; f / D 2. a2CC nE #.a; f / C a2E ı.a; f / D a2b C For a proof and more results on Zalcman functions, see [177]. Example 4.5 Let f denote any Zalcman function for .R; z0 /. – R.z/ D z2 C i: f is entire and #.i; f / D 12 , #.1 C i; f / D 13 , #.i; f / D 16 . – R.z/ D z C z2 : f is entire and #.Rn . 12 /; f / D 2n (n 2 N).   – R.z/ D 2i z  1z : #.a; f / D 12 for a D 1; 1; 0; 1, with critical orbit: ˙i 7! ˙1 7! 0 7! 1 D R.1/.

4.1.4 The Theorems of Picard and Montel The close relation between Picard- and Montel-type theorems, in other words between the qualitative theory of entire and meromorphic functions on one hand, and the theory of normal families of holomorphic and meromorphic functions on the other, has been known for a long time. Zalcman’s Lemma perhaps provides the most elegant approach to both. We start with a proof of Picard’s first or ‘little’ theorem. Theorem 4.5 Every transcendental meromorphic function assumes every value on the Riemann sphere with at most two exceptions. Proof (Ros [145]) Let f be any transcendental meromorphic function that omits three values, which may be assumed to be 0, 1, and 1. For technical reasons it is required that f 0 .0/ ¤ 0, which is achieved by considering f .z0 C z/ with f 0 .z0 / ¤ 0 instead of f without changing notation. We consider the sequence . fn / of entire p p n n functions fn .z/ D 2 f .2n z/; where fn .0/ is any of the values 2 f .0/: Then fn omits n the values 0 and e2 i=2 (0   < 2n /. If we assume that the sequence . fn / is normal on the plane, each convergent subsequence . fnk / tends to some entire function g, n which by Hurwitz’ Theorem omits every value e2 i=2 (0   < 2n ; n 2 N/, hence omits every value on the circle jwj D 1. This implies that either jg.z/j  1 or else jg.z/j  1 holds on C, hence g is constant by Liouville’s Theorem. This, however, is excluded by g0 .0/=g.0/ D f 0 .0/=f .0/ ¤ 0. On the other hand, if . fn / is not normal at some point z0 , Zalcman’s Lemma yields some non-constant entire limit function n g.z/ D limk!1 fnk .zk C k z/, which again omits the values e2 i=2 , and the same contradiction is obtained.

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In combination with Zalcman’s Lemma, Picard’s Theorem immediately leads to Montel’s Second Normality Criterion. Theorem 4.6 The family F of meromorphic functions f omitting a fixed triple of values is normal on D. Proof If F is assumed to be non-normal at some point z0 2 D, Zalcman’s Lemma applies, yielding some non-constant meromorphic function that omits three values by Hurwitz’ Theorem. This, however, contradicts Picard’s Theorem. Exercise 4.3 The omitted values a; b; c in Theorem 4.6 are assumed to be the same for every f 2 F . They may, however, also vary with f . Suppose that each f 2 F omits three values af ; bf ; cf . Prove that F is normal, provided .af ; bf /.bf ; cf /.cf ; af /  ı > 0: (Hint. Consider the family of cross-ratios . f ; af ; bf ; cf / or use the Bolzano–Weierstraß Theorem.) The step from Montel’s Second Criterion to Picard’s ‘Great Theorem’ is classical. Theorem 4.7 Let f be meromorphic on 0 < jz  z0 j < r with essential singularity at z D z0 . Then on every punctured neighbourhood of z0 , f assumes every value with at most two exceptions infinitely often. Proof Suppose that f takes on three values (0, 1, and 1, say) only finitely often on 0 < jzj < ı0 (z0 D 0 is tacitly assumed), hence omits these values on some punctured disc 0 < jzj < ı < ı0 . Then the sequence of functions fn .z/ D f .2n z/ (2n < ı) is normal on 0 < jzj < 1, and we may assume that . fnk / converges to some limit g. If g is finite it is bounded by M on jzj D 1=2, thus f is bounded by 2M, say, on the circles jzj D 2nk 1 . By the Maximum Principle, f is bounded by 2M on the annuli 2nkC1 1 < jzj < 2nk 1 , hence on 0 < jzj < 2n1 1 , which contradicts the fact that z0 D 0 is an essential singularity. If, however, . fnk / converges to 1, the same argument applies to the sequence .1=fnk / and limit function g D 0.

4.1.5 Normal Functions Suppose f is meromorphic on the unit disc D. Then the functions f ı M, where zei˛ Ca M.z/ D 1CN denotes any Möbius transformation mapping D onto itself, are also azei˛ meromorphic on D. They form the family Ff , and f is called normal if the family Ff is normal. The term ‘normal function’ was coined by Lehto and Virtanen [108].1

1

Normal functions (without the name) were considered much earlier by Noshiro [130].

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121

An equivalent statement is supjzj 1, and consider u.z/ D

j f 0 .z/j j f .z/j.R C log j f .z/j/

.z 2 E; R > 0 arbitrary/:

Then j f 0 .z/j  2 and u.z/  2=R hold on C \ @E, while log u D u2 holds on E n fzeros of f 0 g.2 We fix z0 2 E, set Ez0 D fz W jz  z0 j < Rg, v.z/ D

R2

2R  jz  z0 j2

. log v D v 2 on Ez0 /;

1 is the Poincaré density of eR < jwj < 1 and satisfies log U D U 2 . jwj.R C log jwj/ The chain-rule for the Laplacian yields log u D log j f 0 jC.U 2 ıf /j f 0 j2 D u2 for u D j f 0 j Uıf whenever f 0 .z/ ¤ 0.

2

U.w/ D

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and D D fz 2 E \ Ez0 W u.z/ > v.z/g. If D is empty, u.z0 /  2=R holds. Otherwise w D log u  log v satisfies w D u2  v 2 > 0 on each connected component C of D (note that f 0 .z/ ¤ 0), and has boundary values w < 0 on @C \ @E and w D 0 on @C \ E. Since w is subharmonic on C, this contradicts the maximum principle. Thus D D ; and u.z0 /  2=R holds. The inequality u.z/  2=R can be written as  log j f .z/j  : j f 0 .z/j  2j f .z/j 1 C R Taking the limit R ! 1 yields j f 0 .z/j  2j f .z/j whenever j f .z/j > 1, while j f 0 .z/j  2 is trivially true if j f .z/j  1. The main assertion in Theorem 4.8 follows by integrating j f 0 .z/j < 2˛.1 C j f .z/j/. Remark 4.2 Clunie and Hayman [30] proved T.r; f / D O.rˇC1 / for entire functions with spherical derivative f ] .z/ D O.jzjˇ /; ˇ > 1.

4.2 Applications of the Zalcman–Pang Lemma Bounded entire functions are constant, and locally bounded families of holomorphic functions are normal. Also meromorphic functions omitting a triple of values are constant, and families of meromorphic functions omitting a fixed triple of values are normal. These are two specifications of Bloch’s heuristic principle, which may be vaguely verbalised as follows: Any ‘property’ P, which enforces meromorphic functions (on the plane) to be constant, imposed on a family of meromorphic functions on some domain D makes this family normal. Zalcman’s Lemma was one by-product in a bid to formalise Bloch’s principle. Much more on this principle, also called the Robinson–Zalcman Heuristic Principle, can be found in Schiff [148]. Zalcman’s Lemma is, however, not restricted to verifying or falsifying Bloch’s Principle in particular cases. For example, it may be employed to prove qualitative (‘soft’) versions of theorems on entire and meromorphic functions, or can be used to transfer results which are only known for meromorphic functions of finite order of growth to the general case. With regard to the extensive literature it would be presuming to aim at completeness. Our focus will be on topics that are in some sense representative and, in particular, closely related to the main issues of the present book.

4.2.1 Examples of Bloch’s Principle In accordance with the introductory words we will discuss some examples. Theorem 4.9 Let F be any family of meromorphic functions on some domain D such that every f 2 F assumes each of the values aP.1    q/ always with q multiplicity at least m (omits a if m D 1). Then D1 .1  m1 / > 2 implies normality of F .

4.2 Applications of the Zalcman–Pang Lemma

123

Proof If F is not normal at some point we obtain in the usual way some nonconstant meromorphic function g.z/ D limk!1 fk .zk Ck z/ . fk 2 F ) with bounded spherical derivative. Whenever g.0 / D a there exists some sequence k ! 0 such that fk .zk C k k / D a (k  k0 ) with multiplicity at least m , hence a is totally ramified for g with .a ; g/  1  1=m (orP is a Picard value if m D 1). This, q however, contradicts Nevanlinna’s inequality D1 .a ; g/  2. As a corollary we obtain the following normality criterion. The original result due to Lappan [107] corresponds to normal functions. Corollary 4.1 Let K be a positive number, a .1    5/ distinct complex numbers, and F a family of meromorphic functions on some domain D, such that for every f 2 F , f ] .z/  K holds on f 1 .fa1 ; : : : ; a5 g/. Then F is normal. Proof If F is not normal, the re-scaling process yields a non-constant meromorphic function g.z/ D limk!1 fk .zk C k z/ . fk 2 F ) with ]

g] .0 / D lim k fk .zk C k k / D 0 .0 D lim k ; fk .zk C k k / D a / k!1

k!1

whenever g.0 / D a . Thus each value a is totally ramified or a Picard value for g, again in contrast to Nevanlinna’s theorem. Remark 4.3 If F consists of holomorphic functions only, then three finite values will suffice, since one can always take into consideration the Picard value a4 D 1 with m4 D 1 and 1  1=m4 D 1. Pq Exercise 4.5 To prove that the condition D1 .1  m1 / > 2 as well as the number five are the best possible, consider the sequence fn .z/ D }.nz/ on D, say; } denotes the Weierstraß P-function for some lattice (} 02 D 4.}  e1 /.}  e2 /.}  e3 /). Prove that . fn / is not normal, although the values e1 ; e2 ; e3 ; and e4 D 1 are totally ] ramified .m D 2/ and fn .z/ D 0 whenever fn .z/ 2 fe1 ; e2 ; e3 ; e4 g. As a holomorphic example consider the non-normal sequence of functions fn .z/ D cos.nz/ on D, with a1 D 1 and a2 D 1 (m1 D m2 D 2), and Picard value a3 D 1 .m3 D 1/. For every transcendental meromorphic function f , n  1, and c ¤ 0, f . f .n/  c/ has infinitely many zeros (Hayman’s alternative [78]). This leads to the following normality criterion, see Gu [64]. Exercise 4.6 Prove that the family F of meromorphic functions such that f and f .n/ C an1 f .n1/ C    C a1 f 0 C a0 f  1 is zero-free on D .n  1; a0 ; : : : ; an1 fixed) is normal. We note that a0 ; : : : ; an1 may even depend on f provided ja j < K (0   < n) holds for some K > 0 independent of f . (Hint. Since f 2 F has no zeros, Theorem 4.2 applies with any ˛ < 1. Which ˛ is appropriate?) As we have seen, the Zalcman–Pang Lemma may be used to prove new normality criteria3 inspired by Bloch’s Principle and certain results in Nevanlinna theory.

3

It seems that some criteria of this kind are easier to prove than to apply to non-artificial problems.

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Note, however, that neither Bloch’s Principle nor its reversal is a theorem. There are ‘properties’ P which lead to normality criteria though there exist non-constant meromorphic functions having these properties. Theorem 4.10 ([13, 136, 137]) Let a ¤ 0 and b be complex constants and n  3 an integer. Then the family F of meromorphic functions f such that f 0 C af n  b has no zeros in the common domain of definition, is normal. Proof We may assume a D 1. If F is not normal at z0 , we choose fk 2 F , zk ! z0 ; 1 k ! 0, and ˛ D n1 such that gk .z/ D k˛ fk .zk C k z/ tends to some non-constant meromorphic function g. From n˛ D ˛ C 1 it then follows that g0k .z/ C gk .z/n D k˛C1 Œ fk0 .zk C k z/ C fk .zk C k z/n  never assumes the value k˛C1 b. Since g0 C gn cannot be zero-free by Corollary 4.2 below (n D 3) and Theorem 3.8 (n  4) if g is transcendental, and a fortiori if g is rational, there exists some zero of g0 C gn . The image of D W jz  j < ı under g0 C gn covers some disc jwj < , and so .g0k C gnk /.D/ covers jwj < =2 for k  k0 , say, by Hurwitz’ Theorem. This, however, contradicts k˛C1 b ! 0. Remark 4.4 It is obvious that a D af and b D bf may depend on f , provided 1=K  jaf j  K and jbf j  K. Exercise 4.7 In [138] Pang and Ye proved the following Criterion. Let k  1 and n  k C 3 be integers, and a1 ; : : : ; akC1 holomorphic functions on some domain D. Then the family F of meromorphic functions f such that f .z/n C f .k/ .z/ C a1 .z/f .k1/ .z/ C    C ak .z/f .z/ C akC1 .z/ ¤ 0 on D, is normal. For a proof assume non-normality at some point and apply Zalcman–Pang Rek scaling with ˛ D n1 2 .0; 1/ to obtain a non-constant meromorphic function g n such that  D g C g.k/ has no zeros. By Exercise 3.2, n D k C 3 is necessary, and g satisfies g0 D c.g  ˇ1 /.g  ˇ2 /. Deduce that g.k/ D Pk .g/, where Pk is a polynomial of degree k C 1 with simple zeros at ˇ1 and ˇ2 . Prove that gn C Pk .g/ D .g  ˇ1 /n1 .g  ˇ2 /n2 with n1 C n2 D n and n1 ˇ1 C n2 ˇ2 D 0 is impossible. Remark 4.5 By Exercise 3.9 there exist transcendental meromorphic functions such that f 0  f 3  1 resp. f 0 C f 4  1 is zero-free. It is obvious how to modify these functions such that this holds for f 0 C af n C b .n D 3 resp. n D 4 and ab ¤ 0). More importantly, for n D 4 these functions satisfy some fixed Riccati equation w0 D c0 C c1 w C c2 w2 with constant coefficients (see Theorem 3.8 for a D 1 and b D 1), hence the meromorphic functions with the property f 0 Caf 4 Cb ¤ 0 form a normal family (f ] .z/  jc0 j C jc1 j C jc2 j). This matches the following modification of Bloch’s principle, which was suggested in [13], namely to replace the above condition ‘any property P, which enforces meromorphic functions to be constant’ with the less restrictive ‘any property P, which enforces the family of meromorphic functions with this property to be normal on C’.

4.2 Applications of the Zalcman–Pang Lemma

125

4.2.2 From Finite to Infinite Order The Zalcman–Pang Lemma can also be used to prove theorems on entire and meromorphic function in two steps. In the first part the proof is given for functions of finite order or even functions having bounded spherical derivative (so-called Yosida functions), while the general case is reduced to the special one by applying the Zalcman–Pang Lemma. Of course, the results obtained this way can only be qualitative. As an example, we quote the following results due to Pang and Zalcman [139]. Theorem 4.11 Let f be a transcendental entire function all of whose zeros have multiplicity at least k  1. Then f .k/ f n .n  1/ assumes every non-zero value infinitely often. Proof We first suppose that f has bounded spherical derivative, hence f is of exponential type (j f .z/j  AeBjzj holds). If f .k/ f n assumes some value a ¤ 0 only finitely often, then f must have infinitely many zeros. For otherwise, f .z/ D P1 .z/e˛z and f .k/ .z/ D P2 .z/e˛z hold with polynomials P1 and P2 , hence also f .k/ .z/f .z/n D P3 .z/e.nC1/˛z . Entire functions of this type, however, assume every non-zero value infinitely often. We now assume that f has infinitely many zeros, each of order at least k. If f .k/ f n assumes a ¤ 0 only finitely often, then f .k/ .z/f .z/n D P4 .z/e z C a if k C n  3 resp. 12 f .z/2 D P5 .z/e z C az C a0 if k D n D 1 has infinitely many multiple zeros, which clearly is impossible. To settle the general case consider any sequence z0 ! 1 such that f ] .z0 / ! 1, and apply the Zalcman–Pang Lemma to f .z/ D f .z0 C z/. Then (some subsequence of) g .z/ D ˛ f .z0 C z C  z/ k with z ! 0;  ! 0; and ˛ D  nC1 > k tends to some non-constant entire .k/

function g of exponential type (this includes polynomials), while h D g gn tends to h D g.k/ gn . By Hurwitz’ Theorem g has zeros of order at least k, hence g 6 0 cannot be a polynomial of degree less than k. In other words, h is non-constant and assumes every non-zero value (even infinitely often if g is transcendental). Suppose h.0 / D a. By Hurwitz’ Theorem there exists some sequence  ! 0 such that h . / D a (  0 ), hence f .k/ f n assumes a at z0 C z C   ! 1. Theorem 4.12 Let k  1 and n  1 be integers, and let F be the family of holomorphic functions f on some domain D, with zeros having multiplicity at least k  1 and such that f .k/ f n omits some fixed value a ¤ 0. Then F is normal. Proof If F is not normal we obtain, similar to the proof of Theorem 4.11, some non-constant entire function g of exponential type, such that either g.k/ gn ¤ a or else g.k/gn  a. The latter is clearly impossible for transcendental functions as well as for non-constant polynomials, while the other contradicts Theorem 4.11. One more result with this kind of proof is Theorem 4.13 ([14]) Let f be non-constant meromorphic and m > `  1. Then . f m /.`/ assumes every finite non-zero value infinitely often. Remark 4.6 For m > ` C 1, see the stronger (semi-quantitative) Theorem 3.7.

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4 Normal Families

Proof If f has only finitely many zeros, then the assertion follows from Theorem 3.5 in [78] (Hayman’s alternative for f m ). We thus may assume that f has infinitely many zeros, and will start by explaining the main idea which leads to a proof if f has finite order of growth. Let h be any transcendental meromorphic function with infinitely many multiple zeros zj , and set g.z/ D z  h.z/=c .c ¤ 0 arbitrary). The zeros zj are parabolic fixed-points of g, that is, g.zj / D zj and g0 .zj / D 1 hold. Fixed points zj of this kind attract some invariant domain Lj under iteration (known as a Leau domain or parabolic basin), that is, the sequence of iterates gn D g ı gn1 tends to zj 2 @Lj as n ! 1, locally uniformly on Lj , and Lj contains some critical or asymptotic value of g .j D g. j / with g0 . j / D 0, j 2 Lj , or g.z/ ! j as z ! 1 on some path j tending to infinity in Lj ).4 Since the domains Lj are mutually disjoint, the critical and asymptotic values j are mutually distinct and g has infinitely many critical or asymptotic values. If in the present case g has only finitely many asymptotic values, then g has infinitely many critical values. If, however, the number of asymptotic values is infinite, then also the number of critical points is infinite by a deep theorem due to Bergweiler and Eremenko [14].5 If g and so h has finite order, h has infinitely many critical points and g0 . j / D c holds. Since the zeros of f are multiple zeros of h D . f m /.`1/ , this argument applies if f has finite order of growth. We now assume that f has infinite order and . f m /.`/ assumes the value c D 1, say, only finitely often. Consider the sequence of functions fn .z/ D 2kn f .2n z/ with k D `=m on 14 < jzj < 2, say. This sequence cannot be normal, since otherwise ] fn .z/  M holds on 12 < jzj < 1, say, which implies f ] .2n z/  2.k1/n M < M on 12 < jzj < 1 and f ] ./ < M on jj > 1 in contrast to our hypothesis that f has infinite order. The Zalcman–Pang Lemma with ˛ D k 2 .1; 0/ yields a non-constant meromorphic function F.z/ D lim jk fnj .zj C j z/ D lim jk 2knj f .2nj .zj C j z// .zj ! z0 ; j ! 0/ j!1

j!1

of finite order. Since . fnm /.`/ .z/  1 D . f m /.`/ .2n z/  1 has only a fixed number of zeros, namely z D 2n  with . f m /.`/ ./ D 1, and z0 is non-zero, .F m /.`/  1 is zero-free on C (note that . fnmj /.`/ .zj C j z/ tends to .F m /.`/ .z/ on C n fpoles of Fg). This contradicts the first part of the proof, provided F is transcendental. Now suppose that F is rational. Then .F m /.`/  1 ¤ 0 on C implies F.z/m D z` =`Š C c`1 z`1 C   

.z ! 1/;

which is clearly impossible since 1  ` < m.

4

For rational functions g this is a basic fact in iteration theory, see [174], Chap. 3, §5, or any other textbook in this field. Abel’s functional equation  ı g D  C 1 has a local solution (the inverse to Abel’s function being discussed in Sect. 4.1.3) on some subdomain Pj of Lj (called a petal), and analytic continuation of  leads to singularities arising from critical values of g in Lj . In the transcendental case the singularities may also come from asymptotic values in Lj . 5 Corollary 3 ([14]) Meromorphic functions of finite order having only finitely many critical points cannot have infinitely many asymptotic values.

4.2 Applications of the Zalcman–Pang Lemma

127

As a corollary we obtain the following completion of Theorem 3.8. Corollary 4.2 For every transcendental meromorphic function g, g0 C g3 has infinitely many zeros. Proof Theorem 4.13 applies to f D 1=g, m D 2, and ` D 1. Hence . f 2 /0  2, say, has infinitely many zeros, and so has g0 C g3 .

4.2.3 From Normal Families to Differential Equations Meromorphic solutions to (implicit) first-order algebraic differential equations have finite order of growth. This was proved by Gol’dberg [56] (see also Exercise 3.2), and generalised by Bergweiler [11] as follows. Theorem 4.14 Let PŒw be a differential polynomial, that is, a finite sum of terms qM .z; w/MŒw, where qM 6 0 may be any polynomial in w with rational coefficients, and each monomial M has the form MŒw D .w0 /r1 .w00 /r2    .w.m/ /rm for some r1 ; : : : ; rm , with reduced weight ıM D r1 C 2r2 C    C mrm < n. Then every meromorphic solution to the differential equation w0n D PŒw has finite order of growth. Proof Assuming that w has infinite order, the spherical derivative tends to infinity faster than any power of z on some sequence z D z0k ! 1. Then the sequence of functions w.z0k C z/ is not normal at z D 0, and by Zalcman’s Lemma there exist zk with z0k  zk ! 0 and k D 1=w] .zk / ! 0, such that (some subsequence of) wk .z/ D w.zk C k z/ tends to some non-constant meromorphic function w, and wk satisfies X   qM zk C k z; wk knıM MŒwk : w0n k D ./

Now zsk k ! 0 for every integer s, n > ıM , and wk ! w./ , locally uniformly on C n fpoles of wg, yields w0n  0 in contrast to the fact that w is non-constant. Exercise 4.8 Prove a quantitative version of Theorem 4.14 as follows. Suppose that the coefficients in qM .z; w/ do not grow faster than jzjt for some t  0. Then w] .z/ D O.jzjt / holds, hence w has order of growth at most 2t C 2. Not every attempt is successful. The most general second-order differential equations without ‘movable singularities’ other than poles have the form w00 D L.z; w/w02 C M.z; w/w0 C N.z; w/;

(4.5)

the most prominent examples are Painlevé’s equations I, II, and IV, which will be discussed in Chap. 6. We assume that Eq. (4.5), where L, M, and N are rational functions in both variables, possesses a transcendental meromorphic solution.

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4 Normal Families

Unfortunately, Theorem 4.14 does not apply to Eq. (4.5). If w has infinite order, the method of proof of Theorem 4.14 leads to some limit function w that solves w00 D .w/w02 with ` D P0 =P for some polynomial P of degree ` D degw L (  0 and P D 1 if L  0), hence w0` D const  P.w/. What can be shown is `  6 (Bergweiler [11]), but nothing else; actually deg L  4 is known. Example 4.6 In the case of equation w00 D z C 6w2 we try again. Since w has poles of order two with principal part .z  p/2 we may consider any sequence wk .z/ D k˛ w.zk Ck z/ with 1 < ˛  2 and k D 1=w] .zk /. Assuming zk k2C˛ ! 0 on some sequence zk ! 1 and wk ! w, we obtain w00 D 6w2 if ˛ D 2 and w00 D 0 if ˛ < jaj jaj ] ] 2. In this case w D az C b and 1CjazCbj 2 D w .z/  w .0/ D 1Cjbj2 hold for every z. This is only possible if b D 0 and w D az, and may be interpreted as follows: for 1 1 every  > 0, w] .z/ D O.jzj 4 / holds outside the discs jz  qj < ıjqj 4 (ı > 0 1 arbitrary) about the zeros of w. We note that w] .z/ D O.jzj 4 / without exception 5 would lead to the sharp estimate T.r; w/ D O.r 2 /. Though the latter is true, the 3 former is not. Actually, the best one can prove is w] .z/ D o.jzj 4 / (Chap. 6). Exercise 4.9 State and prove an analogous result for w00 D ˛ C zw C 2w3 .

4.2.4 From Differential Equations to Normal Families Normality of any family F of meromorphic functions on some domain D is not only equivalent to local boundedness of the family F ] (Marty’s Criterion), but also follows from the fact that the family 1=F ] of reciprocals 1=f ] is locally bounded. It suffices to consider the unit disc D. Theorem 4.15 ([59, 178]) Let F be a family of meromorphic functions on D and assume that f ] .z/   > 0 holds for every f 2 F . Then F is normal. Proof It is, of course, not always the case (although one might believe) that the Zalcman–Pang method provides the fastest way to prove normality criteria. Actually the first authors [59] successfully used this method. We will prove the explicit estimate f ] .z/ 

2= : .1  jzj/2

Since f ] is non-zero, f is locally univalent and its Schwarzian derivative (see Sect. 1.5.4) is holomorphic on D. This implies the representation f D w1 =w2 ; where w1 and w2 form a fundamental set of the linear differential equation w00 C 12 Sf .z/w D 0;

4.3 The Yosida Classes

129

normalised by W.w1 ; w2 / D 1. From f 0 D 1=w22 and f ] D .jw1 j2 Cjw2 j2 /1 it then follows that jw1 .z/j2 C jw2 .z/j2  1= holds. To prove the assertion we just remark that the Cauchy–Schwarz inequality, applied to 1 D jw1 w02  w01 w2 j, yields f ] .z/ D

1  jw01 .z/j2 C jw02 .z/j2 : jw1 .z/j2 C jw2 .z/j2

p p The standard Cauchy estimate: jw.z/j  1=  on D implies .1jzj/jw0 .z/j  1=  then gives the desired estimate.  1 C z i Exercise 4.10 Prove that f .z/ D ( > 0, f .0/ D 1) has spherical and 1z  2 C 22 2 Schwarzian derivative f ] .z/ D and S .z/ D ; f j1  z2 j j f .z/j C j f .z/j1 .1  z2 /2   on D and f ] .x/ D with f ] .z/ > f ] .˙i/ D on 1 < x < 1.6

2 cosh 2  1  x2 (Hint. log f ] obeys the Minimum Principle; j f .z/j D e =2 holds on jzj D 1, ˙Im z > 0.)

4.3 The Yosida Classes Entire and meromorphic functions having bounded spherical derivative are known as Yosida functions, in higher dimensions also called Brody curves. They were introduced by Yosida [210] (classes A and A0 ). In this section Yosida functions and their generalisations will be discussed. They occur in a natural way in the context of algebraic differential equations. Unless otherwise stated, all functions are assumed to be non-constant meromorphic on C.

4.3.1 Yosida Functions Yosida functions may be equivalently defined by the fact that the translates fh .z/ D f .z C h/

6 (W. Krauss, Über .1  jzj2 /2 den Zusammenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung, Mitt. Math. Sem. Giessen 21 (1932), 1–28), and sufficient that jSf .z/j  2 (Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. .1  jzj2 /2 55 (1949), 545–551). The present example (E. Hille, Remarks on a paper by Zeev Nehari, Bull. Amer. Math. Soc. 55 (1949), 552–553) shows that Nehari’s criterion is sharp.

6

For f to be univalent in the unit it is necessary that jSf .z/j 

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4 Normal Families

form a normal family on C. Of particular interest are the Yosida functions such that every limit function f.z/ D lim fhk .z/

(4.6)

hk !1

is non-constant (class A0 in [210]). Then these limits are also Yosida functions. Yosida’s definition was extended in [179] to a wider class of meromorphic functions as follows: f belongs to the class Y˛;ˇ with ˛ 2 R and ˇ > 1, if the family . fh /jhj>1 of functions fh .z/ D h˛ f .h C hˇ z/

(4.7)

is normal on the plane and at least one of the limit functions (4.7) is non-constant, and to the Yosida class Y0˛;ˇ , if all limit functions are non-constant; Y00;0 coincides with Yosida’s class A0 ; it contains the elliptic functions.

4.3.2 A Modified Spherical Derivative The expression f ]˛ .z/ D

jzj˛ j f 0 .z/j jzj2˛ C j f .z/j2

(4.8)

occurs quite naturally in the context of nonlinear algebraic differential equations. Exercise 4.11 Prove that jzjj˛j f ] .z/  f ]˛ .z/  jzjj˛j f ] .z/ holds for jzj  1. (Hint. First assume ˛  0, afterwards consider 1=f .) The next theorem will show how normality of the family of functions (4.7) is related to the modified spherical derivative (4.8). Theorem 4.16 The family of functions (4.7) is normal on C if and only if f ]˛ .z/ D O.jzjˇ /

(4.9)

holds. Moreover, (4.9) implies f ] .z/ D O.jzjj˛jCˇ /: ]

Proof Normality of the family . fh / at z D 0 implies that the family . fh .0//jhj>1 is bounded, that is, ]

fh .0/ D

jhj˛ˇ j f 0 .h/j jhj˛ˇ j f 0 .h/j D 2˛ D jhjˇ f ]˛ .h/  C 2˛ 2 1 C jhj j f .h/j jhj C j f .h/j2

.jhj > 1/;

4.3 The Yosida Classes

131

and thus (4.9) holds. To prove the converse we note that jhj˛ˇ j f 0 .h C hˇ z/j ˇ zj2˛ C j f .h C hˇ z/j2 h!1 jh C h D lim sup jzjˇ f ]˛ .z/ < 1

]

lim sup fh .z/ D lim sup h!1

(4.10)

z!1

.z D h C hˇ z/ holds, uniformly with respect to jzj < R. This shows that the ] continuous function .h; z/ 7! fh .z/ is bounded on jhj > 1; jzj < R, and the family . fh /jhj>1 is normal on C by Marty’s Criterion. Remark 4.7 For f 2 Y˛;ˇ any finite and non-constant limit function (4.6) is a Yosida function (f 2 Y0;0 ) with f] .z/  lim supz!1 jzjˇ f ]˛ .z/: Exercise 4.12 In order that at least one limit function is non-constant it is necessary and sufficient that the right-hand side limsup in (4.10) is positive. Prove that the condition lim supz!1 jzjˇ f ]˛ .z/ < 1 combined with lim inf sup jjˇ f ]˛ ./ > 0 z!1 jzj 0/

implies f 2 Y0˛;ˇ , and vice versa. Exercise 4.13 Given f 2 Y˛;ˇ , prove that F./ D  m f . n / (n 2 N, m 2 Z) belongs to YmC˛n;.ˇC1/n1 . Discuss the case ˛ 2 Q. Exercise 4.14 It is known [118] that the Weierstraß elliptic functions }.zI g2 ; g3 /; } 0 .zI 0; g3 /; } 2 .zI g2 ; 0/; and } 3 .zI 0; g3 / resp: } 02 .zI 0; g3 / may be written as fn .zn / with n D 2; 3; 4; 6, respectively. Prove that the corresponding meromorphic functions fn belong to Y00; 1 1 . n

4.3.3 A Modified Ahlfors–Shimizu Formula Although w D z˛ f need not be single-valued, w] is well-defined, and an elementary calculation shows that w] .z/ D f ]˛ .z/ C O.jzj1 /. Now a closer inspection of the proof of the Ahlfors–Shimizu Formula yields N.r; f / C

1 4

Z

2

0

log.1 C r2˛ j f .rei /j2 / d D

1

Z

r 1

A˛ .t; f /

dt ; t

(4.11)

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4 Normal Families

Z with A˛ .t; f / D

jzj 0 holds; equivalently, 4ı . p/ \ 4ı .q/ D ; holds for p 2 P, q 2 Q, and ı > 0 sufficiently small. In many cases the poles of solutions to algebraic differential equations are simple with residues  D . p/ and are ˇseparated from each other. Re-scaling along any sequence of poles yields wp .z/ D p˛ w. p C pˇ z/ D . p/pˇ˛ =z C    ; hence for w 2 Y˛;ˇ it is necessary that . p/  jpj˛ˇ as p ! 1, and w D O.jzj˛ / outside Pı hold. In combination, these conditions are also sufficient for f 2 Y˛;ˇ . Theorem 4.18 Let f be meromorphic and suppose that the poles p of f are ˇseparated and simple with residues  D . p/ satisfying j. p/j  jpj˛ˇ as p ! 1. Assume also that f .z/ D O.jzj˛ / holds outside Pı . Then f 2 Y˛;ˇ . Proof We first assume j. p/j D 1, hence ˛ D ˇ. The well-known Cauchy estimate rj f 0 .z/j  max j f ./j .z … Pı ; r D ı2 jzjˇ / jzjDr

yields f 0 .z/ D O.jzj2ˇ /, hence f ]ˇ .z/ D O.jzjˇ / outside Pı . The Cauchy estimate applied to f .z/ D

 C 0 C    zp

on 4ı . p/

gives 0 D O.jpjˇ /, and the Maximum Principle applied to the regular function F D f 0 C f 2  20 f yields jF.z/j D O.jzj2ˇ /, j f 0 .z/j D O.jzj2ˇ C j f .z/j2 / and f ]ˇ .z/ D O.jzjˇ / on 4ı . p/ as p ! 1:

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4 Normal Families

In the general case, g.z/ D f .z/=. p/ satisfies g.z/ D O.jzj˛.˛ˇ/ / D O.jzjˇ / on @4ı . p/, hence jg0 .z/j D O.jzj2ˇ Cjg.z/j2 / holds on 4ı . p/. In terms of f this means j f 0 .z/j D O.j. p/jjzj2ˇ C j f .z/j2 j. p/j1 / Cj f .z/j2 / D j. p/j1 O.jzj2ˇC2.˛ˇ/  ˇ˛ 2˛ 2 D .jzj C j f .z/j / O jzj

on 4ı . p/;

hence f ]˛ .z/ D O.jzjˇ / holds on Pı in any case, while f .z/ D O.jzj˛ / and the Cauchy estimate imply f 0 .z/ D O.jzj˛Cˇ / and f ]˛ .z/ D O.jzjˇ / outside Pı . Exercise 4.15 Suppose that the poles and zeros of f 2 Y˛;ˇ are ˇ-separated from Q the zeros and poles of fQ 2 Y˛;ˇ Q , respectively. Prove that f f 2 Y˛C˛;ˇ Q .

4.3.5 The Class Y0˛;ˇ Much more is known about the functions f 2 Y0˛;ˇ , ˛ 2 R, ˇ > 1. We just list the most important properties. For detailed proofs, see [179]. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

supjzj 1 arbitrary) contains poles and zeros. There exists an M > 0 such that each ‘local unit disc’ 41 .z0 / contains at most M poles and zeros; in particular, the multiplicities are bounded. The poles of f are ˇ-separated from the zeros. f .z/  jzj˛ holds on C n .Pı [ Qı /. f ] .z/ D O.jzjˇj˛j / and f 0 .z/=f .z/ D O.jzjˇ / hold on C n .Pı [ Qı /. f 0 2 Y0˛Cˇ;ˇ if and only if the poles are ˇ-separated. The value distribution of f takes place in Pı if ˛ < 0, and in Qı if ˛ > 0. Y00;0 contains every limit f D limhk !1 fhk for f 2 Y0˛;ˇ : Y00;0 is universal.

Most of the results also hold in Y0˛;1 ; 2. has to be replaced with T.r; f /  log2 r. The interested reader is also referred to [43, 53–55, 112, 206]. 1 2 Example 4.7 ([7]) The differential equation w02 D 1z 2 w.4w  g2 /; where g2 is chosen such that is one of the periods of the corresponding P-function (with } 02 D 4} 3 g2 }), has a solution such that w.sin / D }./; w and w.k/ belong to Y00;1 and Y0k;1 , respectively. The limit functions w D limhk !1 whk are non-constant and 1 2 satisfy w02 D  .zC1/ 2 w.4w  g2 /; hence are given by w.z/ D }.i log.z C 1/ C c/

with essential singularity at z D 1.

4.3 The Yosida Classes

135

Remark 4.9 One may also consider meromorphic functions f on sectors S W j arg z 0 j < . Restricting the sequences .hk / to j arg z  0 j <   ı, ı > 0 arbitrary, leads to the classes Y˛;ˇ .S/ and Y0˛;ˇ .S/. The solutions to several important classes of differential equations belong to appropriate classes Y˛;ˇ . It is an open problem to determine the solutions w that belong to some hybrid class between Y0˛;ˇ and Y˛;ˇ as follows: the plane is divided into finitely many alternating sectors S and ˙ , such that w 2 Y0˛;ˇ .S / while w.z/z˛ has a constant limit as z ! 1 on every closed sub-sector of ˙ , and thus has an asymptotic expansion. Sometimes the sectors S degenerate to smaller domains like j arg z j < .jzj/ where .r/ ! 0 as r ! 1, and even to rays arg z D  ; more in Chaps. 5 and 6.

4.3.6 Yosida Classes and Riccati Differential Equations The theory of generalised Yosida functions is most effective in the context of algebraic differential equations. This will be demonstrated in the case of the Riccati differential equations w0 D P.z/  w2 ; where, for the present, P.z/ D zn C an1 zn1 C    C a0 is any polynomial. From jw0 j  jP.z/j C jwj2 D O.jzjn C jwj2 / it follows that w is a candidate for Y n2 ; n2 . Suppose whk ! w 6 1. Then w0hk ! w0 , locally uniformly on C n fpoles of wg, and w satisfies w0 D 1  w2 ; hence the possible finite limit functions are w  ˙1 and w D coth.z C z0 /; the limit w  1  n2 2 does not occur since vhk D 1=whk satisfies vh0 k D 1  hn k P.hk C hk z/vhk , which n n n n excludes vhk ! 0. Thus w 2 Y 2 ; 2 and 1=w 2 Y 2 ; 2 . Exercise 4.16 Prove that to  > 0 there exists an r0 > 0 such that each disc 44 . p/ W n jz  pj < 4jpj 2 about some pole p of w with jpj > r0 contains exactly three poles n n n p, p C . i C 1 /p 2 ; and p  . i C 2 /p 2 , and two zeros p C . 2 i C 3 /p 2 and n p  . 2 i C 4 /p 2 , with jj j < : The same is true if the terms ‘zero’ and ‘pole’ are permuted. In particular, the poles and zeros of w are n2 -separated from each other. (Hint. Re-scale about any sequence . pn / of poles and consider the poles and zeros of coth z; note that < 4 < 32 .) Henceforth we assume that ı > 0 is chosen so small that the discs n

4ı . p/ W jz  pj < ıjpj 2

. p 2 P/ and 4ı .q/ .q 2 Q/

are mutually disjoint; again P and Q denote the set of non-zero poles and zeros of w, respectively.

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4 Normal Families

Exercise 4.17 Let the closed simple curve r be obtained by modifying the circle Cr W jzj D r as follows: r D Cr , if Cr does not intersect any disc 4ı . p/. If, however, Cr \ 4ı . p/ ¤ ; for some pole p, replace this sub-arc of Cr with the part of @4ı . p/ outside Cr if jpj  r, and inside Cr otherwise. Prove that r has length n O.r/ and deduce from resp w D 1 and jwj D O.jzj 2 / on r that n

n.r; w/ D O.r 2 C1 /

n

and T.r; w/ D O.r 2 C1 /:

Note that for w 2 Y n2 ; n2 , T.r; w/ D O.rnC2 / had to be expected! Remark 4.10 The construction of r works in a much more general context: Let .4j / be any sequence of mutually disjoint discs with centres zj ! 1. Then a simple closed curve r with winding number 1 about points zj with jzj j  r and 0 if jzj j > r can be constructed which has length O.r/ and avoids the discs 4j . Exercise 4.18 Deduce from Exercise 4.16 that each pole of w of sufficiently large modulus belongs to a unique sequence . pk / of poles such that  n2

pkC1 D pk ˙ . i C o.1//pk

(4.13)

holds with fixed sign ˙. Sequences of this kind are called strings or, more precisely, .˙ i; n2 /-strings. n

The recursion (4.13) can easily be resolved. Set qk D pk2

C1

to obtain

n 2 C1 D qk ˙ .. C 1/ i C o.1//; qkC1 D qk .1 ˙ . i C o.1//q1 k / 2 n

2

hence qk D ˙Œ. n2 C 1/ i C o.1/k and pk D Œ˙.. n2 C 1/ i C o.1//k nC2 :

holds for some . Exercise 4.19 (Continued) Prove that arg pk ! O D 2C1 nC2 O The rays arg z D  are called Stokes rays; asymptotically they attract all poles. Exercise 4.20 Prove that each string has counting function r 2 C1 n C o.r 2 C1 /: n . 2 C 1/

n

n.r; . pk // D

In combination with Exercise 4.17 deduce that there are only finitely many strings, n and show that on each sector Sı W 1 C ı < arg z <   ı; w.z/z 2 tends to some  D  .w/ 2 f1; 1g as z ! 1. (Hint. Re-scale along any sequence in Sı and look for poles of the limit functions w D limhk !1 whk .) P P   Actually the number of strings is nC1 is even, and m D0 .1/  if n Dm .1/  n if n D 2m  1 is odd (due to the discontinuity of z 2 along arg z D ); more on Riccati equations and the Yosida class Y n2 ; n2 in Chap. 5.

Chapter 5

Algebraic Differential Equations

One of the most difficult problems in the theory of Algebraic Differential Equations is to decide whether or not the solutions are meromorphic in the plane. In case this question has been answered satisfactorily, which by experience requires particular strategies adapted to the equations under consideration, there remain several major problems to be solved: to determine the Nevanlinna functions, the distribution of zeros and poles, zero- and pole-free regions, asymptotic expansions on polefree regions, and solutions deviating from the ‘generic’ case. This program will be pursued in the subsequent sections on linear, Riccati, and implicit first-order differential equations.

5.1 Linear Differential Equations Throughout this section we will consider linear homogeneous differential equations LŒw D w.n/ C an1 .z/w.n1/ C    C a1 .z/w0 C a0 .z/w D 0

(5.1)

with entire coefficients. The solutions are entire functions and the Wronskian determinant W.w1 ; : : :R; wn / of any fundamental set w1 ; : : : ; wn is an exponential function, W.z/ D e an1 .z/ dz . The operator L may be rediscovered as a formal determinant, LŒw D

W.w1 ; : : : ; wn ; w/ : W.w1 ; : : : ; wn /

© Springer International Publishing AG 2017 N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, DOI 10.1007/978-3-319-59800-0_5

(5.2)

137

138

5 Algebraic Differential Equations

5.1.1 The Order of Growth By Theorem 1.7 all solutions have finite order of growth if the coefficients in L are polynomials. In the particular case of constant coefficients the order of growth is at most one, this following from the well-known form of the distinguished fundamental set, which consists of functions zj ez . The converse is also true. Theorem 5.1 Suppose every solution to equation (5.1) has finite order resp. order of growth at most one. Then the coefficients in L are polynomials resp. constants. Proof We will prove by induction on n that if (5.1) has meromorphic coefficients and n linearly independent meromorphic solutions of finite order of growth resp. order at most one, then m.r; a / D O.log r/ resp. m.r; a / D o.log r/ .0   < n/ holds. This is obvious if n D 1, where a0 D w0 =w. To reduce the order of the equation we set w D w1 y. The new equation for u D y0 is then u.n1/ C bn2 .z/u.n2/ C    C b1 .z/u0 C b0 .z/u D 0; n X  j  . j/ =w1 .1    n  1/: b1 D a C  aj w1 jDC1

Then m.r; b1 / D O.log r/ holds by hypothesis, and assuming m.r; aj / D O.log r/ for  < j  n yields m.r; a /  m.r; b1 / C

n X 

  . j/ m.r; aj / C m r; w1 =w1 C O.1/ D O.log r/:

jDC1

In the second case (order  1) one has just to replace O.log r/ with o.log r/. Theorem 5.1 is a special case of the following Theorem of Frei [51]. Theorem 5.2 Suppose that the coefficients ajC1 ; : : : ; an1 are polynomials, while aj is transcendental. Then there are at most j linearly independent solutions of finite order. Exercise 5.1 Frei’s Theorem may be proved by induction on n: adapt the above proof to show that if m.r; a / D O.log r/ holds for  D k; : : : ; n  1, and if k C 1 linearly independent solutions w1 ; : : : ; wkC1 have finite order of growth, then also m.r; ak1 / D O.log r/ holds. In any case prove that m.r; a / D O.log max rT.r; w // 1n

holds outside some exceptional set of finite measure. To determine the possible orders of growth there are two methods in common. The first one employs the Wiman–Valiron Theory, see Sect. 1.5.3. Suppose aj .z/ D

5.1 Linear Differential Equations

139

Aj z˛j C    as z ! 1, and let w be any entire transcendental solution. Then the central index .r/ of w approximately solves the reduced equation yn C

n1 X

Aj x˛j Cnj yj D 0 . y  .r/; x D zr /

(5.3)

jD0

with zr any point on jzj D r such that M.r; w/ D jw.zr /j, and r 62 F, where F has finite logarithmic measure. The order of growth of w coincides with one of the non-positive (actually negative) slopes of the associated Newton–Puiseux polygon. Exercise 5.2 Prove that the solutions to w000 C zw0  .z C 1/w D 0 have order of growth either 1 or 32 . One solution is w1 D ez of order 1. Derive the differential equation for y D u0 with w D uez to prove that any solution w 6 cez has order of growth 32 .

5.1.2 Asymptotic Expansions The second far-reaching method is based on asymptotic integration. Linear equations (5.1) with polynomial (or rational) coefficients may be transformed into linear systems zq y0 D A.z/y which were discussed in Sect. 1.4. The procedure is not unique, in particular, the rank q C 1 depends on the transformation, and it may be difficult to determine the smallest possible integer q. If Theorem 1.9 is translated into the context of equation (5.1) we obtain: Theorem 5.3 Equation (5.1) has a formal fundamental system ı

w .z/ D eP .z

1=p /

z H .z; log z/

.1    n/;

(5.4)

where – – – –

p is a positive integer, P is a polynomial with deg P D d and P .0/ D 0,  is some complex number, and H .z; t/ is a polynomial in t, its coefficients have asymptotic expansions in z1=p .

Given any sector S W j arg zj < h with sufficiently small central angle, there exists ı some fundamental system w1 ; : : : ; wn such that w is represented by w on S.

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5 Algebraic Differential Equations

Remark 5.1 In this form Theorem 5.3 goes back to Sternberg [187]. The first ı 1=p approximation w .z/  eP.z / z .log z/ with P.z1=p / D bz% C    (we omit the index ) may be differentiated arbitrarily often: dk ı 1=p w.z/  .b/k eP.z / zCkk .log z/ : k dz The triples .P ;  ;  / are mutually distinct if different branches of z1=p are well taken into account; in particular, logarithmic terms may occur only if .P ;  / D .P ;  / for some  ¤ . The similarity to linear differential equations with constant coefficients is apparent: solutions zj e!z (j  1) occur if and only if ! is a multiple characteristic root. It is obvious that the orders of growth of transcendental solutions w are among the positive numbers d =p, that is T.r; w/ rd =p holds for some : In particular, there always exist solutions of maximal order of growth %max D max1n d =p: Exercise 5.3 Prove that the leading terms b d of the polynomials P may be determined as follows: the simplified algebraic equation (5.3) has n branches y .x/ D c x  C    as x ! 1. Then d D p  and b D pc =d if

 > 0. Example 5.1 To determine the essential terms, namely the number p and the leading terms of the polynomials P (or more) it is sometimes more convenient to set y D w0 =w, derive a differential equation for y, and determine the first terms in the asymptotic expansions. For w000 C z2 w D 0 this leads to y00 C 3yy0 C y3 C z2 D 0: To determine the leading term we plug in y  kz˛ , y0  ˛kz˛1 , y00  ˛.˛  1/kz˛2 to obtain k3 z3˛ C z2 D O.jzjmaxf2˛1;˛2g /, hence 3˛ D 2, p D 3, and k D 1, say. Given any sector of central angle 23  we obtain three mutually distinct solutions ı

y  z2=3  23 z1 C    ; hence three linearly independent solutions w .z/ D 3 5=3 e 5 z z2=3 .1 C O.jzj1 //, one for each branch of z1=3 ; logarithmic terms do not occur, and every non-trivial solution has maximal order of growth %max D 53 .

Exercise 5.4 ([3]) Employ Wiman–Valiron theory to compute the possible orders of the solutions to w000 C z2 w00 C zw0 C w D 0; and determine the principal terms of ı ı ı the solutions w (see Theorem 5.3). Prove that the solution represented by w2  w3 on j arg zj < ı has zeros  ek (k ! 1). ı

Solution w1  ez

3 =3

z3 .1 C

10 3 z 3

ı

C    /, w2;3  z˙i .1 

7˙9i 3 z 39

C    /.

5.1.3 Sub-normal Solutions Asymptotic integration and Wiman–Valiron theory applied to Eq. (5.7) provide a finite set of possible orders of growth; % D 0 is reserved for polynomial solutions

5.1 Linear Differential Equations

141

(% D 0 means .r/ D O.1/). Let S be any sufficiently small sector and let w , ı  2 J, denote the solution that is represented on S by w with maximal degree deg P D p%max : Obviously, these functions are linearly independent, hence every fundamental system contains at least as many solutions of maximal order of growth as are formal solutions with maximal degree of P . Nothing more can be said in general. The non-trivial solutions of order of growth less than %max are called sub-normal, in combination with the trivial solution they form a linear subspace. Necessary for the existence of sub-normal solutions, but not sufficient, is that the Newton–Puiseux diagram has at least two non-positive slopes. For example, this is the case in Exercise 5.2. Example 5.2 The existence of sub-normal solutions is in some sense exceptional. The Newton–Puiseux method applied to equation w000 C zw0  .z C 1 C /w D 0 indicates linearly independent solutions having orders of growth 1; 32 ; 32 . The method of Example 5.1 yields y00 C 3yy0 C y3 C zy  .z C 1 C / D 0 having solutions 1 1 y1  1 C z1  3z2 C    and y2;3  iz 2  12 C 38 iz 2  . 2 C 34 /z1 C    , one ı

1

for each branch of z 2 , hence w1 D ez z .1 C O.jzj1 // and ı

w2;3 D exp

2

3 iz

3 2

1  3 1  12 z C 34 iz 2 z 2  4 .1 C O.jzj 2 //:

A simple compactness argument shows that solutions of order of growth one (if any) are given by w D Q .z/ez with some polynomial Q of degree . An elementary computation shows that this happens for every  2 N0 with Q0 .z/ D 1, Q1 .z/ D z C 3, Q2 .z/ D z2 C 6z C 12; etc. In any other case the non-trivial solutions have maximal order of growth %max D 32 .

5.1.4 The Phragmén–Lindelöf Indicator Let w be any solution to (5.1) of order of growth % > 0. Then hw ./ D lim sup r!1

log jw.rei /j r%

(5.5)

is called the Phragmén–Lindelöf indicator or just indicator of w. Given any angle ı O w is represented on S W j arg z  j O < ı by some linear combination Pn c w ,  D1 O with coefficients c D c .w; /. If the polynomials P of degree p% have mutually distinct leading coefficients b , an elementary computation yields hw ./ D maxfRe b ei% W c ¤ 0; deg P D p%g O < ı (with the convention that the maximum of the empty set is zero). Then on j  j hw ./ is continuous and piecewise continuously differentiable. The discontinuities

142

5 Algebraic Differential Equations

Oj of h0w ./ are called Stokes directions (of w of order %); they occur if the maximum is attained for  D  on 0 <   O < ı 0  ı and for  D  ¤  on 0 < O   < ı 0  ı. If O is not a Stokes direction, log jw.z/j D hw ./r% C O.r% /

.0 < < 1=p/

(5.6)

holds as z D rei ! 1 on S. We mention without proof that this remains true if S O is a Stokes direction and S is replaced with S n j fz W jz  zj j < rj g, such that P 1 jzj j 0 satisfying (5.6) with error term o.r% / outside some so-called C0 -set j fz W jz  zj j < rj g with 1 P jzj j 0. In particular, the deficiency ı.0; w/ is either less than one or else 0 is a Borel or even Picard exceptional value of w. Prove also that the counting function of zeros outside the union of sectors j arg z  Oj j < ı is O.r% /. Remark 5.3 If the polynomials P are not mutually distinct one has to estimate from P 1=p below functions f represented by finite sums eQ .z / z H .z; log z/ on sectors S, but outside small neighbourhoods of the zeros; f behaves like an entire function of order of growth at most %  and may have infinitely many zeros with counting function O.r% /. Nevertheless, the statements in Exercise 5.5 also remain valid in

5.1 Linear Differential Equations

143

this case. Bank [3] seems to have been the first to notice that there exist solutions having infinitely many zeros outside arbitrarily small sectors j arg z  Oj j < about the Stokes directions, a phenomenon that does not occur if n D 2. Example 5.3 Exponential polynomials w.z/ D

m X

p .z/e! z

D1

(p 6 0 polynomials, ! mutually distinct) satisfy some Eq. (5.1) with constant coefficients. Prove that w has indicator hw ./ D max1m Re ! ei : We mention R 2 1 without proof that 2 0 hw ./ d is the circumference of the so-called indicator diagram I, the convex hull of the points !N 1 ; : : : ; !N m (see [140], Vol. II, p.163). Prove that ı.0; w/ D 0 if and only if 0 2 I.

5.1.6 Exceptional Fundamental Sets We will now discuss the question of whether there may exist fundamental systems w1 ; : : : ; wn that are exceptional in the sense that E D w1    wn has fewer zeros than expected (N.r; 1=E/ D o.r%max /). The case n D 2 is very easy to deal with on one hand, but is not very instructive on the other. This is true in many respects, most phenomena occur in the case of n  3 only. Exercise 5.6 For w00 C a0 .z/w D 0 (a0 6 0), E D w1 w2 , and W.w1 ; w2 / D c ¤ 0 derive the differential equation 2EE00  E02 C 4a0 .z/E2 D c2 ; and deduce m.r; 1=E/ D O.log r/. To prove that either N.r; 1=E/ maxfT.r; w1 /; T.r; w2 /g or else a0 .z/  2 and w1;2 .z/ D c1;2 e˙z holds, compare the central index w .r/ of any solution w 6 0 with the central index E .r/ if E 6 const.: prove E .r/  2w .r/. If, however, E is a non-zero constant, then a0 .z/ is also a non-zero constant. The goal of this section is to transfer this alternative to the case n > 2. Theorem 5.4 Let w1 ; : : : ; wn be some fundamental set to Eq. (5.7) such that ı.0; w / D 1 holds for every w of maximal order of growth %max : Then L has constant coefficients. 2

Exercise 5.7 The differential equation w000 4z2 w0 12zw D 0 is solved by w1 D ez 2 and w2 D ez . Expand W.w1 ; w2 ; w3 / D c to obtain the linear inhomogeneous equation 4zw003 C 4w03 C 16z3 w3 D c; for c ¤ 0 deduce m.r; 1=w3 / D O.log r/ and .r/  2r2 (central index of w3 ). By Theorem 5.4 the zeros of w3 have exponent of convergence 2. The proof of Theorem 5.4 requires some preparation. We first note that by Exercise 5.5, zero is a Borel (or Picard) exceptional value for every w of maximal

144

5 Algebraic Differential Equations q

order q D %max I q is a positive integer, and every w has the form w .z/ D eb z g .z/, where g is an entire function of order of growth less than q (or a polynomial), with b D 0 if %.w / < q. The functions w with fixed b D b span the subspace Vb . q

Lemma 5.1 Let w.z/ D g.z/ebz with b ¤ 0 and %.g/ < q be any solution to Eq. (5.1). Given any sufficiently small sector S, let wO 1 ; : : : ; wO n denote the solutions ı ı that are represented on S by the formal solutions w1 ; : : : ; wn , and assume P .z1=p / D q bz C    exactly for  D 1; : : : ; m. Then w; wO 1 ; : : : ; wO m are linearly dependent. Proof Suppose that w.z/ 

m P D1

cwO  .z/ D

n P

DmC1 iq

c w O  .z/ 6 0 holds on SI then the

left-hand side has indicator h./ D Re be (of order q), while the right-hand side has not. This proves c D 0 for m <   n. Lemma 5.2 Suppose that Vb is spanned by w1 ; : : : ; wm . Then W.w1 ; : : : ; wm / has only finitely many zeros, and the operator MŒw D

W.w1 ; : : : ; wm ; w/ D w.m/ C sm1 .z/w.m1/ C    C s0 .z/w W.w1 ; : : : ; wm /

has rational coefficients s .z/ D

 m .bqzq1 /m C    

at z D 1:

Proof The coefficients s of M have only finitely many poles if and only if the Wronskian W.w1 ; : : : ; wm / has only finitely many zeros, and then, indeed, are rational functions, this following from the proof of Theorem 5.1 and Exercise 5.1, which show that m.r; s / D O.log r/. Since the algebraic equation ym C sm1 .x/ym1 C    C s0 .x/ D 0 has m solutions y .x/ D bqxq1 C    with the same leading term, hence has the simplified equation . y  bqxq1 /m D 0; the shape of the rational functions s follows from the binomial theorem. To prove that W.w1 ; : : : ; wm / has only finitely many zeros we consider any sufficiently small sector S and denote by wO 1 ; : : : ; wO m the solutions that occur in Lemma 5.1. Then w1 ; : : : ; wm and wO 1 ; : : : ; wO m span the same linear space, and this implies W.w1 ; : : : ; wm / D cW.wO 1 ; : : : ; wO m / on S for some c D c.S/ ¤ 0. To compute W.wO 1 ; : : : ; w O m / we will have a closer look 1=p Œ0 at a single solution wO  ; it is represented on S by eP .z / z H .z; log z/ with .k/ P .z1=p / D bzq C Q .z1=p / (deg Q < deg P ), and the k th derivative wO  has the 1=p Œk representation eP .z / z Ck.q1/ H .z; log z/ (at each step the factor qbzq1 occurs Œk in addition; the constant qb is incorporated into H ). Thus W.wO 1 ; : : : ; wO m / has P 1=p m 1 the representation eP.z / z H.z; log z/ with  D D1  C 2 .q  1/m.m  1/ Pm and P D D1 P ; H.z; t/ is some polynomial in t with coefficients having asymptotic expansions in z1=p . On S, H.z; log z/ represents some analytic function h.z/ D c.1 C o.1//.log z/K zN=p (K; N integers, z ! 1 on S), which obviously

5.1 Linear Differential Equations

145

has only finitely many zeros. A simple compactness argument then shows that W.w1 ; : : : ; wm / has only finitely many zeros on the plane. Proof of Theorem 5.4 Let Vb be the vector space spanned by the solutions w .z/ D q ebz g .z/ (1    m) with q D %max , b ¤ 0, and %.g / < q: The Wronskian W.w1 ; : : : ; wn / is constant and W.MŒwmC1 ; : : : ; MŒwn / D

W.w1 ; : : : ; wn / W.w1 ; : : : ; wm /

(5.8)

holds by Exercise 1.13. The right-hand side of (5.8) has no zeros and at most finitely many poles by Lemma 5.2. We will show that the left-hand Wronskian has at least m.n  m/.q  1/ zeros, which leads to the following alternative: – m D n, hence W.w1 ; : : : ; wn / D enbz C is non-constant. – q D %max D 1, hence L has constant coefficients. q

The solution space V of LŒw D 0 is the direct sum Vb ˚ Vˇ1 ˚    ˚ Vˇk , where q Vˇ is spanned by the functions w .z/ D eb z g .z/ with b D ˇ ¤ b (ˇ ¤ b mutually distinct; ˇ D 0 is admitted, V0 collects the sub-normal solutions). Given any sufficiently small sector S, Vˇ is also spanned by the functions represented on q 1=p S by wO ` with b` D ˇ . Let u.z/ D eˇ z CQ.z / z H.z; log z/ denote any solution 1 . j/ with ˇ ¤ 0 and p deg Q < q. From u .z/ D .ˇ qzq1 C o.jzjq1 //j u.z/ and   sj .z/ D mj .bqzq1 C o.jzjq1 //mj it easily follows that m X  m q1 MŒu D C o.jzjq1 //mj .ˇ qzq1 C o.jzjq1 //j u.z/ j .bqz jD0

D .ˇ  b C o.1//m qm z.q1/m u.z/: Similarly we obtain

d dz MŒu

D .ˇ  b C o.1//m qm z.q1/m u0 .z/, etc., hence

W.MŒwmC1 ; : : : ; MŒwn / D .C C o.1//zm.nm/.q1/W.umC1 ; : : : ; un / Q with C D kjD1 qm .ˇ  b/m ¤ 0. Since W.MŒwmC1 ; : : : ; MŒwn / has at most finitely many poles and W.umC1 ; : : : ; un / has no poles and only finitely many zeros on S, a compactness argument shows that W.MŒwmC1 ; : : : ; MŒwn / D r.z/eQ.z/ holds, where Q is some polynomial and r is a rational function which has a pole of order at least m.n  m/.q  1/ at infinity, hence also has at least m.n  m/.q  1/ finite zeros.

146

5 Algebraic Differential Equations

5.1.7 Fundamental Sets with Zeros Along the Real Axis Let g be an entire function with Nevanlinna characteristic T.r; g/ r% . The zeros of g are said to be asymptotically distributed along the rays arg z D Oj

.O1 < O2 <    < Ok < OkC1 D O1 C 2/

if for every > 0 the counting function of zeros on the union of sectors Oj C < arg z < OjC1  is o.r% /. For entire functions of completely regular growth the distribution of zeros has an effect on the order of growth as follows. Theorem 5.5 Let g be an entire function of completely regular growth of order %, and suppose that its zeros are asymptotically distributed along the rays arg z D Oj . Then ı.0; g/ D 0 implies min .OjC1  Oj /  =%: 1jk

Proof The indicator of g is piecewise trigonometric, that is, hg ./ D Re cj ei% holds R 2 R 2  on Oj < arg z < OjC1 , and ı.0; g/ D 0 yields 0 hC g ./ d > 0 hg ./ d D 0: Thus hg is never negative, and there is at least one interval .Oj ; OjC1 / where hg ./, hence cos.% C j / is positive. This implies %.OjC1  Oj /  . Remark 5.4 In [33] a similar result is proved: Suppose the zeros of some entire function g (not necessarily of completely regular growth) are distributed along the rays arg z D Oj and % is sufficiently large (depending on the geometry of the rays). Then ı.0; g/ > 0. In other words, % cannot be arbitrarily large if ı.0; g/ D 0. Theorem 5.6 Let w1 ; : : : ; wn be any fundamental set of Eq. (5.7). Then each of the conditions below implies that the coefficients of L are constants. – T.r; w1    wn / D o.r%max /; – the zeros of the solutions w of order %max are asymptotically distributed along the real axis. Proof Set E D w1    wn . In the first case, ı.0; w / D 1 holds for every solution w of maximal order, this following from N.r; 1=w /  N.r; 1=E/ D o.r%max /. By Theorem 5.4 the coefficients of L are constants. In the second case the entire function E satisfies  1  W  m r; D m r; C O.1/ D O.log r/ E w1    wn (the Wronskian W D W.w1 ; : : : ; wn / is constant), and E is either a polynomial and Theorem 5.4 applies, or else is a transcendental function of order of growth

5.2 Riccati Equations

147

%.E/  1 by Theorem 5.5. In case of %max  1, Theorem 5.1 applies. If, however, %max > 1 holds, then Theorem 5.4 applies since N.r; 1=w /  N.r; 1=E/ D O.r/ D o.r%max / holds for every solution w . Exercise 5.8 We note that Theorem 5.6 fails to hold if in (5.1) arbitrary rational coefficients are admitted. To give an example, let !1 ; : : : ; !n be non-zero and 2 mutually distinct complex constants, and set w .z/ D e! z : Prove that the Wronskian W.w1 ; : : : ; wn / has only finitely many zeros, hence the functions w form a zero-free fundamental set of some Eq. (5.1) with non-constant rational coefficients. Remark 5.5 Theorem 5.4 was proved by Frank [45] under the hypothesis that E D w1    wn has only finitely many zeros, that is, if 0 is a Picard exceptional value for every w , and by Brüggemann [18] under the weaker assumption that 0 is a Borel exceptional value for each w , thereby solving a problem posed by Frank and Wittich. Brüggemann [19] also settled a conjecture of Hellerstein and Rossi (see [17]) by proving the second part of Theorem 5.6, assuming that the non-real zeros of E have exponent of convergence less than %max . In the present form, Theorem 5.6 is due to the author [171–173]. There is an extensive literature on second-order equations, where naturally much more is known; we mention [3– 5, 38–40, 45, 50, 67, 70, 74, 81, 82]. In the next section we will consider second-order equations via a new approach to Riccati equations.

5.2 Riccati Equations The basic facts concerning the value distribution of the solutions to Riccati differential equations w0 D a.z/ C b.z/w C c.z/w2

.c 6 0/

(5.9)

with polynomial coefficients are well understood due to the pioneering work of Wittich (see his book [202], Chapter V, p. 73–80). The solutions are meromorphic in the complex plane, and every non-rational solution has Nevanlinna characteristic n T.r; w/ r 2 C1 ; the non-negative integer n can easily be computed from the coefficients a; b, and c. Equation (5.9) is closely related to some second-order linear differential equation (transformation c.z/w D u0 =u), hence it would be possible to take the detour via this also well-understood linear equation. We will, however, use a new approach which was initiated in Sect. 4.3 and also applies to higher-order equations and systems.

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5 Algebraic Differential Equations

5.2.1 A Canonical Form The change of variables w1 D cw C 12 b C 12 .c0 =c/ transforms (5.9) into w01 D A.z/  w21 ; A.z/ D 14 b2  ac  12 b0 C 34 .c0 =c/2 C 12 b.c0 =c/  12 .c00 =c/ D an zn C an1 zn1 C    as z ! 1: Replacing the variables z and w1 with zQ D z= and w.Q Q z/ D w1 .z/ (nC2 an D 1), we eventually obtain the canonical form, which, of course, will be written Q -free as w0 D P.z/  w2 ;

(5.10)

P.z/ D zn C cn1 zn1 C    as z ! 1:

This equation was the object in Exercises 4.16–4.20, where P was assumed to be a polynomial, while now P is a rational function. This, however, is not really restrictive since we are interested in asymptotic results. Of course, we will only consider solutions that are meromorphic in the plane. This is the case if the coefficients in (5.9) are polynomials. The method applies if n  1, though n D 1 never occurs in the polynomial case. Exercise 5.9 Prove that w0 D 2z1 .z C w C w2 / has the meromorphic solution w D p p 3 2 z tan z of order of growth 12 . Deduce the canonical form w0 D z1  16 z  w2 : Note, however, that just one solution is single-valued. Exercise 5.10 The degrees of the coefficients a; b; c in (5.9) may be arbitrarily large compared with n. Prove that the Riccati equation w0 D z2m1 C 2zm1  .m  1/zm2 C 2.zm C 1/w C zw2 has the canonical form w0 D 1 C z1 C 34 z2  w2 . The main result on the solutions to the canonical equations (5.10) may be stated as follows. Theorem 5.7 Let w be any transcendental meromorphic solution to the canonical Riccati equation (5.10). Then – w belongs to the Yosida class Y n2 ; n2 ; – the poles of w are distributed in .˙i; n2 /-strings; each string . pk / n

– satisfies the iterative scheme pkC1 D pk ˙ .i C o.1//pk 2 ; – is asymptotic to some ray arg z D O D 2C1 nC2 ; at most one for each ; n C1 2 r ; – has counting function n.r; . pk //  n . 2 C 1/

5.2 Riccati Equations

149

– on ˙ W O1 < arg z < O , w has an asymptotic expansion P n 1 2 .nj/ w  z 2 C 1 . D .w; / 2 f1; 1g/I jD1 cj . /z n r 2 C1 n C o.r 2 C1 /; – w has Nevanlinna characteristic T.r; w/ D `.w/ n 2 . 2 C 1/  where `.w/ denotes the number of strings .1  `.w/  n C 2/. Remark 5.6 The rays  W arg z D O and sectors ˙ W O1 < arg z < O are called Stokes rays and sectors, respectively. The coefficients cj in the asymptotic expansions depend only on , and c2j1 vanishes if n is even. The reader who is acquainted with the field of Complex Dynamics (see, for example, [174]) will notice the close relationship of the iterative scheme with the iteration of the rational function R˙ .z/ D z ˙ izm (assuming n2 D m); R˙ has a parabolic fixed-point at 1 and m C 1 attracting petals around the rays arg.˙izmC1 / D 0 mod 2; convergence to infinity takes place along the rays arg z D O . Proof Most of the above statements may be deduced from Exercises 4.16–4.20; they are based solely on the elementary estimate jw0 .z/j D O.jzjn C jw.z/j2 /: It remains to prove the existence of asymptotic expansions and the fact that there is at most one string along each Stokes ray arg z D O . Re-scaling along any sequence hk ! 1 with O C ı < arg z < OC1  ı yields n n solutions w D limhk !1 whk (wh D h 2 w.h C h 2 z/) to w0 D 1  w2 without n poles, hence w D ˙1. Since the cluster set1 of z 2 w.z/ as z ! 1 on each sector n O C ı < arg z < OC1  ı is connected, it follows that z 2 w.z/ !  2 f1; 1g. In other words, n

n

w.z/ D  z 2 C o.jzj 2 / D

0 .z/

n

C o.jzj 2 /

holds as z ! 1, uniformly on O C ı < arg z < OC1  ı. To prove that w has an asymptotic expansion we could apply Theorem 1.11. It is, however, more instructive to give an ad hoc proof. Assuming n

w.z/ D  z 2 C

k X

cj z

nj 2

C o.jzj

nk 2

/D

k .z/

C o.jzj

nk 2

/

(5.11)

jD1

for some k  0, hence w2 C P.z/ D k0 .z/ C o.jzj 2 1 /; we may compare w with P n Œk nj 0 2 to the algebraic equation y2 CP.z/ D the solution y D  z 2 C 1 jD1 cj z k .z/: nk

1

Let f be a bounded holomorphic function on some unbounded domain D. By definition, the cluster set of f as z ! 1 on D is the set of all limits limn!1 f .zn / with zn ! 1 in D. The cluster set is always compact, and connected if D is locally connected at infinity. A sufficient condition for the latter is that any two points a; b 2 D may be joined by a curve in D \ fz W jzj  minfjaj; jbjgg. For example, this is true for sectors, and also for sectors with mutually disjoint ‘holes’ jz  zj j  rj , zj ! 1.

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5 Algebraic Differential Equations

This leads to .w  y/.w C y/ D o.jzj nk n o.jzj 2 1 2 / and n

w D  z 2 C

kC1 X

Œk

cj z

nj 2

nk 2 1

C o.jzj

n

/ and w C y  2  z 2 , hence w  y D

n.kC1/ 2

/D

kC1 .z/

C o.jzj

n.kC1/ 2

/;

jD1

that is, (5.11) holds with k replaced with k C 1. To determine the total number of strings of poles we first assume that n is even, and remind the reader of the construction of the closed simple curve r in Exercise 4.17: r has winding number 1 and 0 about poles on jpk j  r and jpk j > r, respectively, n w D O.jzj 2 / holds on r , and the length of the sub-arc of r contained in any sector j arg z  j < ı is < Kır. Since all but finitely many poles have residue 1, n.r; w/ D

1 2i

Z w.z/ dz C O.1/

(5.12)

r

holds. The contribution of the sector ˙ to the integral in (5.12) is  r

n 2 C1

1 2

Z

O O1

r 2 C1 n C O.ır 2 C1 /; / D .1/  n . 2 C 1/ n

e

i. n2 C1/

d C O.ır

n 2 C1



this proving that w has `.w/ D

nC1 X .1/ 

(5.13)

D0

strings of poles. To determine the number of strings along a single Stokes ray we again apply the Residue Theorem. For  fixed let r denote the simple closed and O O positively oriented curve that consists of the line segment from r0 ei. ı/ to rei. ı/ , O O followed by the sub-arc of r in j arg z  O j < ı from rei. ı/ to rei. Cı/ , the O O line segment from rei. Cı/ to r0 ei. Cı/ , and finally the (short!) circular arc from O O r0 ei. Cı/ to r0 ei. ı/ . Then 1 2i

Z

.1/ r 2 C1 n .   C1 / n C O.ır 2 C1 / 2 . 2 C 1/ n

w.z/ dz D

r

holds and may be interpreted as follows: there is exactly one string of poles along  if  D  C1 D .1/ , no such string if C1 D  , while  D  C1 D .1/C1 never occurs. If n D 2m  1 is odd, the proof runs along the same lines. Since, p however, z is discontinuous across the Stokes ray arg z D , it becomes necessary to enumerate the Stokes rays  from  D m to  D m, so that m W arg z D .2mC1/ denotes the negative real axis. Then m D m means that the asymptotic nC2

5.2 Riccati Equations

151

expansions on ˙m and ˙m are different. The total number of strings is m X

`.w/ D

.1/ 

(5.13’)

Dm

rather than given by (5.13). The final assertion follows from m.r; w/ D O.log r/.

5.2.2 Value Distribution Since coth z has poles z D ki and zeros .k C 12 /i, the zeros of the solutions w to the canonical equation (5.10) ‘separate’ the poles at distance 2 w.r.t. the metric ds D n

C o.1//pk 2 . jzj 2 jdzj: the zeros form .˙i; n2 /-strings .qk / satisfying qk D pk C . i 2 The value distribution of wStakes place in an arbitrarily small neighbourhood of n the set Q of zeros: Qı D q¤0 4ı .q/ with 4ı .q/ D fz W jz  qj < ıjqj 2 g: To determine the distribution of arbitrary -points of transcendental solutions to the general Riccati equation (5.9) it suffices to determine the distribution of the zeros of w   for solutions w to the canonical equation, where .z/ D Czs C    .C ¤ 0 and z ! 1/ may be any rational function. There are five cases to be considered. n

n

n

1. s > n2 . Since w D O.jzj 2 / holds outside and jzj 2 D O.jw.z/j/ inside the union n of discs 4ı . p/ D fz W jz  pj < ıjpj 2 g about the poles, all but finitely many zeros of w   are contained in these discs. To prove that for jpj sufficiently large there is exactly one zero in 4ı . p/ consider f .z/ D .z/=w.z/, which is holomorphic on 4ı . p/ and has exactly one zero, namely p. Since f .z/ tends to infinity as p ! 1, uniformly on @4ı . p/, Rouché’s Theorem applies to f and f  1 on 4ı . p/ (jpj sufficiently large), hence w   D .1  f /=f has exactly one zero on 4ı . p/. In other words, the zeros of w   form strings that ‘follow’ the strings of poles; at infinity they become indistinguishable from the poles in the n metric ds D jzj 2 jdzj. n 2. s < 2 . In the same manner it can be proved that the zeros of w   form strings just like the zeros of w; up to finitely many, all zeros of w   are contained in the union of discs 4ı .q/ about the zeros of w, exactly one in each disc. 3. s D n2 , C ¤ ˙1. Re-scaling along any sequence of zeros k of w leads to the initial value problem w0 D 1  w2 , w.0/ D C ¤ ˙1, hence w.z/ D coth.z C z0 / with z0 D 12 log CC1 . Thus the zeros of w   form strings that are ‘parallel’ to C1 n

n

the strings of poles: k D pk C z0 pk 2 C o.jpk j 2 /. 4. 0 D P.z/  2 . Then w   has only finitely many zeros. 5. s D n2 , C D ˙1, but 0 ¤ P.z/  2 . Again re-scaling along any sequence . k / of zeros of w   leads to the initial value problem w0 D 1  w2 , w.0/ D ˙1, hence w  ˙1. Thus the zeros of w   are ‘invisible’ from the poles since n j k j 2 dist . k ; P/ tends to infinity as k ! 1. Now u D 1=.w  / solves the Riccati equation u0 D 1 C 2.z/u C .z/u2 . D 0  P C 2 6 0/ with normal

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5 Algebraic Differential Equations

form v 0 D P .z/  v 2 and coefficient P D P  20 C 34 . 0 =/2 C . 0 =/  1 . 00 =/ D zn C    : Thus the zeros of w   form strings of the same kind. 2 The transformation w 7! c.z/wC 12 b.z/C 12 .c0 .z/=c.z// provides the essential step leading from (5.9) to the canonical equation (5.10), while the final transformation of the independent variable, which just stretches and rotates the plane, is inessential. We assume for simplicity that the first transformation already leads to the canonical form. Up to finitely many, the zeros of y   are zeros of w   with  D c.z/ C 1 1 0 2 b.z/ C 2 .c .z/=c.z/, and vice versa. Exercise 5.11 Prove that the zeros q and poles p of the solutions to (5.9) are n2 n separated (infq dist .q; P/jqj 2 > 0) if and only if deg b  n2 . Assuming this, prove that the transcendental solutions belong to the Yosida class Y deg cC n2 ; n2 . Exercise 5.12 (Continued) Assuming deg b  n2 , prove that all but finitely many zeros of w   are contained in the discs – 4ı . p/ about the poles of w if deg c > n2 and  ¤ 0, and n n – 4ı . / about the zeros of w  z 2 if c.z/ C 12 b.z/ D z 2 C    , exactly one zero in each disc. If, however, deg b > n2 , prove that – almost all zeros of w   belong to the discs 4ı . p/ about the poles of w, with at most one exceptional , and – w does not belong to any Yosida class. Exercise 5.13 Employ the previous exercise to discuss in detail the value distribution of the solutions to – – – –

w0 w0 w0 w0

D z2 C z3 C 2.z C z2 /w C zw2 , D  34 z2 C .2z C z2 /w C z2 w2 , D z C .2z C z2 /w C .z  34 z3 /w2 , and D 14 z C .2z C z2 /w C .z C z3 /w2 .

5.2.3 Truncated Solutions Every Stokes sector ˙ is ‘pole-free’ in that every solution has only finitely many poles on the closed sub-sectors of ˙ . Conversely, a fixed solution is said to be polefree on some sector S if it has only finitely many zeros on every closed sub-sector. Exercise 5.14 Suppose w is a pole-free solution on some sector S. Adapt the proof n of Theorem 5.7 to show that w has an asymptotic expansion w  z 2 C    on S, with D .w/ 2 f1; 1g.

5.2 Riccati Equations

153

Exercise 5.14 is relevant only if the sector in question contains some Stokes ray  . It turns out that in this case the asymptotics holds on ˙1 [  [ ˙ : Any solution that has no string of poles along the Stokes ray  is called truncated (along  ). For n D 1 truncated solutions are rational. From now on we assume n  0. Theorem 5.8 Given any Stokes ray  there exists a solution that is truncated along n

 . It is uniquely determined by its asymptotics w  z 2 C    , which holds on 4 ˙1 [  [ ˙ with central angle nC2 ; 2 f1; 1g may be prescribed. Proof We choose 2 f1; 1g and set y.z/ D zm w.z/  if n D 2m is even, and y.z/ D zn w.z2 /  if n is odd to obtain zm y0 C mzm1 . y C / D P.z/z2m  . y C /2 ; zn1 y0 C nzn2 . y C / D 2P.z2 /z2n  2. y C /2 ; respectively. Then Theorem 1.10 applies to both equations when written as zq y0 D f .z; y/

with q D m resp: q D n C 1 and lim fy .z; 0/ ¤ 0: z!1

2 there exists some solution having an Given any sector with central angle qC1  1 asymptotic expansion y  c1 z C    . Hence to every sector j arg z  0 j < nC2 (in both cases) there exists some solution to Eq. (5.10) with asymptotic expansion n  , w  z 2 C    ; may be prescribed. In particular, this applies to j arg z  O j < nC2 2 O and then even to S W j arg z   j < nC2 . To prove uniqueness we consider two solutions of this kind, set u D w1  w2 and observe that 1

u0 D .w1 .z/ C w2 .z//u D 2 z 2 .1 C O.jzj 2 //u n

holds on S. Since u.z/ ! 0 this requires Re z 2 C1 > 0 on S, which, however, is 2 impossible on sectors with central angle greater than nC2 ; this proves w1 D w2 . n

Exercise 5.15 For n D 0 the Stokes rays are arg z D ˙ 2 . Suppose w is truncated along arg z D 2 , hence w.z/ D C O.jzj1 / holds as z ! 1 on j arg z C 2 j < . Prove that w is also truncated along arg z D  2 and thus is a rational function. R (Hint. The Phragmén–Lindelöf Principle applies to F.z/ D e zC w.z/ dz on every sector j arg z C 2 j < ı < 4 ; jzj > r0 ; note that w may have finitely many poles with residue ¤ 1.) The number of strings is as large as possible if  D .1/ holds for every . On the other hand,  D .1/ for some  turns out to be exceptional. Theorem 5.9 Suppose w is any solution with the ‘false’ asymptotics w  .1/ z 2 C    n

on ˙ :

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5 Algebraic Differential Equations

Then w is truncated along the Stokes rays 1 and  which bound ˙ . Conversely, if w is truncated along  and has the ‘right’ asymptotics w  .1/ z 2 C    n

on ˙ ;

then w has the ‘false’ asymptotics w  .1/C1 z 2 C    on ˙C1 and is also truncated along C1 . n

Proof It follows from the proof of Theorem 5.7 that  D .1/ implies C1 D  and also 1 D  , hence w is truncated along 1 and  ; this proves the first part. If, on the other hand, w is truncated along  and  D .1/ , then again from the proof of Theorem 5.7 it follows that C1 D  D .1/C1 , and we are in p p n the first case with  replaced with  C 1. (Note that z 2 D zm z with Re z > 0 is assumed if n D 2m C 1 is odd.) Summary – The asymptotic expansions on ˙ of non-truncated (generic) solutions have n leading terms .1/ z 2 . – Truncated solutions are truncated along adjacent pairs of Stokes rays. – To each  there exists some uniquely determined solution with asymptotics w  n .1/ z 2 on ˙ ; w is truncated along 1 and  : – The solutions w are not necessarily distinct. P In any case truncated solutions are truncated along 2d.w/ Stokes rays with w d.w/ D n C 2. – The exceptional solutions w correspond to those solutions to the linear differential equation y00 D P.z/y that are subdominant on some sector S; y D R exp w.z/ dz is called subdominant on S if y tends to zero exponentially as z ! 1 on S. Example 5.4 Gundersen and Steinbart [74] considered the linear differential equation f 00  zn f D 0. They proved among other things that certain contour integrals (generalised Laplace transformations) f .z/ D

1 2i

Z eP.z;w/ dw C

represent solutions having no zeros along given Stokes rays 1 and  . These solutions give rise to the truncated solutions w D f0 =f to the special Riccati equation w0 D zn  w2 , which is invariant under the transformations w.z/ 7! w.z/, nC2 D 1. Obviously there are exactly two solutions which themselves are invariant, namely those which either have a pole or else a zero at the origin. These solutions cannot be truncated, hence there exist n C 2 mutually distinct exceptional solutions. They are obtained from a single one by rotating the plane:  2i  2i w .z/ D e nC2 w0 e nC2 z has a single string of poles along every Stokes ray  except those which bound the Stokes sector ˙ .

5.2 Riccati Equations

155

Example 5.5 The eigenvalue problem f 00 C .z4  /f D 0, f 2 L2 .R/, has infinitely many solutions .k ; fk /, see Titchmarsh [188]. The eigenfunctions fk have only finitely many non-real zeros. For every eigenpair .; f / D .k ; fk /, u.z/ D f .ei=6 z/ satisfies u00  .z4 C ei=3 /u D 0, and w D u0 =u solves w0 D z4 C ei=3   w2 : In general there are 4 C 2 D 6 exceptional solutions w , 0    5. They are, however, not mutually distinct: w2 D w5 is truncated along 1 ; 2 ; 4 , and 5 , has strings of poles along 0 and 3 , and the asymptotics w  z2 C    on 6 < arg z < 7  and w  z2 C    on  56  < arg z < 6 . The exceptional solutions w0 ; w1 ; w3 , 6 and w4 have four strings of poles and a ‘pole-free’ half-plane (Fig. 5.1). Example 5.6 Eremenko and Gabrielov [39] considered the linear differential equation y00 .z3 azC/y D 0: For certain real parameters a and  there exist solutions with infinitely many zeros, almost all are real and negative. Thus w0 D z3  az C   w2 has a solution w D w1 D w1 which is truncated along 2 , 1 , 0 , and 1 with p 3 asymptotics w  z 2 (Re z > 0) on j arg zj < , and mutually distinct solutions w0 , w2 , and w2 , which are truncated along 1 and 0 , 1 and 2 , and 2 and 2 , respectively (Fig. 5.2).

Fig. 5.1 Asymptotics and distribution of poles of generic and exceptional solutions in the eigenvalue case

Fig. 5.2 Asymptotics and distribution of poles of generic and exceptional solutions in the p eigenvalue case. The square-root z is discontinuous along the negative real axis; in the third and fourth case the ray arg z D  is truncated!

156

5 Algebraic Differential Equations

5.2.4 Poles Close to a Single Line Several papers (Eremenko and Gabrielov [39], Eremenko and Merenkov [40], Gundersen [67, 70], Shin [156]) are devoted to the question under which circumstances linear differential equations y00  P.z/y D 0

.P.z/ D an zn C    some polynomial with jan j D 1/

(5.14)

have solutions with all but finitely many zeros on the real axis. From Theorem 5.8 we obtain (see also [40, 67]): Theorem 5.10 Suppose that Eq. (5.14) has a solution whose zeros are asymptotic to the real axis. Then the following is true. – If n is even, then either y has only finitely many zeros, or else n  0 mod 4, an D 1, y has exactly one string of zeros asymptotic to the negative and positive n real axis, respectively, and y0 =y  ˙iz 2 C    holds on ˙Im z > 0. – If n D 2m  1 is odd, then either an D 1 and y has exactly one string of zeros 1 asymptotic to the negative real axis with asymptotics y0 =y  .z/m z 2 C    on j arg zj < , or else an D 1 and y has exactly one string of zeros asymptotic to 1 the positive real axis with asymptotics y0 =y  zm .z/ 2 C    on j arg.z/j < . If P is a real polynomial, y is a (multiple of a) real entire function with all but finitely many zeros real. Proof If y.z/ D P1 .z/eP2 .z/ has only finitely many zeros, then n D 2 deg P2  2 is even, and hardly more can be said (of course, P can be computed explicitly from P1 and P2 ). From now on we assume that y has infinitely many zeros. The change of variables w.z/ D y0 .z/=y.z/ with nC2 an D 1 transforms Eq. (5.14) O into Eq. (5.10) with new coefficient P.z/ D 2 P.z/ D zn C    in place of P, hence the question of whether or not there are solutions y to (5.14) having infinitely many zeros, ‘most’ of them close to the real axis, is transformed into the question of whether or not there are solutions w to Eq. (5.10) having just one string of poles asymptotic to some Stokes ray  W arg z D O D .2C1/ if n is odd, and asymptotic nC2 O

˙ei up to an to the Stokes rays  and CmC1 if n D 2m is even. This yields N D  i nC2 arbitrary .n C 2/-th root of unity. We are free to choose  D e and  D 0 if n D 2m is even, and  D ˙1 and  D m if n D 2m  1 is odd. If n D 2m is even we obtain an D 1, and from the proof of Theorem 5.7 it follows that 0  1 D 2 and .1/mC1 . mC1  mC2 / D 2, hence 0 D 1 and 1 D 1, this implying 1 D 2 D    D mC1 D 1, mC2 D    D 2mC1 D 0 D 1, m D 2k and n D 4k. This proves the first part of Theorem 5.10. If n D 2m  1 is odd we have an D C1 and an D 1 with zeros asymptotic to the negative and positive real axis, respectively, and asymptotic expansions 1

y0 =y  .1/m .˙z/m 2 C    on j arg.˙z/j < :

5.2 Riccati Equations

157

If P is a real polynomial, the zeros of the solution(!) y .z/ D y.Nz/ are also asymptotic to the real axis, hence y and y are linearly dependent since y is uniquely determined up to a constant factor. Thus y is a multiple of a real entire function with all but finitely many zeros real.

5.2.5 Locally Univalent Meromorphic Functions P To construct meromorphic functions with sum of deficiencies ı.a; f / D a2b C 2; Nevanlinna [127] considered locally univalent meromorphic functions f of finite order. They are characterised by the fact that their Schwarzian derivative Sf D . f 00 =f 0 /0  12 . f 00 =f 0 /2 is a polynomial 2P, say. Moreover, f equals the ratio y.zI 0/=y.zI 1/ of linearly independent solutions to the linear differential equation y00 C P.z/y D 0 .deg P D n/:

(5.15)

Using Hille’s method of asymptotic integration ([85], §5.6.), Nevanlinna was able to show that f has finitely many deficient values a with deficiencies ı.a ; f / D P 2d (d 2 N) and d D n C 2. Since Eq. (5.15) is equivalent to the Riccati    nC2 0 2 equation w D P.z/  w via w D y0 =y, an easier way is to ‘count’ poles. Generic n solutions have counting function of poles and Nevanlinna characteristic  Cr 2 C1 . The exceptional (truncated) solutions w , however, have counting function of poles n  and Nevanlinna characteristic  C nC22d r 2 C1 , where d is some positive integer nC2 P such that  d D n C 2. Since f  a and y.zI a/ D y.zI 0/  ay.zI 1/ have the same zeros, which coincide with the poles of w.zI a/ D y0 .zI a/=y.zI a/, it follows that f 2d has the desired properties with a defined by w .z/ D w.zI a / and ı.a ; f / D nC2 ; n C1 note that T.r; f /  Cr 2 holds by Cartan’s Identity.

5.2.6 A Problem of Hayman We return to a problem which we discussed in Sect. 3.2.5, namely to determine those meromorphic functions f such that ff 00 has no zeros. This problem was settled by Langley [104] (solution f .z/ D eazCb and f .z/ D .az C b/n ) in general, and by Mues [119] for functions of finite (lower) order. Mues’ proof matches the topic just being discussed. He considered the meromorphic function F.z/ D zf .z/=f 0 .z/ with non-vanishing derivative F 0 .z/ D

f .z/f 00 .z/ : f 0 .z/2

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5 Algebraic Differential Equations

Since F has only simple poles (at zeros of f 0 ), F is locally univalent and its Schwarzian derivative SF D 2Q is an entire function, and even a polynomial: T.rk ; Q/ D m.rk ; Q/ D O.log rk / holds on some sequence rk ! 1; F has the form w1 =w2 with linearly independent solutions to w00 C Q.z/w D 0. We first consider the case Q 6 0 and may assume Q.z/ D zn C    . From F 0 =F D w01 =w2  w02 =w2 D y1  y2 with y0j D Q.z/  y2j it follows that the plane is divided into finitely many sectors ˙ W O1 < arg z < O such that either F 0 =F  0 (that is, n F 0 =F D o.jzjm / holds for every m 2 N) and F  c , or else F 0 =F  2  z 2 C    with  D ˙1 and F.z/ ! 0 or F.z/ ! 1, exponentially as z ! 1 on ˙ . The entire function g D 1=f has finite order of growth and logarithmic derivative g0 .z/ 1 D zCF.z/ D O.jzj1 /, hence g grows at most logarithmically along every g.z/ ray arg z D  ¤ O . From the Phragmén–Lindelöf Principle applied to the sectors j arg z  O j < ı it follows g is constant. This, however, contradicts SF 6 0, and it remains to discuss the case when SF vanishes identically. 0

.z/ AzCB Exercise 5.16 Suppose F.z/ D CzCD is a Möbius transformation, hence ff .z/ D CzCD 0 holds. From the fact that the residues of f =f are negative integers Cz2 C.DA/zB

deduce f .z/ D eazCb or f .z/ D .az C b/n .

5.3 First-Order Algebraic Differential Equations This section is devoted to implicit first-order differential equations 0

P.z; w; w / D

q X

P .z; w/w0 D 0 .q  2/;

(5.16)

D0

where P.z; x; y/ is an irreducible polynomial over the field C.z/. Equations of this type are completely understood. With the help of the so-called Second Theorem of Malmquist and the investigations of the previous section we will draw a comprehensive picture of the solutions.

5.3.1 Malmquist’s Second Theorem Malmquist’s Second Theorem shows that the so-called Fuchsian conditions for the absence of movable singularities other than poles are necessary for the existence of some transcendental meromorphic solution. These conditions may be verified algebraically. Towards Malmquist’s Theorem we will start with a simple example. By D.z; x/ we denote the discriminant of the polynomial P.z; x; y/ with respect to y.

5.3 First-Order Algebraic Differential Equations

159

Example 5.7 The differential equation z2 w02 C P1 .z; w/w0 C P0 .z; w/ D 0 with P1 .z; w/ D 4z2  4z4 C .2z C 4z3 /w C .1 C z2 /w2 ; P0 .z; w/ D 4z2 C 8z4 C 8z6  .4z C 8z3 C 16z5 /w .6z2  6z4 /w2 C .4z C 2z3 /w3 C w4 ; D.z; w/ D 16z4 .w2 C 4zw  4z2 /..1  z2 /w C 2z3 /2 has three rational solutions: 0 .z/ D z, which p does not solve the discriminant equation D.z; w/ D 0, and 1;2 .z/ D .2 ˙ 2 2/z, which do. On the other hand, 3 the solution 3 .z/ D z2z 2 1 to D.z; w/ D 0 does not solve the differential equation. The solutions 1;2 are called singular. The general solution is transcendental and 2 Cz2 given by w D t tCz with t0 D z2 C 1  t2 ; 0 is obtained from t.z/ D z. Theorem 5.11 (Malmquist’s Second Theorem) Suppose Eq. (5.16) has a transcendental meromorphic solution. Then – degx P  2q  2; in particular, Pq is independent of x; – if x D .z/ solves D.z; x/ D 0, and y D .z/ C b.z/ k=m C    with b.z/ 6 0, .k; m/ D 1; and m  2 solves P.z; .z/C; y/ D 0 in a neighbourhood of  D 0, k  m  1 and D 0 hold. Q v; v 0 / D v 2q P.z; 1=v; v 0 =v 2 / D 0 with – an analogous condition holds for P.z; Q Q discriminant D.z; v/; D.z; 0/ D 0 corresponds to the ‘solution’ D 1. Conversely, these conditions are sufficient in order that the solutions have no movable singularities except poles, hence are meromorphic if no fixed singularities exist. Remark 5.7 The necessity of the first condition was proved in Theorem 3.3 with the help of Eremenko’s Lemma. For the second condition the reader is referred to Eremenko’s ‘algebraic’ paper [34], and also to Golubev [58], II. §2, where the Fuchsian conditions for the absence of so-called ‘movable singularities’ other than poles are derived. The second condition says that certain solutions to D.z; x/ D 0 also solve (5.16). Solutions of this kind are called singular. In Example 5.7 we have k D 1 and m D 2 for the singular solutions 1;2 , but k D m D 1 for 3 , which solves the discriminant equation but not the differential equation.



MAPLE code t1:=1+zˆ2-tˆ2; r1:=tˆ2+zˆ2; r2:=t+z; r:=r1/r2;

t0 D 1 C z2  t2 ; w D r.z; t/ D

s:=diff(r,z)+diff(r,t)t1; s1:=numer(s); s2:=denom(s); P:=resultant(xr2-r1,ys2-s1,t); Delta:=discrim(P,y); solve(Delta,x);

r1 .z;t/ r2 .z;t/

decoded 2 Ct2 D z zCt s .z;t/

w0 D s.z; t/ D s12 .z;t/ P.z; w; w0 / D 0 compute the discriminant and its roots

160

5 Algebraic Differential Equations

5.3.2 Binomial Differential Equations The so-called binomial differential equations w0q D R.z; w/

(5.17)

were considered in [6, 7, 160, 208, 211],2 and the general degree-two equation .w0 C B.z; w//2 D R.z; w/ in [161] (R resp. B and R rational). It turns out that there are five canonical equations as follows: 1: 2: 3: 4: 5:

.w0 C b.z/w/2 D a.z/w.w  c.z//2 w02 D a.z/.w  1 /.w  2 /.w  3 /.w  4 / w03 D a.z/.w  1 /2 .w  2 /2 .w  3 /2 w04 D a.z/.w  1 /2 .w  2 /3 .w  3 /3 w06 D a.z/.w  1 /3 .w  2 /4 .w  3 /5

(5.18)

In any case a, b, and c are not identically vanishing rational functions, and  ¤  ( ¤ ) holds. The discriminant D.z; w/ is just given by the respective right-hand side. Note that in the first case the root w D c.z/ of D.z; w/ is not necessarily a singular solution. The first equation may be transformed into some Riccati equation 0 Cb.z/w for t D wwc.z/ , hence a.z/w D t2 , while for a.z/ D const. the solutions to the other equations are specific elliptic functions; they are closely related to the Schwarz–Christoffel conformal mappings of rectangles and certain regular triangles onto the upper half-plane. The Malmquist conditions are obviously fulfilled. Example 5.8 Binomial differential equations occur in a quite different context. Let b R be any non-linear rational function and suppose P there exists some map    W C ! N [ f1g such that .a/ deg.a/ R D .R.a// and a2b 1  1=.a/ D 2; where C deg.a/ R denotes the local degree of R (the multiplicity of the solution to R.z/ D R.a/ at z D a). Then O D .b C; / is called a parabolic orbifold. Any parabolic orbifold has a ramified covering map F W C ! O; it omits a if .a/ D 1, is branched exactly over those a with 1 < .a/ < 1, and has local degree deg.z/ f D . f .z//. It turns out that every Zalcman function f .z/ D limk!1 Rnk .zk C k z/ (see Sect. 4.1.3) is a ramified covering map f W C ! O; it satisfies the binomial

2 The authors seemed to be unaware of Malmquist’s Second Theorem, or were in doubt about his reasoning. One reason might be that several elegant and transparent proofs of his First Theorem were known, but none for his Second. Eremenko’s commented in [35]: “The paper [42] (Malmquist’s paper [114] here) has had practically no influence on the work of other authors [: : :]”.

5.3 First-Order Algebraic Differential Equations

161

differential equation w0m D c

Y

.w  a/m.a/

1 0 holds on S (always 2 f1; 1g). n (Hint. y D t   satisfies y0 D .t.z/ C .z//y with t.z/ C .z/  2 z 2 C    .) To proceed we note that the general assumption degw P0 D 2q implies m.r; w/ D O.log r/, hence r2 .z; t.z// has a non-trivial asymptotic expansion and Exercise 5.20 applies to p D r2 . Almost all poles of w are simple; they arise from poles of t if deg r1 D deg r2 C 1 on one hand, and the solutions  D .z/ to the equation O r2 .z; / D 0 which do not solve  0 D P.z/   2 on the other. The classification of zeros of t   and their distribution in the plane (see Sect. 5.2.2, where  may be any rational function) remains valid for algebraic functions , and this proves the second assertion. The proof of the third assertion follows from Valiron’s Lemma T.r; w/  qT.r; t/ and T.r; t/ 

n C 2  2dt n C1 r2 : . n2 C 1/2 

To prove the fourth assertion  we note that  the Wittich–Mokhon’ko Lemma (see also Exercise 3.5) yields m r; 1=.w  / D O.log r/ and ı. ; w/ D 0; provided the rational function does not solve Eq. (5.16). If, however, solves (5.16) but is not singular, the algebraic equation r.z; /  .z/ D 0, equivalently p.z; / D r1 .z; /  .z/r2 .z; / D 0, has solutions  D .z/ that also solve Eq. (5.26). This implies that there are at most two deficient rational functions of this kind. Now p factors into p1 p2 , such that all solutions to p1 .z; / D 0, but none to p2 .z; / D 0 also solve the Riccati equation (5.26). To proceed we refer to Exercise 5.21 Let p.z; / be any polynomial over C.z/ such that none of the solutions  D .z/ to p.z; / D 0 solves (5.26). Prove that p.z; t.z// has equally many strings of poles and zeros (strings of k-fold zeros will be counted k times),   n 1 1 hence m r; p.z;t.z// D N.r; p.z; t.z///N r; p.z;t.z// CO.log r/ D o.r 2 C1 / holds. Exercise 5.21 applies to p D p2 and p D r2 , hence (note that m.r; t/ D O.log r/)  p .z; t.z//   r .z; t.z//  n 2 2 C m r; D o.r 2 C1 / m r; p2 .z; t.z// r2 .z; t.z// holds, and w  .z/ D  m r;

p2 .z;t.z// p .z; t.z// r2 .z;t.z// 1

1 w



 D m r;

implies  1 n C o.r 2 C1 /: p1 .z; t.z//

166

5 Algebraic Differential Equations

Since p1 .z; t.z// has only finitely many zeros, the First Main Theorem yields  m r;

1 w



D T.r; p1 .z; t.z/// C o.r 2 C1 / D 1q deg p1 T.r; w/ C o.r 2 C1 /; n

n

hence ı. ; w/ D 1q deg p1 . On the other hand, singular solutions have p1  1 and ı. ; w/ D 0. This proves the fourth assertion. To prove the fifth assertion we will now clarify the relation between ramified functions and rational solutions. It is obvious that w  .z/ has infinitely many multiple zeros if and only if r.z; / D

.z/

(5.29)

has .mj C 1/-fold solutions  D j .1  j  s/ which do not solve (5.26). The zeros of t  j .z/ are arranged in n C 2  2dt strings, and w  .z/ has n C 2  2dt strings of .mj C 1/-fold zeros. This shows #. ; w/ D 1q .m1 C    C ms /. Also for z fixed, the rational function r./ D r.z; / has at most 2q  2 critical values, hence there are at most 2q  2 rational functions such that (5.29) has multiple roots. Finally, let .z/  0, say, be a singular solution, and recall that P.z; x; y/ is the resultant of the polynomials x r2 .z; /  r1 .z; / and y s2 .z; /  s1 .z; / w.r.t. ; in particular, the resultant of r1 .z; / and s1 .z; /, namely P.z; 0; 0/, vanishes identically, hence r and s have common roots. From s.z; /  rz .z; / C O   2 / and rz .z; .z// C r .z; .z// 0 .z/  0 it follows that r .z; /.P.z/ O  .z/2   0 .z// s.z; .z//  r .z; .z//.P.z/ holds for any solution to r.z; /  0. In other words, the common roots of r and s coincide with the multiple roots of r, and we are in the first case. This proves the fifth assertion and also finishes the proof of Theorem 5.13.

5.3.6 Some Examples of Genus Zero We will now discuss several examples to illustrate various features of Theorem 5.13. Example 5.9 Singular solutions need not be single-valued since the differential equation (5.16) may have fixed singularities. This is the case with w03 C 7z2 w02 C Œ54zw2  21zw C z C 8z4 w0 C Œ27w4  27w3 C .9  104z3 /w2  .1 C 4z3 /w  .z3 C 16z6 / D 0; D.z; w/ D .4z3 C 27w2 /D3 .z; w/2

.degw D3 D 3/:

The non-singular solutions are given by w D t.t2 C z/, where t is any solution to t0 D z  t2 . The roots of D3 .z; w/ do not solve the differential equation, and

5.3 First-Order Algebraic Differential Equations

167

q 4 3 the singular solutions .z/ D  27 z are not single-valued. The origin is a fixed singularity of the differential equation, which, however, is invisible at first glance. Regarding we refer to the second condition in Malmquist’s Second Theorem. Example 5.10 For q D 2, regular (non-singular) rational solutions are deficient, while ramified rational functions and singular rational solutions coincide and are not deficient. This is not necessarily true if q > 2, where ‘regular-singular’ rational solutions may occur. With the exception of .z/ D z C 1, the solutions to the Riccati equation t0 D z2 C 2z C 2  t2 are transcendental meromorphic functions, and w D r.z; t/ D .t  .z//t2 satisfies some irreducible differential equation w03  6.z C 1/ww02 C PO 1 .z; w/ww0 C PO 0 .z; w/w2 D 0 with PO 1 .z; 0/ 6 0, degw PO 1 D 1, and PO 0 .z; 0/ 6 0; degw PO 0 D 2. The discriminant is D.z; w/ D w3 .27w C 4.z C 1/3 /D1 .z; w/2 . Rational solutions are 0 and .z/ D 4  27 .zC1/3 ; is singular since r.z; / .z/ D . C 13 .zC1//.  23 .zC1//2 ; while 0 is simultaneously deficient and singular. It is easily seen that ı.0; w/ D #.0; w/ D 1 2 1 3 , .0; w/ D 3 , ı. ; w/ D 0, #. ; w/ D . ; w/ D 3 , and ı.1; w/ D 0, 2 #.1; w/ D .1; w/ D 3 ; 1 is completely ramified, and may be viewed as a ‘singular solution’, while 0 is deficient and completely ramified. The procedure to be described in Sect. 5.4 will show that w 2 Y3;1 can be immediately read off the differential equation. tz Exercise 5.22 For t0 D 1 C z2  t2 and w D t2 tCz compute the corresponding 3 03 0 0 O O differential equationp z w CP2 .z; w/ww CP1 .z; w/ww CPO 0 .z; w/w2 D 0; prove that 1 2 1;2 .z/ D 2 .11 ˙ 5 5/z and 1 are singular solutions and 0 is a regular-singular solution. Verify #.0; w/ D ı.0; w/ D #.1; w/ D .1; w/ D #. 1;2 ; w/ D . 1;2 ; w/ D 13 and .0; w/ D 23 .

Exercise 5.23 Singular solutions need not be single-valued (cf. Example 5.9), so it makes no sense to ask whether a many-valued singular solution may be deficient or ramified for w. Many-valued singular solutions, however, form cycles 1 ; : : : ; s Q under analytic continuation, and p.z; / D sjD1 .  j .z// is a polynomial in  over C.z/. Prove that v D p.z; r.z; t// satisfies some first-order equation Q.z; v; v 0 / D 0 of degree qs, and 0 is a singular solution that is ramified for v. Example 5.11 The strings of poles need not be n2 -separated from each other. Let t t4 be any solution to t0 D z2  t2 . Then w D .tz/.t2z/.tz 2 / solves some Eq. (5.16) of degree four. According to the classification of the zeros of t  .z/,  rational, w has four different types of poles distributed in .˙i; 1/-strings along (some of) the Stokes rays arg z D .2 C 1/ 4 . They correspond to 1. the poles p of t; they form the set P.t/ and are arranged in (two or four) strings; 2. the zeros of t.z/  z2 ; they are contained in Pı .t/, exactly one belongs to 4ı . p/ D fz W jz  pj < ıjpj1 g for jpj sufficiently large.

168

5 Algebraic Differential Equations

3. the zeros Q of t.z/  2z; they form strings that are ‘parallel’ to the strings of the first kind, Qk D pk  . 12 log 3 C o.1//p1 k ; O 4. the zeros of t.z/  z; they are ‘invisible’ from the poles of the first kind in the metric ds D jzjjdzj, that is, j Ok jdist. Ok ; P.t// ! 1 as k ! 1.

5.3.7 Equations of Genus One The solutions to first-order algebraic differential equations of genus one are locally given by w.z/ D r.z; t.z/; t0 .z//; t.z/ D }.˚.z//; and ˚ 0 .z/ D

p a.z/:

The main problem arising here is that the coefficients are algebraic rather than rational functions, and ˚ may (and will) have algebraic and logarithmic singularities. p On the other hand, all coefficients are functions of some fixed root p z in some p neighbourhood of infinity, and this also holds for t.z/ D .z; w.z/; w0 .z// D f . p z/: Then W D f . / satisfies W 02 D A. /.4W 3 g2 W g3 / with A. / D p2 2p2 a. p / D A0 2ˇ C    on j j > r0 , and from } ] 1, which holds outside arbitrarily small euclidean discs about the critical points of }, it follows that sup j j r0 and T.r; w/ log2 r (ˇ D 1) 2ˇC2

resp. T.r; w/ r p (ˇ > 1) holds. Exercise 4.7 shows that the first case actually may occur. In the second case, however, the order of growth 2ˇC2 is restricted to p integer multiples of 12 or 13 , see Eremenko [35] for a proof. This result is related to the question of whether elliptic functions may be written as f ı Q, where f is meromorphic and Q is a non-linear polynomial. The answer given by Mues [118] is that Q.z/ D a C b.z  z0 /d and d 2 f2; 3; 4; 6g is necessary. Each case may occur: Exercise 5.24 Let }.zI g2 ; g3 / denote the Weierstraß P-function with differential equation } 02 D 4} 3  g2 }  g3 . Prove that 1. 2. 3. 4.

}.zI g2 ; g3 / D f2 .z2 / and w D f2 satisfies w02 } 0 .zI 0; g3 / D f3 .z3 / and w D f3 satisfies w03 } 2 .zI g2 ; 0/ D f4 .z4 / and w D f4 satisfies w04 } 3 .zI 0; g3 / D f6 .z6 / and w D f6 satisfies w06

D 14 z1 .4w3  g2 w  g3 /: D 12 z2 .w2  g3 /2 : 1 3 3 D 16 z w .4w  g2 /3 . 1 5 4 D 64 z w .4w  g3 /3 .

Remark 5.9 First-order equations are discussed in [185] in more detail. Eremenko considered the same class of equations (5.16). Among other things he gave a proof of Malmquist’s Second Theorem in [34]. He also proved in [35] that the solutions

5.4 Differential Equations and the Yosida Classes

169

have order of growth % D k=2 mean type .k 2 N) in case of genus zero, and % D k=2 or % D k=3 mean type .k 2 N0 ) otherwise, where k D 0 means T.r; w/ log2 r.

5.4 Differential Equations and the Yosida Classes Let w be any transcendental solution to the algebraic differential equation Q.z; w; w0 ; : : : ; w.n/ / D 0: We set z D h C hˇ z; wh .z/ D h˛ w.h C hˇ z/; w0h .z/ D h˛ˇ w0 .h C hˇ z/, etc., .n/ this leading to Q.h C hˇ z; h˛ wh ; h˛Cˇ w0h ; : : : ; h˛Cnˇ wh / D 0: Taking the limit  ˛ ˛Cˇ ˛Cnˇ limh!1 h Q.h; h w0 ; h w1 ; : : : ; h wn / for suitably chosen then yields some autonomous differential equation Q.w; w0 ; : : : ; w.n/ / D 0; which is solved by every limit function w D limhk !1 whk . Apart from the fact that the real parameters ˛ and ˇ are arbitrary, the method is by no means justified. Nevertheless it can be justified if w belongs to the Yosida class Y˛;ˇ .

5.4.1 Application to First-Order Differential Equations For Eq. (5.16) the formal re-scaling process leads to Briot–Bouquet equations P.w; w0 / D 0:

(5.30)

In the case of genus zero, neither the parametrisation (5.22) nor the Riccati equation (5.26) are at hand. Thus the problem arises how to determine the parameter n as well as the asymptotic expansions exclusively from (5.16), and also possible values of ˛, if any, such that w 2 Y˛; n2 . 1. Asymptotic expansions. To determine the possible leading terms of the m m asymptotic series w  az 2 C    , hence w0  m2 az 2 1 C     m2 z1 w, exclusively from the differential equation in question (it followsPfrom (5.22) that m must be 1 k  Q y/ D P.x; y; m x1 y/ D 2q some integer), consider P.x; D0 A .1 C O.jxj //x y I 2 0 Q w/ may be viewed as an approximation to P.z; w; w /. The Newton–Puiseux P.z; method applies to the simplified equation 2q X D0

A xk y D 0:

(5.31)

170

5 Algebraic Differential Equations

As x ! 1 the solutions have leading terms aj xj (aj ¤ 0), and the possible leading terms of the asymptotic expansions for w are among the terms aj zj with 2j 2 Z. Some terms may also point to singular solutions .z/ D aj zj C    . 2. The parameters. To select the ‘true’ parameters ˛ and ˇ out of a variety of candidates (the procedure is not unique) it is reasonable to postulate that Eq. (5.30) has maximal degree degw0 P D q and ˇ D n2 is as small as possible (the ‘local unit discs’ 41 . p/ W jz  pj < jpjˇ should be as large as possible). We assume Pq .z; w/  1. For w0 and w1 fixed, consider ˚.h; w0 ; w1 / D hq.˛Cˇ/ P.h; h˛ w0 ; h˛Cˇ w1 / D P.w0 ; w1 / C .h; w0 ; w1 / q

with P.w0 ; w1 / D w1 C    and degw1  < q. Then ˛ and ˇ (as small as possible) have to be adjusted in such a way that .h; w0 ; w1 / tends to zero as h ! 1. 3. Is w 2 Y˛;ˇ ? Having determined the possible parameters one has to prove w]˛ .z/ D

jzj˛ jw0 .z/j D O.jzjˇ /: jzj2˛ C jw.z/j2

This may be done by using well-known estimates for the roots of ordinary polynomials applied to (5.16), where P is regarded a polynomial in w0 . Example 5.12 Consider z2 w02 C P1 .z; w/w0 C P0 .z; w/ D 0 with P1 .z; w/ D .2z  2z3 /w  14 .2  z2 /w2 ; z4 /w2 C .4z C 5z3 /w3  14 .2 C 3z2 /w4 ; P0 .z; w/ D 2z5 w C .1 C 31 4 4 D.z; w/ D 16z w.w  8z/..2 C 7z2 /w C 4z3 /2 :

(5.32)

1. The reduced equation yx2 .32x3 C 124x2 y C 80xy2  12y3 / D 0 has solutions y D 0; 8x;  31 x; x. The first pair corresponds to the singular solutions w D 0 and w D 8z, while the second pair determines the principal terms of the asymptotic expansions w   13 z C    and w  z C    2. For any choice of ˛ and ˇ the principal part of ˚.h; w0 ; w1 / has the form 1 w21  .2h1ˇ w0  h˛ˇ w20 /w1 C 2h3˛2ˇ w0 4 31 22ˇ 2 3 C h w0 C 5h1C˛2ˇ w30  h2˛2ˇ w40 : 4 4 Obviously, ˇ  1 is necessary. Choosing ˇ D 1, the terms  14 h˛1 w20 w1 and 2h1˛ w0 enforce ˛ D 1. The corresponding Briot–Bouquet equation is P.w; w0 / D w02  .2w  14 w2 /w0 C 2w C with solutions w D 0; 8;  31 ; 1 (again!), and w D

31 2 w 4

C 5w3  34 w4 D 0

coth2 z coth z2

(normalised).

5.4 Differential Equations and the Yosida Classes

171

3. To prove jw0 j D O.jzj2 C jwj2 / (˛ D ˇ D 1) we start with 1

jw0 j D O.maxfjP1 .z; w/=z2 j; jP0 .z; w/=z2 j 2 g/ 3 1 1 3 D O.maxfjzjjwj; jwj2 j; jzj 2 jwj 2 ; jzj 2 jwj 2 g/: Applying various Hölder inequalities then shows jw0 j D O.jzj2 C jwj2 /. Re-scaling along any sequence of poles (wp .z/ D p1 w. p C p1 z/) yields non-constant limit functions w D =z C    (the residues satisfy 2  14 3  34 4 D 0). To exclude whk ! 1 as hk ! 1 we consider the differential equation v02 C .2v  14 /v0 C 2v3 C

31 2 4 v

C 5v 

3 4

D0

for v D 1=w; which has no trivial solution. This proves w 2 Y1;1 .5 4. The distribution of poles. Almost all poles of w are simple (degw P0 D 4) and are distributed in .˙i; 1/-strings asymptotic to the rays arg z D .2 C 1/ 2 . The poles of w occur in pairs p and pQ D p C . 12 log 3 C o.1//p1 , this following from the distribution of the poles of coth z, the zeros of coth z  2, and Hurwitz’ Theorem. Example 5.13 The same procedure applied to w02  z2 .4w  w2 /w0 C 4z4 w C .8z2  z2 /w2 C .4  6z2 /w3  .1  z2 /w4 D 0 yields ˛ D 0 and ˇ D 2, which, however, does not reflect the properties of the t2 transcendental solutions w D t1 with t0 D z2  t2 , hence t 2 Y1;1 . The reason for the failure of the method is that the zeros q and poles p of w that correspond to the zeros and one-points of t ‘collide’ (infp jqjjq  pj ! 0 as q ! 1). Exercise 5.25 Discuss the differential equation z2 w02 C Œ.2z  4z2 /w  .1  z/w2 w0 C Œ4z3 w C .1  z2 C 8z3 /w2 C .4z  6z2 C 4z3 /w3  .1  z C z2 /w4  D 0 in the spirit of Example 5.12. In particular, prove that the transcendental solutions belong to Y0; 1 by showing that ˛ D 0, ˇ D 12 (as small as possible) is 2

1

necessary, and w] .z/ D O.jzj 2 / follows exclusively from the differential equation. Derive w02 D 4w.w C 1/2 for every limit function (solutions w D 0; 1 and

O One cannot expect to re-construct the parametrisation w D r.z; t/ and t0 D P.z/  t2 . The fact coth2 z t2 2 O suggests w D and P.z/ D z C a z C a0 , actually that t; w 2 Y1;1 and w D coth 1 z2 t2zc0 a1 D a0 D c0 D 0.

5

172

5 Algebraic Differential Equations

w D  coth2 z), and deduce the essential properties exclusively from the differential equation, namely: 1

– the poles of w occur in pairs p and pQ D p C o.jpj 2 / and are distributed in .˙i; 12 /-strings along arg z D .2 C 1/=3; – #.0; w/ D #.4z; w/ D 12 ; 1 – the possible asymptotic expansions are w  z 2  z1 ˙    . Solution D.z; w/ D z4 w.w  4z/..1  z C 2z2 /w C 2z2 /2 , w D

t2 0 tz , t

D z  t2 .

Resume There is a clear distinction between the solutions to equations of genus zero and genus one. In the first case the poles are arranged in finitely many strings along regularly distributed rays, while in the second case the poles locally form a lattice and globally ‘fill’ the plane. The trigonometric and elliptic functions, solutions to certain Briot–Bouquet equations P.w; w0 / D 0, are prototypal.

Chapter 6

Higher-Order Algebraic Differential Equations

In this chapter we will extend the investigations of the previous chapter to secondorder algebraic differential equations and two-dimensional Hamiltonian systems whose solutions are meromorphic functions. Having established the Painlevé property for distinguished equations and systems, we will draw a comprehensive picture of the solutions. This includes detecting and describing the distribution of zeros and poles, zero- and pole-free regions, and asymptotic expansions on pole-free regions, and characterising the so-called sub-normal solutions. As in the preceding chapter, a crucial role is played by the method of Yosida Re-scaling. It establishes the central discovery that the first, second, and fourth Painlevé transcendents belong to the Yosida classes Y 1 ; 1 , Y 1 ; 1 , Y1;1 , respectively. 2 4

2 2

6.1 Introduction Higher-order algebraic differential equations ˝.z; w; w0 ; : : : ; w.n/ / D 0

(6.1)

and systems x0 D P .z; x1 ; : : : ; xn /

.1    n/

(6.2)

all of whose solutions admit unrestricted analytic continuation up to ‘fixed singularities’ are said to have the Painlevé property. In the absence of fixed singularities the solutions are meromorphic functions. In most cases the evidence of the Painlevé property follows from Painlevé’s Theorem 1.3.9, applied to the equation under consideration itself or to some often ingeniously transformed equation or system. This will exemplarily be discussed in Sect. 6.2. For distinguished equations and © Springer International Publishing AG 2017 N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, DOI 10.1007/978-3-319-59800-0_6

173

174

6 Higher-Order Algebraic Differential Equations

systems having the Yosida property we will determine the value distribution of generic and exceptional solutions.

6.1.1 Painlevé Tests As a rule, within any proper class of nonlinear algebraic differential equations and systems, the equations having the Painlevé property are still the exception. To sort the wheat from the chaff there are several Painlevé tests in common, which provide necessary conditions for the validity of the Painlevé property. A simple version is to test whether or not for all (or countably many) p there P j exists some (formal) solution 1 jDk cj .z  p/ with pole at p (ck ¤ 0; cj D cj . p/). Equations and systems with this property are said to pass the Painlevé test for poles. Usually it is not hard to determine the principal terms ck .z  p/k from the differential equation. For example, the meromorphic solutions (if any) to w00 D a.z/ C b.z/w C 2w3 with polynomial coefficients have simple poles with residues 1 or 1. The differential equation passes the Painlevé test if and only if a0 .z/ D b00 .z/  0 (cf. Example 3.1). Exercise 6.1 Realise the Painlevé test for – the Hamiltonian system x0 D Hy .z; x; y/, y0 D Hx .z; x; y/ with Hamiltonian H.z; x; y/ D 13 .x3 C y3 / C c.z/xy C b.z/x C a.z/y; a, b, and c polynomials. Solution. a0 D b0 D c00 D 0 [99, 100] is necessary and sufficient. – w00 C 3ww0 C w3 C z.w0 C w2 / D 0 (special case of Eq. (VI). in [94]). – w00 C 3ww0 C w2 C zw  z D 0. Example 6.1 In Painlevé’s fourth equation w00 D

w02 ˇ C 32 w3 C 4zw2 C 2.z2  ˛/w C 2w w

(IV)

w D 0 may be singular. One way to check this is to apply the Painlevé pole test to the differential equation for v D 1=w. The direct and natural way is to prove or disprove that there are always (formal) solutions w D c1 .z  q/ C c2 .z  q/2 C    . This could be called the Painlevé test for zeros. Actually, Eq. (IV) passes the Painlevé test for zeros; the coefficient c2 D 12 w00 .q/ remains undetermined and is completely 0 2 free, while p c1 D w .q/ is restricted to c1 C 2ˇ D 0. Prescribing q, w.q/ D 0, 0 00 w .q/ D 2ˇ, and w .q/ determines a unique solution.

6.1 Introduction

175

6.1.2 The Painlevé Story Painlevé [134, 135] classified the second-order differential equations w00 D R.z; w; w0 /

.R rational in w; w0 /

(6.3)

without movable singularities other than poles. We will not make use of the terms ‘fixed’ and ‘movable’ singularity, and thus will not give a precise definition. To give an idea, suppose the solution defined by initial values w0 D w.z0 / and w00 D w0 .z0 / has a singularity at z . Then solutions with initial values close to w0 and w00 likewise have a singularity of the same type in some neighbourhood of z : the singularity 02 ‘moves’ with the initial values. For example, w00 D w2w1 is solved by w D 2 C1 w 1

tan log.z  c/, with poles z D c C e.kC 2 / and a logarithmic singularity at z D c; all singularities ‘move’ with c. Like many other stories, ours does not tell the whole truth, since Painlevé did not detect the corresponding class of differential equations in its entirety. At any rate, he and subsequently Fuchs, Gambier, and others established a list of 50 canonical differential equations which should have this property. The list in [94] starts with I. w00 D 0 and ends up with the master equation L., which is now known as Painlevé’s equation (VI). Actually 44 equations in this list may be reduced to known equations which have this property, such as linear and Riccati equations, equations for elliptic functions, etc., or to one of the remaining six Eqs. (I)–(VI). For example, 2ww00 D w02 C 4aw3  2zw2  1

XXXIV.

(the numbering follows Ince [94]) may be reduced to v 00 D 2v 3 C zv  a  aw D v 0 C v 2 C 12 z if a ¤ 0, and to w000 C 2zw0 C w D 0 if a D 0.

1 2

via

6.1.3 The Painlevé Transcendents Painlevé’s equations (I)–(VI) form two subgroups: (III), (V), and (VI), which have fixed singularities at z D 0, and z D 1 additionally for (VI), and w00 D z C 6w2

(I)

w00 D ˛ C zw C 2w2

(II)

2ww00 D w02 C 3w4 C 8zw3 C 4.z2  ˛/w2 C 2ˇ

(IV)

without fixed singularities. The Painlevé property will be verified in the next section. Any transcendental solution to one of these equations is called Painlevé

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6 Higher-Order Algebraic Differential Equations

transcendent, more precisely, first, second, and fourth transcendent.1 Our treatment follows the classical ideas until the method of Yosida Re-scaling enters the stage. It will turn out that the respective first integrals W W w02 D 4w3 C 2zw  2W;

W0 D w

w02 D w4 C zw2 C 2˛w  W; w02 D w4 C 4zw3 C 4.z2  ˛/w2  2ˇ  4wW;

(I0 )

W 0 D w2

(II0 )

W 0 D w2 C 2zw

(IV0 )

are even more interesting than the transcendents itself. We note that Painlevé’s equations may be treated in a quite different way, based on the Riemann–Hilbert method and the method of isomonodromic deformations. This approach is strongly linked to the fields of Special Functions and Mathematical Physics and differs from ours concerning methods and issues. The reader is referred to the monograph by Fokas, Its, Kapaev, and Novokshënov [44]. Exercise 6.2 For first, second, and fourth transcendents compute in detail the first terms of the formal series expansions about poles: 1 w D .z  p/2  10 p.z  p/2  16 .z  p/3 C h.z  p/4 C    1 1 p.z  p/3 C    W D .z  p/  14h  30

w D .z  p/1  16 p.z  p/  14 .˛ C /.z  p/2 C h.z  p/3 C    7 2 p  13 p.z  p/ C    W D .z  p/1 C 10h  36

(I&I0 )

(II&II0)

w D .z  p/1  p C 13 . p2 C 2˛  4/.z  p/ C h.z  p/2 C    W D .z  p/1 C 2h C 2.˛  /p C 13 . p2 C 4˛  2/.z  p/    (IV&IV0 ) ( D ˙1). All equations have one degree of freedom: the coefficient h D h. p/ remains undetermined and may be prescribed. Exercise 6.3 Prove that Painlevé’s equations (I) and (II) pass Painlevé test for P Pthe 1 n2 n1 poles. Write z D t C p, w D 1 a t .a D 1/ and w D .b20 D 1/ 0 nD0 n nD0 bn t to obtain .n2  5n  6/an D .n2  3n  4/bn D

n2 P kD2 P

6ak ank

.n  7/;

2bj bk b` C pbn2 C bn3 .n  5/;

jCkC`D n j; k; ` < n

1 The original meaning of transcendent was different, namely: The solutions are transcendental functions of the ‘two constants of integration’.

6.1 Introduction

177 3

respectively. Assuming jpj > 1 and ja6 j  Kjpj 2 resp. jb4 j  Kjpj2 , prove that 1

n

jan j  M n 2 jpj 4

1

n

and jbn j  M n 2 jpj 2

.n  1/

hold for some M D M.K/.

6.1.4 Hamiltonian Systems Painlevé’s equations are strongly connected with certain Hamiltonian systems x0 D Hy .z; x; y/; y0 D Hx .z; x; y/: This will be discussed in several exercises. Exercise 6.4 Consider the Hamiltonian system with solution .x; y/ for H.z; x; y/ D yx2  12 zy  12 y2 C .˛  12 /x and let p be a pole of x. Prove that p is simple, with residue either 1 and p is a zero of y, or else with residue 1 and p is a double pole of y. Prove that x and y separately solve x00 D ˛ C zx C 2x3 2yy00 D y02  4y3  2zy  .˛  12 /2

(Painlevé (II)) (related to XXXIV. in [94]):

For .z/ D H.z; x.z/; y.z// prove 002  403 C 2z02 C 20 D 14 .˛  12 /2 . Exercise 6.5 (Continued, see [60], p. 100) For u D x C

˛

1 2

x0  x2  12 z

.˛ ¤ 12 /

(6.4)

deduce u0 C u2 C 12 z D y and u00 D ˛  1 C zu C 2u3 . In other words, the Bäcklund transformation (6.4) transforms Painlevé’s equation (II)˛ into (II)˛1 . Exercise 6.6 For H.z; x; y/ D xy2 C yx2 C 2zxy C 2x C 2y, hence x0 D 2xy  x2  2zx  2; y0 D 2xy C y2 C 2zy C 2; prove that x and y separately solve 2xx00 D x02 C 3x4 C 8zx3 C 4.z2 C   2/x2  4 2 2yy00 D y02 C 3y4 C 8zy3 C 4.z2 C   2/y2  42

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6 Higher-Order Algebraic Differential Equations

(Painlevé (IV)). In other words, the Bäcklund transformations yD

x0 C 2 C 2zx C x2 2x

and x D

y0  2  2zy  y2 2y

transform one Painlevé (IV) equation into the other. The system x0 D y2  zx  ˛; y0 D x2 C zy C ˇ

(IV)

with time-dependent Hamiltonian H.z; x; y/ D 13 .x3 C y3 / C zxy C ˇx C ˛y;

(6.5)

was recently discovered by Kecker [99, 100]. It admits trivial and non-trivial Bäcklund transformations M! W .x; y; ˛; ˇ/ 7! .!x; !y; N !˛; !ˇ/ N .! 3 D 1/ and 8 !˛  !ˇ N C1 ˆ ˆ x.z/  !N ˆ ˆ < !x.z/ C !y.z/ N z !˛  !ˇ N C1 B! W .x; y; ˛; ˇ/ 7! y.z/ C ! ˆ ˆ !x.z/ C !y.z/ N z ˆ ˆ : .!ˇ  !; N !˛ N C !/

(6.6)

The poles of solutions .x; y/, if any, have residues .!; !/. N Remark 6.1 In [100] systems with more general Hamiltonians H.z; x; y/ D an0 .z/xn C a0m .z/ym C

X

a .z/x y

nCm 1) is a pole with residues .!; !/, N hence x.z/ D P1 1 n1 n1 a t and y.z/ D b t hold with z D p C t; a D !; and b0 D !: N 0 nD0 n nD0 n Derive the nonlinear recursion .n  1/an  2!b N nD 2!an C .n  1/bn D

n1 P kD1 n1 P

bk bnk  pan1  an2 .n  3/: ak ank C pbn1 C bn2

kD1

Then a1 ; b1 , a2 , and b2 may be computed explicitly by solving 2!a1 D p!; N 2!b N 1D p!, a2  2!b N 2 D b21  pa1  a0  ˛; and 2!a2 C b2 D a21 C pb1 C b0 C ˇ; while a3 and b3 remain undetermined (cf. Exercise 6.7), but satisfy ja3 j C jb3 j D O.jpj3 / (this will be proved in Lemma 6.5). Assuming 1

jan j < M n 2 jpjn

1

and jbn j < M n 2 jpjn

(6.7)

for 1  n  3, where M is independent of p, prove that (6.7) also holds for n  4 if M > 1 is chosen sufficiently large. (Hint. Estimate the right-hand side and use Cramer’s rule.)

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6 Higher-Order Algebraic Differential Equations

6.1.5 Theorems of Malmquist Type The Painlevé property for Eq. (6.3) means that every solution admits analytic continuation in C n S, where S denotes the set of ‘fixed singularities’. If S is empty, every solution is meromorphic on the plane. On the other hand, Theorems of Malmquist type require just one admissible resp. transcendental solution (in case of rational coefficients). One obstacle for theorems of Malmquist type is that meromorphic solutions that also solve some first-order algebraic differential equation P.z; w; w0 / D 0 have to be excluded. Accounting for the complexity of (6.3), the fact that the output should contain at least 50 normalised equations, and the above mentioned side condition, the problem seems inaccessible. However, the first step in Painlevé’s analysis leads to equations w00 D L.z; w/w02 C M.z; w/w0 C N.z; w/;

(6.8)

where L, M, N are rational functions with respect to w. It is thus quite natural to start with equations of this type, assuming that L, M, and N are rational in both variables and (6.8) has a transcendental meromorphic solution that does not satisfy any firstorder algebraic differential equation. A first general result concerning the degrees of L; M and N w.r.t. w is as follows: If (6.8) is written in lowest terms as D.z; w/w00 D C.z; w/w02 C B.z; w/w0 C A.z; w/ with polynomials A; B; C; D w.r.t. w, then degw D  4; degw .wC  2D/  3; degw B  4; and degw A  6 holds; the bounds are sharp, see [170] and also the next example. Q Example 6.2 Let v 6 const satisfy v 02 D 4jD1 .v  cj / (cj ¤ ck for j ¤ k). Then w.z/ D 1=.v.ez / C z/ satisfies some Eq. (6.8) with coefficients L.z; w/ D

Pw .z; w/ Pz .z; w/ w2 Pz .z; w/ ; M.z; w/ D 1 C ; N.z; w/ D w2 C ; 2P.z; w/ P.z; w/ 2P.z; w/

Q and P.z; w/ D 4jD1 ..z C cj /w  1/: Since w has infinite order of growth it cannot satisfy some first-order algebraic differential equation. If additional conditions are imposed on the coefficients L, M, and N, the state of the art is as follows. 1. w00 D L.z; w/w02 is impossible [167]. P 2. w00 D N.z; w/ implies degw N  3. If N.z; w/ D 3jD0 aj .z/w j is a polynomial, the Painlevé test for poles leads to resonance conditions for the coefficients ([201]). If a3 is a non-zero constant resp. a3  0 and a2 is a non-zero constant,

6.1 Introduction

181

these conditions lead to equations with constant coefficients or Painlevé’s second resp. first equation as canonical representatives. 3. w00 D M.z; w/w0 C N.z; w/, where M 6 0 and N are polynomials w.r.t. w. Necessary conditions are degw M  1 and degw N  3. In case of degw M D 1 there are two canonical equations: .i/ w00 C 2ww0 D A.z/ C .B.z/2  B0 .z//w C B.z/w2 ; .ii/ w00 C 3ww0 D A.z/ C B.z/w C C.z/w2 C D.z/w3 .D 6 0/: The first equation is equivalent to the weakly coupled system u0 D A.z/ C B.z/u; w0 D u  B.z/w  w2 I if A and B are polynomials, then u is entire and w is meromorphic [102, 163]. The second equation (also [163]) with D.z/  1 is closely related to equation VI. in the list of 50 equations in [94]. Remark 6.3 There are three canonical equations W 00 D M.z; W/W 0 C N.z; W/

.M 6 0/

in the list of 50 equations in [94]. Adapted to our preferred form (transformation w D W C .z/ to obtain M.z; w/ D kw, k D 1; 2; 3) these are w00 C 2ww0 D 12 .qq0  q00 / 00

0

3

00

.hence w0 C w2 D 14 q2  12 q0 C const/: 2

0

3

w C 3ww D .2q  q / C 3.q  q /w  w : w00 C ww0 D 12q0  12qw C w3 :

V. VI. X.

In V. and VI., q is an arbitrary meromorphic function, while in X., q is a solution to some equation q00 D 6q2 C  with  D 0,  D 1, or  D z (Painlevé (I)). It is apparent that none of the Eqs. V., VI., and X. covers Eq. 3.(i) with C 6 0. Also the 0 0 2 2 3 equation w1 w001 D 2w02 1  2w1  C.z/w1 C .C .z/  C.z/ /w1  A.z/w1 , which is obtained from 3.(i) by the transformation w1 D 1=w, cannot be reduced to one of the remaining 40 equations with L 6 0.2 4. w00 D M.z; w/w0 CN.z; w/, where at least one of M 6 0 and N is not a polynomial w.r.t. w. There is only one canonical equation, ww00 D Œ1 C Aw C 3Cw2 w0  1  Aw C Bw2 C .C0  AC/w3  C2 w4

(LI.)

.A; B; C rational, C 6 0), equivalent to the weakly coupled Riccati system u0 D B C 2C C Au  u2 ; w0 D 1 C uw C Cw2 :

2 This could be a hint that the list of 50 is incomplete. We note, however, that in [94] and other texts like [58], ‘many-valued’ transformations and solutions are on the agenda, while here only meromorphic solutions and rational transformations are admitted.

182

6 Higher-Order Algebraic Differential Equations

Remark 6.4 Meromorphic solutions to LI., if any, have finite order of growth [165]. For the proof just set v D 1=w to obtain vv 00 D 2v 02 C .3C C Av C v 2 /v 0 C C2 C .AC  C0 /v  Bv 2 C Av 3 C v 4 ; which can be written as v 4 D ˝.z; v; v 0 ; v 00 / with degree d˝ < 4 and weight d˝ D 4. Since v has simple poles with residue 1 (w D 0 implies w0 D 1), Theorem 3.4 applies. Now assume that A, B, and C are polynomials. Then u is transcendental meromorphic with infinitely many simple poles with residue 1, and these poles may cause difficulties: w is meromorphic if and only if the poles of u are zeros of v with v 0 D C=2. Problem. Determine all polynomials A; B; C such that Eq. LI. has meromorphic solutions or even the Painlevé property. The proofs of the Malmquist-type theorems are too elaborate and involved to be presented here; the interested reader is referred to the literature cited above. We will just discuss the case w00 D N.z; w/ D P.z; w/=Q.z; w/: From Valiron’s Lemma and T.r; w00 /  3T.r; w/ C S.r; w/ it follows that degw N D maxfdegw P; degw Qg  3. We want to show that Q is independent of w. To this end we assume degw Q D q  1, and note that Q.z; w.z// can have only finitely many zeros (they are also zeros of the resultant of P and Q w.r.t. w). By the Tumura–Clunie Theorem for Q.z; w.z//, the algebraic equation Q.z; !/ D 0 can have at most two mutually distinct solutions. If this number is two, w solves some Riccati equation w0 D a.z/ C b.z/w C c.z/w2 with ‘small’ coefficients, again by the Tumura–Clunie Theorem, and equating coefficients shows that a, b, and c are rational functions. Otherwise we may assume that Q.z; w/ D wq with 1  q  3, hence we have to consider the differential equation wq w00 D P.z; w/ .degw P  3; P.z; 0/ 6 0/ having a transcendental solution with finitely many zeros. This, however, contradicts m.r; 1=w/ D S.r; w/, which easily follows from P.z; 0/ 6 0.

6.2 The Painlevé Property The step from Painlevé tests to the Painlevé property is enormous and realised only for a few differential equations and systems. In this section we will show, representative for other equations, that System (IV), hence also Painlevé (IV), and Painlevé’s equations (I) and (II) have this property. Whenever Painlevé’s Theorem fails, the argument is more or less the same; we refer to [58, 88– 90, 99, 100, 132, 151, 152, 175]. For example, the pattern to prove that an arbitrary polynomial system x0 D P.z; x; y/ D

X

ajk .z/xj yk ;

y0 D Q.z; x; y/ D

X

bjk .z/xj yk

(6.9)

6.2 The Painlevé Property

183

has the Painlevé property is as follows. Suppose .x; y/ is a meromorphic solution on jzj < R with singularity at z0 D Rei . Then jx.z/j C jy.z/j ! 1 holds as z ! z0 on Œ0; z0 /;

(6.10)

and the goal is to prove that z0 is a pole. To reach the goal one tries to determine a bi-rational transformation u D R.z; x; y/; v D S.z; x; y/; x D R1 .z; u; v/; y D S1 .z; u; v/; such that Painlevé’s Theorem applies to the transformed polynomial system O u; v/; v 0 D Q.z; O u; v/: u0 D P.z; To this end one tries to construct a so-called Lyapunov function ˚.z/ D V.z; x.z/; y.z// that is bounded on Œ0; z0 /; usually this is implied by some differential inequality j˚ 0 j  A C Bj˚j:

(6.11)

The procedure, if any, is not unique. As a rule, the rational functions R, S, and V have to be constructed in such a way that they remain analytic at the poles of x D x.z/ and y D y.z/. The proof that ˚ is bounded requires some side conditions like jx.z/=y.z/j  K on Œ0; z0 /. If this is not the case, Œ0; z0 / has to be deformed into some path from 0 to z0 on which these side conditions hold. In order that a bound for ˚ can be derived from (6.11) it is required that is rectifiable.

6.2.1 The Painlevé Property for System (IV) and Painlevé (IV) For System (IV) we will now implement the details in the above pattern. Theorem 6.1 The solutions to System (IV) are meromorphic in the plane. Proof Suppose .x; y/ is a solution to System (IV) on the disc D W jzj < R, but has a singularity at z0 D Rei on the boundary. Then (6.10) holds. Following [99] we consider the Lyapunov function ˚.z/ D H.z; x.z/; y.z// C x.z/2 =y.z/

H.z; x; y/ C x2 =y and also H.z; x; y/  y2 =x are regular at every (common) pole.

(6.12)

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6 Higher-Order Algebraic Differential Equations

to obtain ˚ 0 C 3.q.z/=y.z//˚ D 3q.z/3 C 2ˇq.z/2 C ˛q.z/ with q.z/ D x.z/=y.z/: Suppose for the moment that jx.z/j jy.z/j

is bounded on Œ0; z0 /:

(6.13)

  Then also 1=jy.z/j D O .jx.z/j C jy.z/j/=jy.z/j D O.jq.z/j C 1/ is bounded, hence (6.11) holds and ˚ is bounded. This implies jx.z/3 C y.z/3 j D O.jx.z/j2 / C O.jy.z/j2 / and y.z/  !x.z/ as z ! z0 for some third root of unity; we may assume ! D 1. Now H.z; x; y/ C x2 =y with y  x as x ! 1 can only be bounded if y D x C z C .˛ C 1  ˇ/=x C O.1=jxj2/

.x ! 1/:

Then u D 1=y tends to 0 and v D y.˛ C 1  ˇ C y.x C y  z// remains bounded as z ! z0 on Œ0; z0 /, and Painlevé’s Theorem applies to the polynomial system‘ u0 D P.z; u; v/; v 0 D Q.z; u; v/I u, v, y D 1=u and x D vu2  .˛ C 1  ˇ/u C z  1=u admit analytic continuation along Œ0; z0 , u has a zero and x a pole at z0 . It is, however, quite unlikely that (6.13) holds on Œ0; z0 /. On the other hand, the above argument works if (6.13) holds on some rectifiable path joining 0 to z0 in D [ fz0 g. For the existence of , which finishes the proof, see Lemma 6.1. Lemma 6.1 There exists some rectifiable path joining 0 to z0 in D [ fz0 g on which x=y is bounded. Proof The deformation of Œ0; z0 / into usually requires quite technical procedures. We will give a proof using a simple re-scaling technique, which also works for general polynomial systems. We assume that (x; y/ solves System (IV) on jzj < R and (6.10) holds, and set E D f 2 Œ0; z0 / W jy./j  12 jx./jg. We may assume that E accumulates at z0 since otherwise we are done. Then x./ ! 1 as  ! z0 on E, and E contains no pole. For  2 E set x.z/ D x.zI ; / D x. C z/ and

 Substituting y D x C z C .˛ C 1  ˇ/=x C C=x2 C    , where v is viewed as a function of the complex variables x, y, and z gives v D .˛ C 1  ˇ/z C C C O.jxj1 / as x ! 1. ‘ P.z; u; v/ D 1 C zu  .˛ C 1 C C z2 /u2 C 2z u3  2 u4 C 2vu3  2zvu4 C 2 u5  v 2 u6 and Q.z; u; v/ D  .˛ C 1 C z2 / C 2 2 zu  zv C 2.˛ C 1 C C z2 /vu  3 u2  6 zvu2  3v 2 u2 C 4 2 vu3 C 4zv 2 u3  5 v 2 u4 C 2v 3 u5 with D ˛ C 1  ˇ are obtained with the help of MAPLE as follows (x1, y1, v1 symbolise x0 ; y0 ; v 0 ; 10 is just a large number to estimate the total degrees of P and Q): beta:=alpha+1-gamma; x1:=-yˆ2-zx-alpha; y1:=xˆ2+zy+beta; y:=1/u; x:=vuˆ2-(alpha+1-beta)u+z-1/u; P:=mtaylor(-y1uˆ2,[u,v],10); Q:=mtaylor(solve(x1-diff(x,z)-diff(x,v)v1-diff(x,u)P,v1),[u,v],10).

6.2 The Painlevé Property

185

y.z/ D y.zI ; / D y. C z/ to obtain the system x0 D y2  . C z/x C ˛2 ; y0 D x2 C . C z/y C ˇ2 with initial values x.0/ D x./, y.0/ D y./, which depends on the parameters  2 E and . For  D 1=x./ we obtain by taking the limit  ! z0 x0 D y2 ; y0 D x2

.x.0/ D 1; jy.0/j  12 /:k

The solutions are elliptic functions which parametrise the algebraic curve  3 C 3 D 1 C y.0/3 and depend analytically on y.0/. Thus given r > 0 sufficiently small there exists a K > 1 such that jx.z/=y.z/j  K holds on jzj D r, uniformly w.r.t. y.0/, jy.0/j  12 : In other words, the family .x ; y / 2E with x .z/ D x.zI ; 1=x.// and y .z/ D y.zI ; 1=x.// is normal on the disc jzj < 2r, say, and jx .z/=y .z/j  2K holds on jzj D r for  2 E0 D E \ f W 0  jj < Rg. The discs jz  j  rj./j are contained in jzj < R, since otherwise z0 would not be a singularity. For  2 E0 this implies jx.z/=y.z/j  2K on jz  j D rj./j: ToSobtain a curve of finite length on which x=y is bounded we set C D Œ0 ei ; z0 / [  2E0 fz W jz  j  rj./jg: The boundary of C consists of two curves ˙ (envelopes), symmetric to Œ0; z0 / and of length at most R, and jx.z/=y.z/j is bounded on ˙ ; also ˙ terminates at z0 since x./ ! 1 as  ! z0 on E; zeros of y on Œ0; 0 ei / may be avoided by small semi-circles, and poles are irrelevant since jx=yj D 1 holds at every pole. Remark 6.5 Lemma 6.1 can be adapted to polynomial systems (6.9) as follows. Suppose one has to prove that x=y is bounded on some rectifiable path , where jx.z/jCjy.z/j ! 1 as z ! z0 on Œ0; z0 / is assumed. Set ˛jk D ajk .z0 /, ˇjk D bjk .z0 /, I D f. j; k/ W .ajk .z/; bjk .z// 6 .0; 0/g, E D f 2 Œ0; z0 / W jy./j  12 jx./jg, 1 p1 p D maxI . jCk/,  D x./ D x. Cz/, P ,  D  j k , x.z/ P and y.z/ D y. Cz/ 0 Q y/ D jCkDp ˛jk x y ; y0 D Q.x; Q y/ D jCkDp ˇjk xj yk as  ! z0 to obtain x D P.x; Q on E,  ! 0, with x.0/ D 1 and jy.0/j  12 : We just need Q.1; 0/ D ˇp;0 ¤ 0 to p1 obtain jx.z/=y.z/j  K on each circle jz  j D rj./j ( 2 E0 , r fixed). From Theorem 6.1 we obtain Theorem 6.2 The solutions to Painlevé (IV) are meromorphic. Proof By v D x C y  z, System (IV) is transformed into Eq. .Iƒ/ on p. 179, and every solution is obtained this way. Hence the solutions are meromorphic in the plane, and so are the solutions to Painlevé (IV) when written as 2ww00 D w02 C 3w4 C 8zw3 C 4.z2  (in .Iƒ/ set w.z/ D av.bz/ with b D k

pi

3

.˛ C ˇ//w2 C 49 .1 C ˛  ˇ/2

(IV00 )

p 4 4=3 and a D  12 b3 ).

The point z0 is ‘blown up’ to a finite circle jzj < r. This is a common technique in differential equations, which also forms the basis of the Zalcman- and Yosida Re-scaling and Painlevé’s socalled ‘˛-method’. Nihil novi sub sole [there is nothing new under the sun].

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6 Higher-Order Algebraic Differential Equations

6.2.2 The Painlevé Property for Painlevé (I) and (II) Theorem 6.3 The solutions to Painlevé (I) and Painlevé (II) are meromorphic. Proof Let w be any solution to Painlevé (I) and (II), respectively, defined in some neighbourhood of the origin, say. If w is assumed not to be meromorphic in C then there exists a circle jzj D R such that w is meromorphic inside, but has a singularity at z0 on that circle. Then jw.z/j C jw0 .z/j ! 1

as z ! z0 on Œ0; z0 /:

(6.14)

The functions ˚ D 2W w0 =w and ˚ D W w0 =w, where W denotes the respective first integral, are regular at poles of w and will serve as Lyapunov functions; they satisfy ˚0 C

˚ z  q.z/ D 2 w w

and ˚ 0 C

˚ ˛  q.z/ D 2 w w

.q D w0 =w2 /;

respectively. If q is unbounded on Œ0; z0 / we have to show in both cases that Œ0; z0 / can be deformed in fz W jzj < Rg [ fz0 g into some rectifiable path joining 0 to z0 , on which q is bounded; then also 1=jwj2 D O..jw0 jCjwj2 /=jwj2 / D O.jq.z/jC1/, a p fortiori 1=jwj and also ˚ is bounded.p In the first case we set ./ D 1= 3 w0 ./ and w.z/ D 2 w. C z/, and ./ D 1= w0 ./ and w.z/ D w. C z/ in the second, and assume that  2 E D f W jw./j2  14 jw0 ./jg accumulates at z0 . As  ! z0 and ./ ! 0 we obtain w00 D 6w2 with w.0/ D 0; w0 .0/ D 1/, and w00 D 2w3 with jw.0/j  12 ; w0 .0/ D 1/; respectively. In both cases we obtain jw0 .z/j=jw.z/j2  K on some circle jzj D r independent of w.0/, and jw0 .z/j=jw.z/j2  2K, say, holds on jz  j D rj./j. Just as in the proof of Lemma 6.1 this permits us to deform Œ0; z0 / into some rectifiable path on which ˚ is bounded. Moreover, not only does (6.14) hold, but w.z/ even tends to infinity as z ! z0 on : In the second case we set u D 1=w to obtain .u0  12 u3 /2 D 1 C zu2 C 2˛u3  ˚.z/u4 C 14 u6 from w02 D w4 C zw C 2˛w  .˚ C w0 =w/. Since u ! 0 as z ! z0 and ˚ is bounded on , u0 tends to 1, say, and u0 D 1 C 12 zu2 C .˛ C 12 /u3 C vu4 D P.z; u; v/ holds, where v D  12 ˚ C O.juj/ is bounded and u tends to zero as z ! z0 . In combination with w00 D ˛ C zw C 2w3 we obtain v 0 D  12 .˛ C 12 /z  .˛ C 12 /2 u  zuv  3.˛ C 12 /vu2  2v 2 u3 D Q.z; u; v/:

6.2 The Painlevé Property

187

Painlevés Theorem applies to the polynomial system u0 D P.z; u; v/; v 0 D Q.z; u; v/, and z0 is a pole of w with residue 1 (assuming u0 ! 1 leads to a pole of w with residuep1). In the first case the final proof runs along the same lines. This time, u D 1= w is meromorphic on some neighbourhood of n fz0 g (note that contains no zeros of w, and poles may be avoided by small circular arcs), tends to zero as z ! z0 on , and satisfies u02  12 u5 u0 D 1 C 12 zu4  14 ˚.z/u6 : Again assuming u0 ! 1 as z ! z0 along leads to u0 D 1 C 14 zu4 C 14 u5 C vu6 D P.z; u; v/I the function v D  18 ˚.z/ C O.juj2 / is bounded as z ! z0 on and satisfies 1 2 z u v 0 D Q.z; u; v/ D  16

3 2 16 zu

 zvu3  18 u3  54 vu4  3v 2 u5 :

By Painlevé’s Theorem, u has a simple zero and w has a double pole at z0 . Remark 6.6 The Painlevé property for equation (I) and (II) conceivably may also be deduced from the Painlevé property of the master equation (IV) resp. (VI) by a method called ‘coalescence’. Actually there is a whole cascade of coalescence .VI/ ! .V/ ! .III/ # # .IV/ ! .II/ ! .I/ ([94], p. 346). We will describe the step (II) ! (I). Let w be the local solution to w00 D z C 6w2 ; w.z0 / D w0 ; w0 .z0 / D w00 ;

(I)

 any non-zero complex number, and y the solution to y00 D 415 C xy C 2y3 ; y.x0 / D w0 C 5 ; y0 .x0 / D 1 w00 ;

(II )

where x0 D 2 z0  610 : Clearly y.x/ D y.xI / depends on  and !.z/ D w.zI / D 1 y.xI /  6 ; z D 2 x C 612 ;   solves ! 00 D z C 6! 2 C 6 z! C 2! 3 and has the same initial values as w at z D z0 : Applying the theorem on analytic dependence on parameters, w.z/ D w.zI 0/ D lim!0 w.zI / follows. One has, however, to take care about the accumulation points of the poles of y.xI / as  ! 0, which can hardly be controlled.

188

6 Higher-Order Algebraic Differential Equations

6.3 Algebraic Differential Equations and the Yosida Classes Just as in the first-order case, the distribution of values of the solutions to higherorder differential equations and systems is strongly coupled with the distribution of poles. To determine the Nevanlinna functions, the distribution of poles, and the asymptotic expansions of transcendental solutions, our strategy will be to prove that the solutions belong to certain Yosida classes Y˛;ˇ . There are two possible ways to get an idea of which parameters .˛; ˇ/ are worth considering: to study special solutions, and to apply the so-called Yosida test. Our goal will be to prove the following theorem, a homage to Painlevé and Yosida, and certainly the main result of the whole chapter. Painlevé–Yosida Theorem The solutions to System (IV) and Painlevé (IV) belong to the Yosida class Y1;1 , while first and second Painlevé transcendents belong to Y 1 ; 1 and Y 1 ; 1 , respectively. 2 4

2 2

The proof will be given in Sect. 6.3.4. The step from the Yosida and Painlevé test to the Yosida and Painlevé property is likewise technical and intricate.

6.3.1 Special Solutions Airy Solutions The solutions to the so-called Airy equations w0 D ˙. 12 z C w2 /

(6.15)

also solve Painlevé (II) with parameters ˛ D ˙ 12 . In particular, these solutions belong to Y 1 ; 1 . By w D u0 =u, Eq. (6.15) is transformed into the linear differential 2 2

equation u00 C 12 zu D 0, which is also known as an Airy equation when written as v 00 C zv D 0.

Exercise 6.9 Prove that w00 D ˛ C zw C 2w3 and w0 D a.z/ C b.z/w C c.z/w2 (rational coefficients) simultaneously hold if and only if ˛ D ˙ 12 and w solves some Airy equation (6.15). (Hint. c.z/ D ˙1 is necessary and simplifies matter.) For ˛ ¤ ˙ 12 the Bäcklund transformations (see also Exercise 6.5) w 7! w C

˛

1 2

w0  w2  12 z

and w 7! w 

˛C

1 2

w0 C w2 C 12 z

(6.16)

transform Painlevé’s equation (II) into the same equation with parameter ˛  1 in place of ˛. Any solution to Painlevé (II) that is obtained from some solution to (6.15) by repeated application of (6.16) and the trivial transformations w 7! w.Nz/, w 7! w, and w 7! !w.!z/ with ! 3 D 1 is called an Airy solution.

6.3 Algebraic Differential Equations and the Yosida Classes

189

Exercise 6.10 Prove that Airy solutions have the form w D R.z; w0 /, where R is some rational function and w0 solves w00 D 12 z C w20 , say. Thus w satisfies some first-order algebraic differential equation P.z; w; w0 / D 0 of genus zero and has 3 Nevanlinna characteristic T.r; w/ r 2 : It is obvious (and algebraically trivial) that second transcendents satisfy some first-order algebraic differential equation if and only if their first integrals do. Weber–Hermite Solutions The poles of any solution .x; y/ to System (IV) have residues .!; !/ N with ! 3 D 1. We are interested in the distribution of residues, and, in particular, what will happen if one or even two kinds of residues are missing. Example 6.3 To determine the transcendental solutions to (IV) having just poles with residues .1; 1/ we note that x C y  z is holomorphic and vanishes at every pole of .x; y/ (see Exercise 6.7), hence vanishes identically. Thus x and y separately solve Riccati equations x0 D ˛  z2 C zx  x2

and y0 D ˇ C z2  zy C y2

(6.17)

with ˛  ˇ C 1 D 0. In particular, x and y belong to the Yosida class Y1;1 : The differential equations (6.17) are closely related to the Weber–Hermite equations w0 D ˙ .2zw C w2 /

(6.18)

Any solution to System (IV) that is obtained from some solution to (6.17) by repeated application of the Bäcklund transformations (6.6) combined with the trivial transformations .x; y/ 7! .!x; N !y/ with ! 3 D 1 is called a Weber–Hermite solution. Weber–Hermite solutions have Nevanlinna characteristic T.r; x/ r2 , T.r; y/ r2 ; and x and y separately satisfy first-order algebraic differential equations. Equivalently, x and y are algebraically dependent over C.z/.

6.3.2 The Yosida Test To test whether the solutions to the algebraic differential equation ˝.z; w; w0 ; : : : ; w.n/ / D 0

(6.19)

belong to some Yosida class, real parameters ˛ and ˇ > 1 have to be determined such that h ˝.h; h˛ w0 ; h˛Cˇ w1 ; : : : ; h˛Cnˇ wn /  Q.w0 ; w1 ; : : : ; wn / as h ! 1 holds for some . This formally leads to the autonomous differential equation Q.w; w0 ; : : : ; w.n/ / D 0:

(6.20)

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6 Higher-Order Algebraic Differential Equations

The procedure is far from being unique. A reasonable postulate is not to destroy the ‘structure’ of (6.19). In particular, Eq. (6.20) should also have order n. Moreover, and even more important, ˇ should be as small as possible. Taking this into account the procedure will be called Yosida test. It was already discussed in Sect. 5.4.1 in the context of implicit first-order differential equations. Example 6.4 For Painlevé’s first equation w00 D z C 6w2 the procedure yields w2  h1˛2ˇ  6w20 h˛2ˇ D Q.w0 ; w2 / C o.1/ .h ! 1/; hence 2ˇ  maxf1  ˛; ˛g. The smallest value of ˇ is obtained if ˛ D 1  ˛, thus ˛ D 12 , ˇ D 14 , and w00 D 1 C 6w2 . Question. Does w belong to Y 1 ; 1 ? 2 4

In the context of algebraic differential systems (6.2) it is appropriate to admit different parameters ˛ for different components, but one and the same ˇ. Example 6.5 Consider the Hamiltonian system (see Exercise 6.4) x0 D y C x2 C 12 z; y0 D 2xy  .a  12 / with Hamiltonian H.z; x; y/ D 12 y2  .x2 C 12 z/y  .a  12 /x: It is assumed that x and y are meromorphic (which is, indeed, the case). Replace z; x; x0 , y; and y0 with h; xh˛1 , x1 h˛1 Cˇ , yh˛2 ; and y1 h˛2 Cˇ to obtain x1 h˛1 Cˇ C yh˛2  x2 h2˛1  12 h D o.h˛1 Cˇ /; y1 h˛2 Cˇ C 2xyh˛1 C˛2 C .a  12 / D o.h˛2 Cˇ /: To preserve the structure we need ˇ  maxf˛2  ˛1 ; ˛1 ; 1  ˛1 ; ˛2 g; the smallest value ˇ D 12 is obtained if ˛1 D 12 and ˛2 D 1. With this choice we obtain the autonomous Hamiltonian system x0 D x2  y C 12 , y0 D 2xy with Hamiltonian H.x; y/ D 12 y2  .x2 C 12 /y. Question. Is it true that x 2 Y 1 ; 1 and y 2 Y1; 1 ? 2 2

2

Exercise 6.11 Realise the Yosida test for – w00 D 3ww0  w3  z.w0 C w2 / – w00 C 3ww0 C w2 C zw  z D 0.

(special case of VI. in [94]).

Solution ˛ D ˇ D 1; ˛ D ˇ D 12 . Question. Is w 2 Y˛;ˇ ?

6.3.3 Yosida–Zalcman–Pang Re-scaling The Yosida test as well as the properties of the special solutions indicate that the solutions to (IV) and Painlevé (IV), first and second transcendents belong to Y1;1 , Y 1 ; 1 and Y 1 ; 1 , respectively. To prove this we need a preliminary stage of the 4 2 2 2 method of Yosida Re-scaling. Throughout this section, .x; y/ denotes some fixed

6.3 Algebraic Differential Equations and the Yosida Classes

191

solution to System (IV). For z ¤ 0 not a pole (of x and y) we set r.z/ D minfjzj1 ; jx.z/j1 ; jy.z/j1 g (convention 01 D 1). By Q we denote the set of non-zero zeros of y  z, and set 4ı .z0 / D fz W jz  z0 j < ır.z0 /g and Qı D

S q2Q

4ı .q/:

Given any sequence zn ! 1, the re-scaled functions xn .z/ D rn x.zn C rn z/; yn .z/ D rn y.zn C rn z/

.rn D r.zn //

satisfy x0n D y2n  .rn zn C rn2 z/xn  ˛rn2 ; y0n D x2n C .rn zn C rn2 z/yn C ˇrn2 xn .0/ D rn x.zn /; yn .0/ D rn y.zn /:

(6.21)

Passing to a subsequence, if necessary, we may assume that rn zn ! c, xn .0/ ! x0 , and yn .0/ ! y0 , with maxfjcj; jx0 j; jy0 jg D 1:

(6.22)

Then (6.21) may be considered as a system depending on parameters rn zn , rn2 , and initial values xn .0/ D rn x.zn /, yn .0/ D rn y.zn / on some neighbourhood of .c; 0; x0 ; y0 /. The role of the Zalcman–Pang Lemma (which does not necessarily apply) is taken by the theorem on analytic dependence on parameters and initial values (cf. Sect. 1.3.7). From this theorem it follows that xn ! x and yn ! y, where x and y satisfy the Hamiltonian system x0 D y2  cx; y0 D x2 C cy with

x.0/ D x0 ; y.0/ D y0

(6.23)

and Hamiltonian H.x; y/ D 13 .x3 C y3 / C cxy: We claim that y 6 c. Otherwise the second equation yields x2 C c2  0, while c2 C cx  0 follows from the first equation. This is possible if and only if c D 0, x0 D 0, and y0 D 0, in contrast to (6.22). The non-constant solutions are either elliptic or trigonometric functions, see also the detailed discussion in Sect. 6.4.1. To proceed we need some technical results. Lemma 6.2 jzj C jx.z/j D O.jy.z/  zj/ holds as z ! 1 outside Qı . Proof We first assume y.zn /  zn D o.jzn j/ on some sequence zn ! 1 and obtain the system (6.23) with y0 D c. Since y.z/ 6 c, Hurwitz’ Theorem implies that there exists qn 2 Q such that zn  qn D o.rn /. To prove rn D O.r.qn //, hence zn  qn D o.r.qn // and zn 2 Qı (n  nı /, we set rQn D r.qn / and assume to the contrary that rQn D o.rn / holds (at least on some subsequence). We re-scale along

192

6 Higher-Order Algebraic Differential Equations

.qn /, that is, set xQ n .z/ D rQn x.qn C rQn z/ and yQ n .z/ D rQn y.qn C rQn z/ to obtain limit functions satisfying xQ0 D Qy2  cQ xQ and yQ 0 D xQ2 C cQ yQ , now with cQ D limqn !1 rQn qn D limzn !1 Qrrnn rn zn D 0 and yQ 0 D 0: Again yQ is non-constant with pole , that is, y has some pole pn D qn C o.rn /. This, however, contradicts yn ! y 6 const on the disc jzj < 1, say, which contains a pole pn ! 0 corresponding to pn , and a zero qn ! 0 of yn corresponding to qn . To prove the second assertion we assume jy.zn /  zn j D o.jx.zn /j/ along zn ! 1; the same argument yields (6.23) with y.0/ D y0 D c and zn 2 Qı .n  nı /. Lemma 6.3 For ı > 0 sufficiently small, each disc 4ı .q/ about some q 2 Q contains at most one more q0 2 Q. Proof Let qn 2 Q tend to infinity and assume that qn is accompanied by q0n ; q00n 2 Q, that is, q0n and q00n are contained in 4ın .qn / with ın ! 0. Then the system (6.23) has a solution such that y  c vanishes at the origin at least with multiplicity three. From the second equation (6.23) and y0 .0/ D y00 .0/ D 0 we obtain x20 Cc2 D 2x0 x0 .0/ D 0, while the first equation gives x0 .0/ D c2  cx0 , hence x20 C c2 D 2x0 .c2 C cx0 / D y0  c D 0 and x0 D y0 D c D 0. This again contradicts (6.22). We will use this property of Q to redefine the discs 4ı .q/ as follows: for ı > 0 sufficiently small, 43ı .q/ contains at most one q0 ¤ q. If q0 62 42ı .q/ (or if no such q0 exists) nothing has to be done, since then 4ı .q/ \ 4ı .q0 / D ;. If q0 2 42ı .q/ n 4ı .q/ we replace 4ı .q/ and 4ı .q0 / with 4ı=2 .q/ and 4ı=2 .q0 /, respectively, and finally replace 4ı .q/ [ 4ı .q0 / with 42ı .q/ if q0 2 4ı .q/. Then Q is covered by mutually disjoint discs 4.q/ı .q/ with .q/ varying between 12 and 2, and Lemma 6.2 holds for the ‘new’ Qı ; each disc 4.q/ı .q/ contains at most two zeros of y  z. The main step towards the proof of x; y 2 Y1;1 now consists in estimating the auxiliary function V.z/ D H.z; x.z/; y.z// C

x.z/2 y.z/  z

outside Qı (note the close relationship to the Lyapunov function (6.12)); V has poles at the zeros of y  z and is regular elsewhere; estimating V also means estimating the unknown coefficients h. p/. Exercise 6.12 Prove that V. p/ D 2h. p/ C 13 p3 C 12 .1 C ˛ C ˇ/p holds at every pole with resp x D 1 (and a similar result if resp x D e˙2i=3 ). Differentiating V and arranging the terms appropriately yields V 0 D A.z/ C B.z/V with A.z/ D

(6.24)

x.z2 C ˛/ x.z3 C 3˛z C .1 C 2ˇ C 2z2 /x/ 3x3 C C , B.z/ D 2 yz .y  z/ .y  z/3

3x , x D x.z/ and y D y.z/. Equation (6.24) will be regarded as a linear .y  z/2 differential equation for V with ‘known’ coefficients A and B.

6.3 Algebraic Differential Equations and the Yosida Classes

193

Exercise 6.13 Employ Lemma 6.2 to prove jA.z/j D O.jzj2 /

and jB.z/j D O.jzj1 /

.z 62 Qı /:

We need a better result. Lemma 6.4 Given  > 0 and ı > 0 there exists a K > 0 such that jV 0 .z/j  Kjzj2 C

 jV.z/j jzj

.z … Qı /:

Proof Consider any sequence zn ! 1, zn … Qı , such that jzn j2 D o.jV 0 .zn /j holds. If no such sequence exists, V 0 .z/ D O.jzj2 / holds outside Qı , and we are done. From jB.zn /j D O.jzn j1 /, hence jV 0 .zn /j  O.jzn j2 / C O.jz1 n V.zn /j, it follows that jzn j3 D o.jzn V 0 .zn /j/ D o.jzn j3 C jV.zn /j/, hence jzn j3 D o.jV.zn /j/ D o.jy.zn /  zn j3 /I the latter follows from the definition of V and Lemma 6.2. This eventually implies jB.zn /j D o.jzn j1 /

and jV 0 .zn /j D o.jzn j1 /jV.zn /j:

Lemma 6.5 For ı > 0 sufficiently small, V.z/ D O.jzj3 / holds outside Qı , and, in particular, h. p/ D O.jpj3 /. Proof The estimate of V via some differential inequality is strongly reminiscent of Gronwall’s Lemma. Let z0 … Qı , z0 ¤ 0 be fixed and set M.r/ D maxfjz3 V.z/j W jz0 j  jzj  r; z … Qı g: The maximum is attained at some point zr , and M.r/ and jzr j increase with r. We may assume that jzr j ! 1 as r ! 1, since there is nothing to do otherwise. We join z0 and zr by some path r in fz W jz0 j  jzj  jzr j; z … Qı g of length at most Ljzr j, where L is a universal constant (for example, L D 2); this is possible since Qı consists of mutually disjoint discs. Then Z  jV.t/j  jdtj Kjtj2 C  M.r/jzr j D jV.zr /j  jV.z0 /j C jtj r Z  jV.z0 /j C KLjzr j3 C M.r/ jtj2 jdtj 3

r

 jV.z0 /j C KLjzr j3 C M.r/Ljzr j3 holds, hence M.r/ is bounded (choose  < 1=L). Exercise 6.14 It follows from Exercise 6.3 and h. p/ D O.jpj3 / that for ı > 0 sufficiently small the discsR jz  pj < ıjpj1 are mutually disjoint. By a simple r geometric argument prove 1 t2 dn.t; P/ D O.r2 / and T.r; x/ D O.r4 /:

194

6 Higher-Order Algebraic Differential Equations

We will now sketch the essential steps in the above argument for Painlevé’s first and second equation, leaving the details to the interested reader. 1

1

3

1. Given any sequence zn ! 1, set rn D r.zn / D minfjzn j 4 ; jw.zn /j 2 ; jw0 .zn /j 4 g 1 1 in the first case, and rn D r.zn / D minfjzn j 2 ; jw.zn /j 2 ; jw0 .zn /j1 g in the second, and define sequences wn .z/ D rn2 w.zn C rn z/ resp. wn .z/ D rn w.zn C rn z/ to obtain limit functions w D limn!1 wn (take subsequences if necessary) satisfying w00 D c C 6w2 c D limn!1 rn4 zn

w00 D cw C 2w3 c D limn!1 rn2 zn :

resp:

p We note that w 6 0 resp. w 6 ˙ c follows from maxfjcj; jw.0/j; jw0 .0/jg D 1. 2. Let Q denote the set of non-zero zeros of w and w2  z, respectively. Then for ı > 0 sufficiently small, Q may be covered by countably many mutually disjoint discs 4ı .q/ D fz W jz  qj < ı.q/r.q/g . 12  .q/  2/; and jzj D O.jwj2 /; 3 jw0 j D O.jwj 2 / resp. jzj D O.jw2  zj/; jw0 j D O.jw2  zj/ hold outside Qı D S ı q 4 .q/: w0 2ww0  1 3. The functions V D 2W  and V D 2W  , where W denotes the w w2  z respective first integral, are regular at the poles of w, and satisfy V0 D

zw.z/2  w0 .z/ 1  V w.z/3 w.z/2 0

2

and V 0 D A.z/ 

respectively, with A.z/ D .2zw 6˛z /wC.26z 4. Given  > 0 there exists a K > 0 such that 1

jV 0 j  Kjzj 2 C 

jVj jzj

z C w.z/2 V; .w.z/2  z/2

3 /w2 C.4˛z6w0 /w3 C4z2 w4 C2˛w5 C2zw6

.w2 z/3

resp: jV 0 j  Kjzj C 

:

jVj jzj

3

holds outside Qı ; this implying V.z/ D O.jzj 2 / resp. V.z/ D O.jzj2 / outside Qı . 3 In particular, h. p/ D O.jpj 2 / resp. h. p/ D O.jpj2 / holds by Exercise 6.15 Let w be any first resp. second Painlevé transcendent. Prove that 7 2 V. p/ D 28h. p/ resp. V. p/ D 10h. p/  36 p holds at every pole p. In combination with Exercise 6.3 deduce Rr 1

1

t 2 dn.t; P/ D O.r2 / resp:

Rr 1

t1 dn.t; P/ D O.r2 /:

6.3 Algebraic Differential Equations and the Yosida Classes

195

6.3.4 Yosida Re-scaling We are now prepared to prove the Painlevé–Yosida Theorem (p. 188) for System (IV), say, and note first that for ı > 0 sufficiently small the discs 4ı . p/ D fz W jz  pj < ıjpj1 g  about the poles p ¤ 0 are mutually disjoint by Exercise 6.8; they form the ıneighbourhood Pı D

[

4ı . p/:

p2P

We will now prove jx.z/j C jy.z/j D O.jzj/

.z … Pı /:

(6.25)

Assume that jzn j D o.jx.zn /j C jy.zn /j/ holds for some sequence zn ! 1. Then r.zn / D o.jzn j1 /, and Yosida–Zalcman–Pang Re-scaling yields x0 D y2 ; y0 D x2 ; and c D lim r.zn /zn D 0I n!1

x and y are not constant, this following from maxfjx0 .0/j; jy0 .0/jg D maxfjx.0/j2 ; jy.0/j2 g D 1: Moreover, from x2 x0 C y2 y0 D 0 it follows that x3 C y3  x.0/3 C y.0/3 , hence x and y have some pole .3 Then x and y have some pole pn D zn C r.zn / C o.r.zn // D zn C o.jzn j1 /I this implies zn D pn C o.jpn j1 / 2 Pı and proves (6.25). To finish the proof we note that the discs 4ı . p/ are mutually disjoint and the residues of x and y have modulus one. By Theorem 4.18, x and y are candidates for the Yosida class Y1;1 . The Yosida limit functions x.z/ D lim xhn .z/ hn !1

and y.z/ D lim yhn .z/ hn !1

(6.26)

are finite (6 1) since x.0/ and y.0/ are finite if inf jhn jdist .hn ; P/ > 0, and x and y have a simple pole at z D 0 if jhn jdist .hn ; P/ ! 0. For fourth transcendents the proof follows from

 The discs 4ı .z0 / D fz W jz  z0 j < ıjz0 j1 g are defined for arbitrary centres z0 , and should not be mixed up with the discs 4ı .z0 / D fz W jz  z0 j < ır.z0 /g. 3 For x.0/3 C y.0/3 D 0, x D !=.z   / holds for some  ¤ 0 and ! with ! 3 D 1. In any other case x and y are elliptic functions.

196

6 Higher-Order Algebraic Differential Equations

Exercise 6.16 Prove that x; y 2 Y1;1 implies v D x C y  z 2 Y1;1 . (Hint. Note that v is regular at poles of .x; y/ with residues .1; 1/, and use Theorem 4.18.) The proof for first and second transcendents is practically the same. For ı > 0 sufficiently small, the discs 4ı . p/ D fz W jz  pj < ıjpjˇ g (ˇ D 14 and ˇ D 12 , respectively; again note the footnote on p. 195) about the non-zero poles ]1

are mutually disjoint (see also Exercise 6.3). For second transcendents, w 2 .z/ D 1 1 O.jzj 2 / follows from jwj D O.jzj 2 / outside Pı , j resp wj D 1 and Theorem 4.18. 1 3 First transcendents satisfy jwj D O.jzj 2 / and jw0 j D O.jzj 4 / outside Pı , and the inequality jwj  jz  pj2 .1 C O.ı 2 // > 0 on 4ı . p/, which follows from 3 Exercise 6.3, implies that both branches of w 2 are meromorphic on 4ı . p/. The 3 Maximum Principle applied to f .z/ D w0 ˙ 2w 2 on 4ı . p/ with boundary values 3 3 3 f .z/ D O.jzj 4 / yields jw0 .z/j D O.jzj 4 C jw.z/j 2 / and 3

3

jzj 4 C jw.z/j 2 1 w .z/ D O.jzj / D O.jzj 4 / jzj C jw.z/j2 ]1 2

1 2

on4ı . p/:

Remark 6.7 The main step in the proof consisted in an appropriate estimate of the respective auxiliary functions V, which were obtained by modifying the corresponding Lyapunov functions in the proof of the Painlevé property. It is quite reasonable that the intimate connection between the Painlevé and the Yosida property holds for a wide class of algebraic differential equations and systems. This could be the topic of future research.

6.4 Value Distribution The Nevanlinna characteristic and counting function of poles of functions f 2 Y˛;ˇ ˇ have magnitude r2C2ˇ , and the limit functions f.z/ D limhn !1 h˛ n f .hn C hn z/ belong to the universal class Y0;0 ; hardly more can be said. Meromorphic solutions to algebraic differential equations that belong to some Yosida class Y˛;ˇ have limit functions that solve autonomous algebraic differential equations and belong to Y0;0 . In many cases they are trigonometric or elliptic functions whose value distribution is completely understood. All one has to do is to retrace their properties to the solutions of the original equation. This will be done exemplarily for the solutions to System (IV) and the Painlevé transcendents.

6.4 Value Distribution

197

6.4.1 The Cluster Set of the Solutions to (IV) In the case of System (IV) the limit functions (6.26) solve the Hamiltonian system x0 D y2  x; y0 D x2 C y

(6.27)

with constant Hamiltonian H.x; y/ D 13 .x3 C y3 / C xy D 13 c: The constant solutions are .x; y/ D .0; 0/ with c D 0, and .x; y/ D .!; !/ N with c D 1, ! 3 D 1. The algebraic curve  3 C 3 C 3 D c

(6.28)

Q – is reducible if c D 1:  3 C 3 C 3  1 D ! 3 D1 . C !  !/ N holds, and the autonomous Hamiltonian system (6.27) may be reduced to x0 D 1 C p x  x2 via p  3  3 1 y D 1  x, with non-constant trigonometric solution x D 2  2 tan 2 z , and similar solutions if y D !N  !x (! D e˙2i=3 ). It is just important to know that 2 the poles form a p -periodic sequence with fixed residue. 3

2

– has genus zero if c D 0, with rational parametrisation  D r.t/ D t3t 3 C1 and 3t  D s.t/ D t3 C1 . The Hamiltonian system (6.27) is solved by the trigonometric functions x D r.ez / and y D s.ez / with poles forming a 2i 3 -periodic sequence, this time with alternating residues. – has genus one if c ¤ 0; 1, and is parametrised by elliptic functions  D x.z/ and  D y.z/ of elliptic order three; they solve x03  3xx02 C x6 C .4  2c/x3 C c2 D 0 (just compute the resultant of the polynomials  3 C 3 C 3  c and 1 C 2 C  with respect to ) and y03 C 3yy02  y6  .4  2c/y3  c2 D 0: To determine the possible constants c in (6.28) consider any sequence hn ! 1 that stays away from P, that is, assume inf jhn jdist .hn ; P/ > 0 and also that the limits x D limhn !1 xhn and y D limhn !1 yhn exist. Then x3 C y3 C 3xy  c holds with c D lim 3h3 n H.hn / hn !1

.H.z/ D H.z; x.z/; y.z//:

(6.29)

The limits (6.29) form the cluster set C.x; y/. The cluster set determines the growth of the Nevanlinna characteristic. Far beyond the estimate T.r; x/ C T.r; y/ D O.r4 / the following holds. Theorem 6.4 Suppose that C.x; y/ contains some cQ ¤ 0; 1. Then T.r; x/ r4

and T.r; y/ r4

hold, at least on some sequence r D rn ! 1.

(6.30)

198

6 Higher-Order Algebraic Differential Equations

Q Proof Let cQ D limhQn !1 3hQ 3 n H.hn / ¤ 0; 1 and set  D

1 2

minfjQcj; jQc  1jg. From

d 3 .z H.z// D 3z4 H.z/ C z3 x.z/y.z/ D O.jzj1 / dz outside Pı it follows that there exists some  > 0 such that jz3 H.z/  cQj <  holds on Dn D fz W jz  hQ n j < jhQ n jg n Pı : Re-scaling along any sequence .hn / with hn 2 Dn yields non-constant limit functions x and y satisfying x3 Cy3 C3xy D c with jc  cQj  , jcj  =2 and jc  1j  =2. In other words, x and y are elliptic functions with common pair of primitive periods .#; # 0 / D .#c ; #c0 / such that j#j  j# 0 j  j# ˙ # 0 j < L. Hence there exists an R > 0 such that every disc jz  z0 j < RjhQ n j1 centred at some z0 with jz0  hQ n j < jhQ n j .n  n0 / contains at least one pole of x and y. This implies that the disc jz  hQ n j < jhQ n j contains at least jhQ n j4 poles, where  > 0 is independent of n (it is almost the same to say that the disc jzj < r contains  r2 gaussian integers k C i`). This proves the assertion with rn D 2jhQ n j, say. To obtain more and better information about the distribution of poles, the cluster set C.x; y/ has to be analysed in detail. Exercise 6.17 For ı > 0 let Cı .x; y/ denote the cluster set of .x; y/ as z ! 1 restricted to C n Pı , that is, in (6.29) only sequences .hn / with jhn jdist .hn ; P/  ı are admitted. Prove that Cı .x; y/ is closed, bounded, and connected. The last assertion is non-trivial and relies on the fact that C n Pı is locally connected at infinity. Exercise 6.18 (Continued) Prove that c D limhn !1 3h3 n H.hn / remains unchanged if hn 2 CnPı is replaced with hQ n 2 CnPı such that jhQ n hn j < Kjhn j1 . (Hint. Prove that xQ.z/ D limj!1 xhQk .z/ D x.z C z0 / and yQ .z/ D limj!1 yhQk .z/ D j

j

y.z C z0 / holds for every convergent subsequence and appropriate z0 .) Exercise 6.19 (Continued) For ı > ; the inclusion Cı .x; y/ C .x; y/ is trivial. Prove thatSCı .x; y/  C .x; y/ if ı >  is sufficiently small, and conclude that C.x; y/ D ı>0 Cı .x; y/ is closed, bounded, and connected. Prove also that Cı .x; y/ contains the limits 1Climpn !1 6p3 n h. pn / for appropriately chosen sequences pn 2 P: (Hint. If dist .hn ; P/ D ıjhn j1 replace hn with pn 2 P such that dist .hn ; P/ D jhn  pn j. Conversely, replace pn 2 P with hn 2 @4ı . pn /.) We mention without proof that the cluster set coincides with the set of accumulation points of the sequence .1 C 6p3 n h. pn //.

6.4.2 Strings and Lattices The value distribution of any trigonometric function (a rational function of e2iz=# ) is parallel to the ray z D t#, while for elliptic functions the distribution of values is regularly spread over the whole plane. Suppose f 2 Y˛;ˇ has only #-periodic

6.4 Value Distribution

199 ˇ

trigonometric limit functions f.z/ D limhn !1 h˛ n f .hn C hn z/. Then the poles of f are arranged in arithmetic sequences .k#/k2Z , so by Hurwitz’ Theorem the poles of f are arranged in sequences . pk / satisfying an asymptotic recursion ˇ

pkC1 D pk C .# C o.1//pk :

(6.31)

Any such sequence is called a string, more precisely, a .#; ˇ/-string. For example, the poles of the solutions to the Riccati equation w0 D zn  w2 form finitely many .˙i; n2 /-strings. Exercise 6.20 Prove that for any .#; st /-string  D . pk / the following holds:   t 1C s – pk D k.1 C st /# sCt .1 C o.1//; (Hint. Consider qk D pk t .) – .s C t/ arg pk D t arg # C o.1/ mod 2 hold as k ! 1; s r1C t . – n.r; /  .1 C st /j#j Remark 6.8 For ˇ some positive integer the analogy with the dynamics of the rational map R. p/ D p C #pˇ is evident (see [174]); R has a parabolic fixed point at infinity with ˇ C 1 invariant petals, and for any p0 in some petal, the iterates pk D Rk . p0 / converge to infinity along some ray .ˇ C 1/ arg z D arg # mod 2. Exercise 6.21 Returning to (IV), set c D 1 C limpn !1 6p3 n h. pn / for some sequence . pn / of poles and Pn D fz W jz  pn j < Rjpn j1 g \ P. Prove that up to an error term o.jpn j1 /, Pn is a finite 0 – lattice pn C .k# C `# 0 /p1 n if c ¤ 0; 1, where .#; # / span the period lattice of the elliptic functions x D limpn !1 xpn and y D limpn !1 ypn ; 2i – arithmetic sequence pn C k#p1 n (# D 3 ) with alternating residues .1; 1/; 2i=3 , if c D 0; .!; !/, N .!; N !/ with ! D e 2 – arithmetic sequence pn C k#p1 (# Dp ) with fixed residues .!; !/ N for some n 3

! with ! 3 D 1, if c D 1:

We say that P locally has a ‘lattice’ and a ‘string structure’, respectively. Two cases are of particular interest. C.x; y/ D f0g. The poles are arranged in .˙ 2i 3 ; 1/-strings with counting function 3 2 r , alternating residues, and arg pk  .2 C 1/ 4 for some . n.r; . pk //  4 2 C.x; y/ D f1g. The poles are arranged in .˙ p ; 1/-strings with fixed residues, counting function n.r; . pk // 

p

3 2 r , 4

3

and arg pk   2 for some .

Remark 6.9 In both cases the question of whether there are only finitely many strings is still open. It is conjectured that there are no solutions with Nevanlinna characteristic strictly between T.r; x/ r2 and T.r; x/ r4 . Exercise 6.22 Prove that C.x; y/ D fcg implies T.r; x/ C T.r; y/ D o.r4 / and c 2 f0; 1g. Thus C.x; y/ is either a continuum or else reduces to f0g or f1g.

200

6 Higher-Order Algebraic Differential Equations

R 1 (Hint. Compute 2i r H.z; x.z/; y.z// dz asymptotically; for the definition of r we refer to Exercise 4.17.) Exercise 6.23 For r > 1 and z D rei … Pı set r ./ D z3 H.z/, and interpolate linearly in the -intervals corresponding to the intersection of jzj D r and Pı . Prove that the family .r /r>1 is bounded and equi-continuous, and deduce that C.x; y/ is the union of closed curves C.rn / W  7! ./ D limrn !1 rn ./. The question of whether C.x; y/ consists of a single curve (that is, limr!1 r ./ exists) is open. Now we can reap the benefits of our work and obtain the following completions of the Painlevé–Yosida Theorem. Theorem 6.5 The transcendental solutions to System (IV) have the following properties: – – – – –

T.r; x/ C T.r; y/ D O.r4 /; r2 D O.T.r; x/ C T.r; y//; m.r; x/ C m.r; y/ D O.log r/; m.r; 1=x/ D O.log r/ if ˛ ¤ 0, and m.r; 1=y/ D O.log r/ if ˇ ¤ 0; x.z/  z jzj and y.z/  z jzj hold outside Pı [ Qı .Q denotes the set of non-zero zeros of .x  z/.y  z//.

Proof Only the second and fourth assertion requires a proof. For the latter, see Exercise 6.24. To prove the second, re-scale along any sequence of poles to obtain elliptic or trigonometric limit functions with periods .#; # 0 / and #, respectively; in the latter case # is either real or purely imaginary. In any case, a simple geometric argument shows that to any pole p of sufficiently large modulus there exists some pole p0 such that jRe pj C jIm pj < jRe p0 j C jIm p0 j and jp0 j < jpj C O.jpj1 /. The assertion n.r; x/  const  r2 then follows from the next exercise. Exercise 6.24 Let .k / be a sequence of positive numbers that tends to infinity and 1 1 satisfies kC1  k C Lk1 . Prove that k  .2L C o.1// 2 k 2 . Exercise 6.25 To prove m.r; 1=x/ D O.log r/ for ˛ ¤ 0 derive some second-order differential equation ˝.z; x; x0 ; x00 / D 0 with ˝.z; 0; 0; 0/ D 4˛ 2 z2 C 4˛ˇ 2 : By symmetry, m.r; 1=y/ D O.log r/ holds if ˇ ¤ 0. In this case and if ˛ D 0 we obtain m.r; 1=x/  m.r; y2 =x/ C 2m.r; 1=y/  m.r; x0 =x/ C m.r; z/ C O.log r/ D O.log r/; hence m.r; 1=x/ C m.r; 1=y/ D O.log r/ holds whenever j˛j C jˇj > 0. The analogue to Theorem 6.5 is Theorem 6.6 Fourth Painlevé transcendents have the following properties. – – – – – –

w 2 Y1;1 and w.k/ 2 Y1Ck;1 ; T.r; w/ D O.r4 /; r2 D O.T.r; w//; m.r; w/ D O.log r/; m.r; 1=w/ D O.log r/ if ˇ ¤ 0; w.z/ C z jzj holds outside Pı [ Qı .Q denotes the set of non-zero zeros of w C z/.

6.4 Value Distribution

201

6.4.3 Weber–Hermite Solutions Weber–Hermite solutions to System (IV) or Painlevé (IV) arise from solutions to certain Riccati equations (which may be reduced to the Weber–Hermite equation (6.18)) by repeated application of Bäcklund transformations for System (IV) resp. Painlevé (IV). Our focus will be on System (IV). The reader will not have any difficulty in transferring the results to the Weber–Hermite solutions to Painlevé (IV). The smallest number of non-trivial Bäcklund transformations (6.6) that are needed to obtain .x; y/ from solutions .Qx; yQ / with xQ C yQ  z D 0 and ˛Q  ˇQ C 1 D 0 is called the order, written ord.x; y/. We will briefly write H.z/ D H.z; x.z/; y.z//. Exercise 6.26 Prove that the coordinates of Weber–Hermite solutions separately solve first-order algebraic differential equations of genus zero and are algebraically dependent over C.z/. (Hint. Prove that x D R.z; xQ /, where R is rational and xQ is an appropriate solution to some equation xQ 0 D ˛Q  z2 C zQx  xQ 2 :) Theorem 6.7 Every solution .x; y/ to System (IV) such that – x and y are algebraically dependent over C.z/, or – x and y separately satisfy first-order algebraic differential equations, or – V D H  13 z3 belongs to Y1;1 has Nevanlinna characteristic O.r2 /. Proof The first two conditions are equivalent. We suppose that x and y satisfy some non-trivial algebraic equation K.z; x; y/ D 0 and write K.z; x; y/ D

n Y

.y  K .z; x//:

D1

Then y.z/ D K .z; x.z// locally holds for at least one , and so by analytic continuation, which transforms K into some K . We may assume that .x; y/ has infinitely many poles p with residues .1; 1/. From Exercise 6.8 and h. p/ D O.jpj3 / it then follows that x.z/ C y.z/ D O.jpj/ (p ! 1 with resp x D 1) holds on 4ı . p/, hence also K .z; x/ D x C O.jzj/ holds as x ! 1, uniformly with respect to z. Re-scaling along any such sequence . pn / yields limit functions x and y satisfying y.z/ D x.z/ C a and also 13 .x3 C .a  x/3 / C x.a  x/ D .a  1/.x2  ax/ C 13 a3 D 13 c; which must be trivial: a D c D 1. This also holds if we consider poles with residues .!; !/, N hence we are in the reducible case and the set of poles P has string structure. Then the corresponding algebraic differential equation P.z; x; x0 / D 0 has genus zero, x has only finitely many strings of poles, and n.r; x/ D O.r2 / holds. This is also true if V 2 Y1;1 since n.r; x/ D n.r; V/ D O.r2 /: Exercise 6.27 Let .x; y/ be any Weber–Hermite solution. Prove that H (hence also V D H  13 z3 ) satisfies some first-order algebraic differential equation.

202

6 Higher-Order Algebraic Differential Equations

(Hint. Use H D 13 .x3 C y3 / C zxy C ˛y C ˇx, H 0 D xy, and K.z; x; y/ D 0 to derive P.z; H; H 0 / D 0 by purely algebraic methods.) Exercise 6.28 Set .Qx; yQ / D B1 .x; y/ (see (6.6) for the definition of the Bäcklund transformation B1 ) with x C y  z 6 0, ˛Q D ˇ  1 and ˇQ D ˛ C 1, and prove (with self-explanatory notation) Q H.z/  H.z/ D xQ .z/  x.z/ D y.z/  yQ .z/: Exercise 6.29 (Continued) Prove by induction on ord.x; y/ that for every Weber– Hermite solution the function V.z/ D H.z/  13 z3 D H.z; x.z/; y.z// C 13 z3 belongs to Y1;1 : (Hint. H D x C ˛z C 13 z3 holds if y D z  x and ˇ D ˛ C 1, that is, if Q D B1 H, noting that the ord.x; y/ D 0. Use Exercise 6.28 for the step from H to H Hamiltonian for .Ox; yO / D .!x; !y/ N is O xO ; yO / D H.z; x; y/ C ˇ.!  1/x C ˛.!N  1/y:/ H.z;

6.4.4 Value Distribution of the Painlevé Transcendents To make a long story short, we will now define and discuss the corresponding cluster sets and transfer the properties of the solutions to the re-scaled differential equations to our main object, the Painlevé transcendents. Yosida Re-scaling yields limit functions w D limhn !1 whn satisfying w00 D 1 C 6w2 ; w00 D w C 2w3 ; and 2ww00 D w02 C 3w4 C 8w3 C 4w2 ; respectively. In the first two cases it follows that w02 D 4w3 C 2w  2c resp. 3

w02 D w4 C w2  c holds, with constants of integration c D limhn !1 hn 2 W.hn / 1 4 resp. c D limhn !1 h2 n W.hn /; this is true if the infimum of jhn j dist .hn ; P/ resp. 1 jhn j 2 dist .hn ; P/ is positive. Exercise 6.30 For any fourth transcendent and limit function w D limhn !1 whn prove that w000 D .6w2 C 12w C 8/w0 and w00 D 2w3 C 6w2 C 8w  2c holds with c D limhn !1 W.hn /h3 n .infn jhn jdist .hn ; P/ > 0/: Deduce w02 D w4 C 4w3 C 4w2  4cw without a new constant of integration! The constants c constitute the respective cluster sets C D C.w/. Just as in case of System (IV), the cluster sets are closed, bounded, and connected, and contain the 3

7 3 limits as pn ! 1 of 14pn 2 h. pn /; 36 10n p2 n h. pn /; and 2pn h. pn /; respectively, for every appropriate sequence of poles . pn /. The statements of Exercises 6.17–6.21 mutatis mutandis remain valid. In any case the limit functions satisfy some simple

6.4 Value Distribution

203

Table 6.1 Exceptional parameters, trigonometric solutions, and periods q q 2 (i) c D ˙i 27 , w D  pi 6 .2 C 3 tan2 . 4 38 .1  i/z//,

 # D .1 ˙ i/ p , 4 6

(ii)

c D 0, c D  14 ,

w D ˙.sinh z/1 , w D ˙ p1 2 tan. pz2 /,

# D 2i, p # D  2,

(iv)

c D 0, 8 c D  27 ,

w D ˙.sinh 2z/1 , w D 8.9 tan2 . pz3 /  3/1 ,

# D i, p # D  3.

Briot–Bouquet equation w02 D P.wI c/ (the analogue to the algebraic curve (6.28)) depending on the parameter c. The 2 3 discriminant of P with respect q to w is 27c C 2, c.4c C 1/, and c .27c C 8/, respectively, with zeros c D solutions are

2 8  27 , c D 0;  14 and c D 0;  27 . The non-constant

– trigonometric functions of degree two if c is exceptional (Table 6.1). – elliptic functions of elliptic order two otherwise; the periods form a lattice Lc which is spanned by .#; # 0 / D .#c ; #c0 / with j#j  j# 0 j  j# ˙ # 0 j and #c0 ! 1 if c approaches some exceptional value. Not every constant solution is also a limit q function. Constant limit functionsqonly occur in the exceptional cases (i) w D  16 , (ii) w D 0 if c D 0 and w D  12

2 if c D  14 , (iv) w D 0 if c D 0 and w D  23 if c D  27 . Note that the constant limit functions are Picard values of the corresponding trigonometric solutions. The analogue to Theorem 6.4 is

Theorem 6.8 Suppose that q the cluster set of some first, second, and fourth tran2 8 scendent contains some cQ ¤  27 , cQ ¤ 0;  41 , and cQ ¤ 0;  27 , respectively. Then 5

T.r; w/ r3 ;

T.r; w/ r 2 ;

and T.r; w/ r4 ;

(6.32)

respectively, holds, at least on some sequence r D rk ! 1. Again we can reap the benefits of our work. Theorem 6.9 First Painlevé transcendents have the following properties: – w 2 Y 1 ; 1 and w.k/ 2 Y 1 C k ; 1 ; 2 4

– – – –

5

2

4 4

T.r; w/ D O.r 2 /; 5 r 2 = log r D O.T.r; w//; m.r; w/  12 log r and m.r; 1=w/ D O.1/; 1 w jzj 2 holds outside Pı [ Qı , where Q is the set of non-zero zeros of w.

204

6 Higher-Order Algebraic Differential Equations

Theorem 6.10 Second Painlevé transcendents have the following properties: – – – – –

w 2 Y 1 ; 1 and w.k/ 2 Y 1 C k ; 1 ; 2 2 2 2 2 T.r; w/ D O.r3 /; 3 r 2 D O.T.r; w//; m.r; w/  . 12 C o.1// log r; 1 w D O.jzj 2 / outside Pı .

In both cases the fourth assertion follows from Theorem 4.17, hence only the third assertion in Theorems 6.9 and 6.10 requires a proof. In the first case we refer to Shimomura [155], and to the proof of Theorem 6.5 in the second. Remark 6.10 The upper and lower estimates for T.r; w/ were proved by Shimomura [153, 155] in both cases. The lower estimate %  52 for first transcendents is due to Mues and Redheffer [121]. Hinkkanen and Laine [91] derived lower estimates for the transcendental solutions to Painlevé (II) and (IV). There are also several published attempts by unnamed authors to obtain upper bounds. They failed for different reasons, one reason being that the ordinary spherical derivative does not qualify as a tool. The approach via the Yosida property (in several steps) can be traced back to the papers [176, 180, 182, 184, 186]. Exercise 6.31 The solutions to Painlevé’s equation 2ww00 D w02 C 8˛w3  2zw2  1 .˛ ¤ 0/

XXXIV.

are given by 2˛w D y0 C y2 C 12 z with y00 D .2˛ C 12 / C zy C 2y3 , see [94], p. 340. Prove that w 2 Y1; 1 and T.r; w/ D 2N .r; y/ C O.log r/ D O.r3 /, where N .r; y/ 2 ‘counts’ the poles of y with residue 1. p 3 Prove also w000 C 2zw0 C w D 0 and .r/  2r 2 (central index) if ˛ D 0.

6.4.5 Airy Solutions Airy solutions to Painlevé (II) are obtained by repeated application of the Bäcklund transformations (6.16) to the solutions to the Airy equations (6.15). They occur for parameters ˛ 2 12 C Z. The analogue to Theorem 6.7 is Theorem 6.11 For every second transcendent each of the following conditions 3 implies T.r; w/ D O.r 2 / W – – – – –

w is an Airy solution. V D W C 14 z2 2 Y 1 ; 1 . 2 2 w solves some first-order algebraic differential equation. V solves some first-order algebraic differential equation. w and V are algebraically dependent over C.z/.

6.4 Value Distribution

205

Proof Airy solutions w D R.z; w0 / (cf. Exercise 6.10) have Nevanlinna characteris3 tic T.r; w/ D O.r 2 /. This also follows from V 2 Y 1 ; 1 , which in combination with 3

2 2

3

resp D 1 yields n.r; w/ D n.r; V/ D O.r 2 / and T.r; w/ D O.r 2 /. Now assume that w also solves P.z; w; w0 / D 0, where P.z; x; y/ D

n Y

.y  G .z; x//

D1

is irreducible. Then w0 D G .z; w/ locally holds for at least one . On the discs 4ı . p/ with resp w D 1, w0 C w2 D O.jpj/ holds as p ! 1 (cf. Exercise 6.3), hence G .z; x/ D x2 C O.jzj/ as x ! 1, uniformly with respect to z. In the same way we obtain w0 D G .z; w/ and G .z; x/ D x2 C O.jzj/ for some other branch if poles with residue 1 are considered. Re-scaling along any sequence of poles 4 2 2 yields w0 D w2 C c and w0 D w2 C c ; hence w02  D w  2c w C c and 02 4 2 2 02 4 2 w D w C 2c w C c ; respectively. Since, however, w D w C w  c always holds this implies c D 12 D c , hence c2 D c2 D 14 and w02 D .w2 C 12 /2 in any case. In particular, the set of poles P has string structure, hence the algebraic curve P.z; x; y/ D 0 has genus zero and P consists of finitely many strings . pk / with p 3 2 32 counting function n.r; . pk //  3 r . Again T.r; w/ D O.r 2 / holds. The same is true if V satisfies some first-order algebraic differential equation or if V and w are algebraically dependent, since then w also satisfies some first-order equation.

6.4.6 The Painlevé Hierarchies The first Painlevé hierarchy is a sequence 2n PI of algebraic differential equations of order 2n; 2 PI coincides with Painlevé’s first equation (I) w00 D z C 6w2 , and 4 PI is given by w.4/ D 20ww00 C 10w02  40w3  8aw  83 z:

4 PI

Shimomura [151, 154] proved that the solutions to 4 PI are meromorphic and have order of growth at least 73 . This is just a special case of a more general result, which says that the meromorphic solutions to 2n PI, if any, have order of growth at least 2nC3 . The Yosida test for equation 4 PI yields 4ˇ D maxf˛ C 2ˇ; 2˛; 0; 1  ˛g; nC1 the smallest value ˇ D 16 is obtained if ˛ D 13 . If w 2 Y 1 ; 1 is true, this implies 7

3 6

T.r; w/ D O.r2C2ˇ / D O.r 3 /, in accordance with and completing Shimomura’s result. Exercise 6.32 Realise the Yosida test for w.6/ D 42w002 C 56w0 w000 C 28ww.4/  280w.w02 C ww00  w3 / C z

6 PI

206

6 Higher-Order Algebraic Differential Equations

and w.4/ D 10w2 w00 C 10ww02  6w5 C zw  

4 PII

(though it is not known whether or not the solutions are meromorphic). Solution ˛ D 14 , ˇ D 18 and ˛ D ˇ D 14 . For meromorphic solutions this suggests 9 5 T.r; w/ D O.r 4 / and T.r; w/ D O.r 2 /, respectively.

6.5 Asymptotic Expansions The solutions to Riccati and implicit first-order differential equations of genus zero have asymptotic expansions on regularly distributed Stokes sectors, while the value distribution takes place on arbitrarily small sectors around the Stokes rays which separate the Stokes sectors. For families of systems and higher-order equations polefree sectors and asymptotic expansions merely occur in exceptional cases, and only for particular solutions. As a rule, the plane is ‘filled’ with poles, but the exception proves the rule.

6.5.1 Pole-Free Sectors and Asymptotic Expansions a) System (IV) Let .x; y/ be any transcendental solution to System (IV) with ‘pole-free’ sector S in the sense that x and y have only finitely many poles on every closed sub-sector of S. Yosida Re-scaling with hn restricted to any closed sub-sector of S leads to limit functions .x; y/ without poles, hence to constants. The possible constants are .0; 0/ and .!; !/, N again with ! 3 D 1, hence we have x.z/ D o.jzj/ and y.z/ D o.jzj/ resp: x.z/ D !z C o.jzj/ and y.z/ D !z N C o.jzj/ as z ! 1 on S. In each case we will prove that, as in the case of first-order equations, this leads to asymptotic expansions on specific sectors ˙1 W j arg z  .2 C 1/ 4 j
r0 Dr is part of a large pole-free sector. To this end we choose r0 > 0 sufficiently large and define the sequence .rn / inductively by rnC1 D rn C 4rn1 (the reason for choosing the factor 4 will soon become clear). By n we denote the largest number such that An D fz W rn  jzj  rnC1 ; O  arg z < n g contains no pole, and note that rn n ! 1 at least like S.rn / ! 1. Re-scaling along any sequence hn ! 1 in A D n0 An yields limit functions x and y that satisfy x3 C y3 C 3xy D 3c, and since A is pole-free, x and y cannot be elliptic functions (which have poles in ‘four directions’) even if hn 2 @A, hence c D limhn !1 h3 n H.hn / D 0 or c D 1. These limits form the cluster set CA .x; y/ f0; 1g of z3 H.z/ as z ! 1 on A. Since A is locally connected at infinity, the cluster set is

208

6 Higher-Order Algebraic Differential Equations

Fig. 6.1 The regions An . The circle jz  pn j D 5jpn j1 about pn 2 @An encloses five poles

O

connected, and from z3 H.z/ ! 0 as z D rei ! 1 we obtain CA .x; y/ D f0g. We may assume that there exists some pole pQ 0 on @A0 , hence arg pQ 0 D 0 . The same is true at least for some subsequence pQ nk 2 @Ank with arg pnk D nk . Re-scaling along the sequence .Qpnk / then yields limit functions x and y that satisfy x3 C y3 C 3xy D 0, hence the algebraic curve (6.28) has genus zero, and x and y have poles z D #, # D 2i : It follows from Hurwitz’ Theorem that to each pole pnk there exist five 3 poles zk;j D pQ nk C j.# C o.1//Qp1 Q n j < 5jQpn j1 , and no nk .2  j  2/ on jz  p others (the reason for choosing the constants ‘4’ and ‘5’ was 4 < 2j#j < 5 < 3j#j). Since zk;2 and zk;2 do not belong to the annulus rnk  jzj  rnk C1 , it follows that Ank 1 and Ank C1 , hence each An .n  n0 ) contains some pole pQ n on its boundary, and .Qpn / is a subsequence of some .˙#; 1/-string . pm /. Strings of this kind satisfy limm!1 arg pm D .2 C 1/ 4 for some integer , hence limn!1 n D .2 C 1/ 4 holds, see Fig. 6.1. O The same argument applies to AQ n D fz W rn  jzj  rnC1 ; n < arg z  g, this showing that in the second case the natural pole-free sectors are given by ˙ W .2 C 1/ 4 < arg z < .2 C 1/ 4 . In the first case the same proof yields pole-free sectors ˙ W  2 < arg z <  2 ; here the strings of poles have type . p ; 1/. 3 Also in the first case the principal terms !z and !z N are known. The argument used in the proof of Theorem 5.7 applies immediately, yielding asymptotic expansions for x and y. In the second case, however, only x.z/ D o.jzj/ and y.z/ D o.jzj/ is known. To determine the true principal terms we consider any closed sub-sector S of ˙ and set maxfjx.z/j; jy.z/jg D jzj. Since jx.z/j C jy.z/j D o.jzj/

.z ! 1 on S/;

(6.35)

we may assume  D .z/ < 12 . From System (IV) and jx0 .z/j C jy0 .z/j ! 0 as z ! 1 on every proper sub-sector S0 S (this following from (6.35) and Cauchy’s coefficient estimates), hence jx0 .z/j C jy0 .z/j < 1, say, we obtain jzx.z/j D jx0 .z/ C y.z/2 C ˛j <  2 jzj2 C j˛j C 1

and jzy.z/j <  2 jzj2 C jˇj C 1:

This yields .   2 /jzj2 < K and  < 2Kjzj2 , hence jx.z/j C jy.z/j D O.jzj1 / and jx0 .z/j C jy0 .z/j D O.jzj2 / as z ! 1 on S; again by Cauchy’s Theorem. From

6.5 Asymptotic Expansions

209

System (IV) it then follows that zx.z/ C ˛ D O.jzj2 / and zy.z/ C ˇ D O.jzj2 /, hence the principal terms are ˛z1 and ˇz1 . From here the proof follows the lines of the proof of Theorem 5.7. b) Painlevé (I), (II), and (IV) For first, second, and fourth transcendents the Stokes sectors and rays ˙ and  are as follows. .i/ .ii/a .ii/b .iv/a .iv/bc

˙ ˙ ˙ ˙ ˙

W j arg z  2=5j < =5 W j arg z  2 3 j < 3 W j arg z  .2 C 1/ 3 j < 3 W j arg z  .2 C 1/ 4 j < 4 W j arg z   2 j < 4

    

W arg z D .2 C 1/=5 W arg z D .2 C 1/ 3 W arg z D .2 C 2/ 3 W arg z D . C 1/ 2 W arg z D .2 C 1/ 4 :

Exercise 6.33 Confirm the following possible expansions on pole-free sectors: .i/ .ii/a .ii/b .iv/a .iv/b .iv/c .i/ .ii/a .ii/b .iv/a .iv/b .iv/c

1 2 z C O.jzj9=2 / D . 6z /1=2  48 1 D ˛z C 2˛.˛ 2  1/z4 C O.jzj7 / z 1=2 D . 2 / C 12 ˛z1 C O.jzj5=2 / D  23 z C ˛z1  14 .3˛ 2  9 2 C 1/z3 C O.jzj5 / D 2z  ˛z1 C 14 .3˛ 2  2 C 1/z3 C O.jzj5 / D z1  12 .2 2  ˛ /z3 C O.jzj5 /:

(6.36)

1 1 z C O.jzj7=2 / D 6. 6z /3=2 C 48 2 1 2 2 D ˛ z Cp ˛ .˛  1/z4 C O.jzj7 / .˛ ¤ 0/ 1 2 D  4 z C ˛ 2.z/1=2 C 18 .1 C 4˛ 2 /z1 C O.jzj5=2 / 8 3 D  27 z C 23 ˛z  16 .3˛ 2 C 9 2  1/z1 C O.jzj3 / D 2˛z C 12 .˛ 2  2 C 1/z1 C O.jzj3 / D 2 z C . 2  ˛ /z1 C O.jzj3 /

(6.37)

w.z/ w.z/ w.z/ w.z/ w.z/ w.z/

W.z/ W.z/ W.z/ W.z/ W.z/ W.z/

(both branches of .z/1=2 and D .ˇ=2/1=2 ). Note that the expansions (ii)a and (iv)c are only significant if ˛ ¤ 0 and ˇ ¤ 0, respectively. Exercise 6.34 (Continued) If ˇ D 0 in (iv)c but w 6 0 set y D w0 =w to obtain y0 D P.z/  12 y2 with P.z/ D 2z2  2˛ C 32 zw.z/ C 2w.z/2  2z2 R2˛: Prove that y has an asymptotic expansion y  ˙2z C    on S. Since w.z/ D e y.z/ dz ! 0 on S requires Re .˙z2 / < 0 for some sign, this yields (iv)c w0 =w  2z C .˛  1/z1 C 14 .˛ 2  4˛ C 3/z3 C    .S ˙0 [ ˙2 / (iv)c w0 =w  2z  .˛ C 1/z1  14 .˛ 2 C 4˛ C 3/z3 C    .S ˙1 [ ˙3 /; 2

and W D O.jzjM /ejRe z j for some M > 0. If ˛ D 0 in (ii)a but w 6 0, prove that (ii)a w0 =w  ˙z1=2  14 z1 

5 5=2 z 32

C

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6 Higher-Order Algebraic Differential Equations 2

and W D O.jzjM /e 3 jRe z

3=2 j

holds on pole-free sectors with ˙Re z3=2 < 0.

The analogue to Theorem 6.12, with exactly the same proof, is Theorem 6.13 Let w be any Painlevé transcendent having an asymptotics expanO Then the Stokes sector ˙ sion (6.36) on some single non-Stokes ray arg z D . which contains that ray is pole-free for w, and the asymptotics expansion holds on ˙.

6.5.2 Truncated Solutions Solutions to System (IV) or any Painlevé equation having no poles on some sector about any Stokes ray  are called truncated (along ). a) System (IV) The following exercises show that truncated solutions exist with prescribed asymptotics. Exercise 6.35 Theorem 1.10 does not immediately apply to System (IV). Suppose ˛ˇ ¤ 0 and set t D z2 , x.z/ D t1=2 u.t/2 , and y.z/ D t1=2 v.t/2 to obtain uP D 

˛ C u2 u2  v 4 ˇ C v2 u4 C v 2 C ; vP D C ; 4u 4tu 4v 4tv

wherePdenotes dtd : Prove that given any half-plane H, Theorem 1.10 now applies to p p the corresponding p system for  D u  ˛ and  D v  ˇ on H. Thus to every quadrant Q D H there exists some solution with the asymptotics (6.34). If H is chosen such that Q contains some Stokes ray arg z D .2 C 1/ 4 , the asymptotics for x, y, and H at least holds on j arg z  .2 C 1/ 4 j < 2 . Exercise 6.36 (Continued) If ˛ D 0 and ˇ ¤ 0 resp. ˛ ¤ 0 and ˇ D 0 show that the transformation t D z2 and x.z/ D t3=2 u.t/2 , y.z/ D t1=2 v.t/2 resp. x.z/ D t1=2 u.t/2 , y.z/ D t3=2 v.t/2 works in the same manner. Exercise 6.37 Prove the analogue for solutions with expansions (6.33) without restriction on ˛ and ˇ. (Hint. Set t D z2 , x.z/ D t1=2 u.t/2 , and y.z/ D t1=2 v.t/2 .) Exercise 6.38 Prove that in any case solutions with pole-free half-plane are uniquely determined by their asymptotics. (Hint. See the proof of Theorem 5.8.) b) Painlevé Transcendents The existence of truncated transcendents is again based on Theorem 1.10. For a proof the Painlevé equations have to be transformed into appropriate systems. Exercise 6.39 Painlevé’s first equation w00 D z C 6w2 is transformed by z D t4=5 , 96 4v vP w.z/ D t2=5 v.t/ into vR D .6v 2 C 1/ C D f .t; v; v/: P Prove that  25 25t2 t

6.5 Asymptotic Expansions

211

p Theorem 1.10 applies to the system xP D y; yP D f .t; x ˙ i= 6; y/ of rank one, hence given any half-plane S, there exists a solution with prescribed asymptotics on 4 S. Choosing S 5 to cover the Stokes rays 1 and  , the asymptotics extends to the sector ˙1 [ 1 [ ˙ [  [ ˙C1 with central angle 65 , and determines the doubly truncated solution w uniquely. (Hint. To prove uniqueness set u D w1  w2 and note p that u tends to zero and satisfies u00 D Q.z/u with Q.z/ D 6.w1 .z/ C w2 .z//  24z C    on S.) Exercise 6.40 Adapt the argument of the previous exercise to Painlevé’s second equation w00 D ˛ C zw C 2w3 with ˛ ¤ 0. 8 4 vP .v C ˛/ C 2 .v 3  v/ C ; 9 9t t v 4 4˛ vP (ii)b z D t2=3 , w.z/ D t1=3 v.t/, vR D .v C 2v 3 / C C 2 : 9 9t 9t t Remark 6.11 Equation (IV) could be dealt with in the same way. It is, however, more convenient to refer to Exercises 6.35–6.38, which show that given any Stokes ray  there exists some fourth transcendent that is truncated along  ; w is uniquely determined by its asymptotics (iv)a , (iv)b , and (iv)c , respectively, which may be prescribed.

Solution (ii)a z D t2=3 , w.z/ D t2=3 v.t/, vR D

Transcendents that are truncated along some Stokes ray arg z D O have an asymptotic expansion on j arg z  O j < , which by Theorem 6.13 extends to ˙ [  [ ˙C1 . In particular, w has the same asymptotic expansion on adjacent Stokes sectors ˙ and ˙C1 .4 The converse is also true. Theorem 6.14 Let w be any Painlevé transcendent that has the same asymptotic expansion (6.36) on adjacent Stokes sectors ˙ and ˙C1 . Then w is truncated along  and the asymptotic expansion holds on ˙ [  [ ˙C1 . R

Proof In any case F.z/ D e W.z/ dz is an entire function of finite order with simple zeros at the poles of w. Since the idea of proof is the same in all cases we will consider exemplarily (iv)b with W.z/ D 2˛z C z1 C O.jzj3 / on ˙ and ˙C1 . D 12 .˛ 2  2 C 1//. Then H.z/ D F.z/e

˛z2 

z D



c C o.1/ as z ! 1 on ˙ cC1 C o.1/ as z ! 1 on ˙C1

p p holds. We set f ./ D H.z/ with z D ei.2C1/ 4  and  > 0 on  > 0: Since f has finite order (logC logC jf ./j D O.log jj/) on j arg j < 2 , say, and limits c˙ as  D re˙iı ! 1 for every 0 < ı < , the Phragmén–Lindelöf Principle yields

4

In cases (i) and (ii) ‘the same expansion’ means that the chosen square-root across  :

p z is continuous

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6 Higher-Order Algebraic Differential Equations

c D cC D c and f ./ ! c on j arg j  ı. Thus the half-plane j arg z.2C1/ 4 j <  2 is pole-free for w, and w has the asymptotic expansion (6.36)(iv)b there. c) Special Transcendents We conclude this section with two examples and a potential example of particular interest. Example 6.6 Equation w00 D zw C 2w3 has a unique solution, named after Hastings and McLeod [76], that is positive and decreasing p on the real line. Moreover, it satisfies w.x/  0 as x ! C1 and w.x/  x=2 as x ! 1. From Theorem 6.13 it follows that the asymptotic expansions w  0 and w.z/  p z=2 C    hold on j arg zj < 3 and j arg z  j < 3 , respectively. The poles are asymptotically restricted to the sectors j arg z  2 j < 6 . Writing w00 D .z C 2w.z/2 /w and noting that w.z/ D o.jzjn / holds on j arg zj < 3 for every n 2 N, this yields w.z/  kAi.z/ for some real constant k, actually k D 1; Ai denotes the Airy function, a special solution to y00 D zy. Even more precisely, 5 5=2 w0 =w  ˙z1=2  14 z1  32 z C    holds as z ! 1 on j arg zj < 3 . Example 6.7 For ˇ D 0 and ˛ real it is conjectured that there exists a unique real solution to Painlevé’s equation (IV)—the so-called Clarkson–McLeod solution [25, 95]—satisfying w.x/  0 as x ! C1 and w.x/  2x C    as x ! 1. If it exists, and existence is supported by numerical experiments and by analogy with the Hastings–McLeod solution, the corresponding asymptotic expansions w  0 (even w0 =w  2z C    by Exercise 6.34) and w  2z C    hold on j arg zj < 4 and j arg z  j < 4 , respectively. Example 6.8 According to Boutroux [16], Painlevé’s equation (I) has five triply truncated solutions (also called tritronquée); for a recent existence proof, see Joshi and Kitaev [98]. Since (I) is invariant under the transformation w 7! a2 w.az/ with a5 D 1, it suffices to provep the existence of a triply truncated solution w0 having p the asymptotics w0 .z/   z=6 with Re z > 0 on j arg z  j < 45 I this solution is truncated along 1 , 2 , and 3 , with poles asymptotically restricted to j arg zj < =5. By Exercise 6.39 there exist transcendents p uniquely determined first p w1 and w2 with asymptotics w1;2 D  z=6 C O.jzj2 / and Re z > 0 on  7 3 z < 95 , respectively. Then u D w1  w2 satisfies 5 < arg z < 5  and on 5  < argp 00 u D 6.w1 .z/ C w2 .z//u D . 24z C O.jzj2 //u on 35  < arg z < 75 : We 4=5 D t1=10 v.t/ on the right half-plane Re t > 0 (t4=5 > 0, set p z D 6t2=5 and u.z/1=10 z=6 D t > 0 and t > 0 if t > 0) to obtain   25vR C 6412 C O.jtj2 / v D 0:

(6.38)

Now every non-trivial solution to (6.38) tends to infinity exponentially as t ! 1 at least on one of the sectors ı < arg t < 2  ı and  2 C ı < arg t < ı. This proves v  0 and u  0, and w0 D w1 D w2 is truncated along 1 , 2 , and 3 . The p pole-free sector is ‘bordered’ by . .1˙i/ ; 14 /-strings of poles (Fig. 6.2). 4 6

6.6 Sub-normal Solutions

213

Fig. 6.2 The solutions w1 and w2 are truncated along arg z D 35 ; arg z D  and arg z D 75 ; arg z D , respectively, and agree on j arg z  j < 2=5 (dark grey). The sector j arg z  j < 4=5 (light and dark grey) is pole-free for w0 D w1 D w2 with asymptoticsp w0 .z/   z=6

Remark 6.12 We note that w0 .Nz/ is also a solution that is truncated along 1 , 2 , and 3 , and hence coincides with w0 by uniqueness. In other words, the tritronquée solution is real on the real axis. We also note that any solution that is truncated along 1 and 2 resp. 2 and 3 coincides with w0 by Exercise 6.39 if it has the same asymptotics as w0 on ˙1 [ 1 [ ˙2 resp. ˙2 [ 2 [ ˙3 . In contrast to Exercise 6.39, the triply truncated solution w0 is not only unique, but also determines its asymptotics. In other words, there wQ that is truncated along 1 , p is no solution p 0 2 , and 3 and is asymptotic to C z=6 with Re z > 0. Otherwise the solution v.z/ D e4i=5 wQ 0 .e2i=5 z/ would be truncated along the Stokes rays 0 , 1 , and 2 , and two of the solutions w0 , wQ 0 , and v would have the same asymptotic expansions on =5 < arg z < 7=5, which by Exercise 6.39 is impossible.

6.6 Sub-normal Solutions As a rule, given any linear differential equation with polynomial coefficients, all non-trivial solutions have maximal order of growth %max . Only under the rarest of circumstances will it happen that transcendental solutions of order less than %max occur. These solutions are called sub-normal. The transcendental solutions to any Riccati equation with polynomial coefficients are normal (not sub-normal). In the present section we will discuss the question of how to define the term ‘subnormal’ for nonlinear higher-order equations and systems, and how to identify the sub-normal solutions. Our goal is to solve this problem exemplarily for Painlevé’s second equation and the Hamiltonian system (IV), hence also for Painlevé (IV). By the way we will settle an old question concerning the deficiency ı.0; w/ for second and fourth Painlevé transcendents with parameter ˛ D 0 and ˇ D 0, respectively.

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6 Higher-Order Algebraic Differential Equations

6.6.1 Sub-normal Second Transcendents By Theorem 6.8, second transcendents with cluster set C.w/ 6 f0;  41 g have Nevanlinna characteristic T.r; w/ r3 as r D rn ! 1; while C.w/ f0;  41 g implies T.r; w/ D o.r3 /. Moreover, the set of poles P has string-structure (P 3 consists of finitely or infinitely many strings of poles). Since r 2 D O.T.r; w// holds anyway it is quite natural to call any second transcendent w sub-normal if 3

T.r; w/ D O.r 2 /

.r ! 1/

holds. Recall that V D W C 14 z2 2 Y 1 ; 1 is sufficient for w to be sub-normal. The 2 2 next exercise will show that V 2 Y 1 ; 1 is necessary for w to be an Airy solution. 2 2

2

d z . 4 C W/ D ˙V 0 Exercise 6.41 It follows from w D ˙. 12 z C w2 / D ˙ dz that V D ˙w C const; hence V 2 Y 1 ; 1 . To prove that this is true for every 0

2 2

Airy solution assume V 2 Y 1 ; 1 for some Airy solution with ˛ ¤ ˛1=2 w0 w2 z=2

2 2

˛1=2 w0 V 0 ;

.˛1/C1=2 : w Q 0 CVQ 0

1 2,

and set

wQ D w C D w C hence w D wQ  Prove that 1 Q Q VQ D V  w  wQ C const and deduce V.z/ D O.jzj 2 / outside Pı , hence VQ 2 Y 1 ; 1 2 2 Q since the discs 4ı .Qp/ are mutually disjoint and respQ VQ D 1; the notation VQ and P is self-explanatory. Airy solutions are sub-normal. The goal of this section is to prove the converse. Theorem 6.15 For second transcendents the following statements are equivalent. – – – – – –

w is sub-normal; w is an Airy solution; w satisfies some first-order algebraic differential equation; w and W (first integral) are algebraically dependent; V D W C 14 z2 2 Y 1 ; 1 ; 2 2 V satisfies some first-order algebraic differential equation.

Sub-normal solutions occur if and only if ˛ 2 characteristic T.r; w/  2˛.w/

p

2 3

3 2

1 2

C Z. They have Nevanlinna

r with .w/ 2 f3; 1; 1; 3g:

Proof We have to show that sub-normal solutions are Airy solutions; everything else has already been proved or discussed in the exercises. Since C.w/ f0;  41 g, we have to consider two cases as follows. a) C.w/ D f0g. For every sequence hn ! 1 the re-scaling method yields limit functions w D limhn !1 whn satisfying w00 D w C 2w3 and w02 D w4 C w2 . The limit functions are w  0 and w D ˙1= sinh.z C z0 /. The zeros of sinh z form a i-periodic sequence, and this leads as usual to the conclusion that the poles of w are arranged in .˙i; 12 /-strings . pk / with alternating residues ˙1 and counting

6.6 Sub-normal Solutions

215 3

2 2 function n.r; pk /  3 r . Each string is asymptotic to some ray arg z D .2 C 1/ 3 , 3 and from T.r; w/ D O.r 2 / it follows that there are only finitely many strings. On 1 the open sectors ˙ W .2  1/ 3 < arg z < .2 C 1/ 3 , w D o.jzj 2 / holds (re-scaling with .hk / restricted to any closed sub-sector yields the limit function w  0). From w3 D o.jzwj/ and w00 D o.jzj3=2 / on every closed sub-sector of ˙ it then follows that zw C ˛ D o.jzwj/ C o.jzj3=2 /, hence w D ˛=z C o.jzj1 / holds. The usual technique then shows that w and W D w4 C zw2 C 2˛w  w02 have asymptotic expansions (6.36) (ii)a and (6.37) (ii)a , one and the same on each sector ˙ . Now

F.z/ D e

R

W.z/ dz 2

is an entire function5 of finite order which satisfies F.z/ D C z˛ .1 Co.1// on every sector .2  1/ 3 C  < arg z < .2 C 1/ 3  . Applying the Phragmén–Lindelöf 2 Principle to F.z/z˛ on the sectors j arg z  .2 C 1/ 3 j <  then shows that F is a polynomial (of degree ˛ 2 ) and w2 D .F 0 =F/0 is a rational function in contrast to our general assumption that w is transcendental. b) C.w/ D f 41 g. The re-scaling method yields limit functions again satisfying w00 D w C 2w3 , but now also w02 D .w2 C 12 /2 with constant and nonp constant limit functions w D ˙i= 2 and w D ˙ p12 tan pz2 , respectively. Thus

the poles of w form .˙ p2 ; 12 /-strings with constant residues and counting function p

3

n.r; pk /  32 r 2 , each being asymptotic to some ray arg z D 0 mod 2=3. Again the number of strings is finite and the constant limit functions lead to asymptotic expansions (6.36) (ii)b and (6.37) (ii)b on ˙ W 2 3 < arg z < 2. C 1/ 3 . Let .w/ denote the difference between the number of strings of poles with residues 1 and 1, respectively, hence 1 2i

Z

p 2 3 3 r 2 C o.r 2 / w.z/ dz D .w/ 3 r

holds by the Residue Theorem. The contribution of the sectors j arg z  2 3 j <  3

p

3

3

and 2 3 C  < arg z < 2. C 1/ 3   is O.r 2 / and ˙.1/ 32 r 2 C O.r 2 /, 1 respectively, this following from the asymptotics on ˙ , w D O.jzj 2 / on r , and the fact that the length of the part of r in j arg z  2 3 j <  is O.r/; the sign ˙ p depends on the branch of z=2. This proves that .w/ 2 f3; 1; 1; 3g. We may assume that .w/ is negative (consider .w; ˛/ in place of .w; ˛/ otherwise). Exercise 6.42 Prove that the asymptotic expansions of w and its Bäcklund trans˛1=2 form wQ D w C w0 w 2 z=2 .˛ ¤ 1=2/ have the same principal term by showing that 1

w0  w2  z=2 D O.jzj 2 /. In particular, .w/ Q D .w/ holds. 5

Also called a  -function, see Okamoto [131].

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6 Higher-Order Algebraic Differential Equations

From VQ D V  w  wQ C const (cf. Exercise 6.41) it then follows that Z Z 1 1 Q .V.z/  V.z// dz D  .w.z/ Q C w.z// dz n.r; w/  n.r; w/ Q D 2i r 2i r p 3 p 3 2r 2 3 2r 2 3 C o.r 2 / D 2.w/ C o.r 2 /: D ..w/ Q C .w// 3 3 In each step w 7! w, Q ˛ 7! ˛Q D ˛  1 the number of strings decreases by 2.w/  2, and after a finite number of steps we end up with ˛ D 12 and w00 D

1 2

C zw C 2w2 :

(6.39)

We note that ˛ 2 12 C N0 is necessary for w to be sub-normal with .w/ < 0, and we have to prove that sub-normal solutions to (6.39) also satisfy w0 D 12 z C w2 . Assuming the contrary, define a meromorphic function u 6 0 locally by w0  w2  12 z D u2 : Exercise 6.43 From (6.39) deduce .z C 2w2  2w0 /w D 2uu0 ; w D u0 =u, and u00 D  12 zuCu3 : To prove that u is meromorphic, transform the differential equation for u into v 00 D zv C 2v 3 : p p Solution v.z/ D u. 2z/,  3 D  12 2. By the first part of thepproof, the function p v in Exercise 36.43 is not sub-normal in contrast to T.r; v/ D T. 3 2r; u/ D O.T. 3 2r; w// D O.r 2 /. Thus w0  w2  12 z D u2  0 holds, and Theorem 6.15 is completely proved. Exercise 6.44 Theorem 6.15 also provides a new proof of Theorem 21.1 in [60]. Second transcendents satisfying some first-order algebraic differential equation are Airy solutions. Prove also that first transcendents do not solve any first-order algebraic differential equation. (Hint. Assuming P.z; w; w0 / D 0, prove that w D Q 3 limhn !1 whn satisfies nD1 .w0  2 w 2 C c / D 0 with 2 D 1, in contrast to 02 3 w D 4w C 2w  2c.) Exercise 6.45 The solutions to y0 D 12 z C y2 generically have three strings of poles, one along each Stokes ray arg z D 2 3 , with residue 1 (.y/ D 3). Prove that the sub-normal solutions to w00 D k C 12 C zw C 2w3 (k 2 N) generically have k strings of poles with residue 1 and k C 1 strings with residue 1 along arg z D 2 3 ; ‘generically’ means ‘up to three exceptional solutions’, in which case the strings of poles are asymptotic to just one Stokes ray (.w/ D 1). In any case, w D Rk .z; y/ holds with degy Rk D 2kC1, and w solves some first-order equation of degree 2kC1 with respect to w0 . Prove also that any two strings of poles  D . pj / and Q D .Qpj / 1 absent themselves from each other: limj!1 jpj j 2 dist . pj ; Q / D 1.

6.6 Sub-normal Solutions

217

Remark 6.13 The special Bäcklund transformation w $ v, cf. [60] Thm. 19.2, maps Airy solutions w to v D 0 (Airy solutions come from out of nowhere). It is worth emphasising the role of the first integral V and the Yosida class Y 1 ; 1 . While 2 2 w 2 Y 1 ; 1 holds for every second transcendent, V 2 Y 1 ; 1 is true if and only if 2 2 2 2 w is sub-normal. In any case V certainly belongs to some Yosida class Y; with 1 2   D   2 ( D  is necessary since V has simple poles with residue 1), this implying n.r; w/ D O.rC1 /. However, re-scaling in Y; with  > 12 leads to limit functions either w D ˙1=z and V D 1=z or else w D 0 and V D const, hence is completely useless.

6.6.2 The Deficiency of Zero of Second Transcendents For  any second transcendent, standard arguments from Nevanlinna theory yield 1 m r; wc D O.log r/ and ı.c; w/ D 0 if j˛j C jcj > 0; in case of ˛ D c D 0 the best known estimate is ı.0; w/  12 , see [60]. We are now able to settle this case completely. Theorem 6.16 Second transcendents have deficiency ı.0; w/ D 0. Proof Of course, the statement is relevant only if ˛ D 0. We use the special transformation of the preceding proof (with w and v interchanged) to switch from w00 D zw C 2w3 to v 00 D 12 C zv C 2v 3 . The poles and zeros of w coincide with the poles of v with residue 1 and 1, or vice versa. For a proof of n.r; 1=w/ D 3 3 n.r; w/ C O.r 2 / and N.r; 1=w/ D N.r; w/ C O.r 2 / we refer to Exercise 6.46 below. Thus N.r; 1=w/ D T.r; w/ C o.T.r; w// holds, at least on some sequence r D rn ! 1, since w is not sub-normal. This proves m.rn ; 1=w/ D o.T.rn ; w// and ı.0; w/ D 0. Exercise 6.46 Let v be any second transcendent and denote by n .r; v/ the number 3 of poles on jzj  r with residue . Prove that jnC .r; v/  n .r; v/j D O.r 2 / is 1 0 always true, and N .r; v/  2NC .r; v/ holds except when v satisfies v D 2 z C v 2 . R 1 (Hint. Use nC .r; v/  n .r; v/ D 2i r v.z/ dz, where r is a simple closed curve 0 like in Exercise 4.17. Consider  D v  12 z  v 2 at poles with residue 1 and residue 1, respectively, and apply the First Main Theorem whenever possible.) Exercise 6.47 Though our focus is on transcendental solutions (and not on rational solutions and other Special Functions results) we note that the method of proof of Theorem 6.15 also applies to rational solutions. Necessary for their occurrence is that ˛ 2 2 N0 , hence also .˛ C k/2 2 N0 for every k 2 Z. Prove that rational solutions exist if and only if ˛ 2 Z, exactly one for each ˛. They are obtained by repeated application of Bäcklund transformations to the trivial solution to w00 D zw C 2w3 .

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6 Higher-Order Algebraic Differential Equations

Exercise 6.48 Let  be the unique rational solution to w00 D ˛ C zw C 2w3 (˛ 2 N), w any transcendental solution, and B the Bäcklund transformation (6.16) which transforms ˛ into ˛  1. Then Q D BŒ is the unique rational solution to wQ 00 D ˛  1 C zwQ C 2wQ 3 , and w Q D BŒw is a transcendental solution. Prove  wQ  Q  D O.log r/; m r; w and deduce ı.; w/ D 0 from Theorem 6.16 by induction on ˛.   0  0  0  Q0  w Q Q (Hint. Prove w and w , where P and PQ D P z; w; w; Q ww D PQ z; w; w; Q wQw w Q Q Q Q are rational in z and polynomials in the other variables.

6.6.3 Sub-normal Solutions to System (IV) and Painlevé (IV) Sub-normal solutions to System (IV) and Painlevé (IV) are defined by the growth condition T.r; x/CT.r; y/ D O.r2 / and T.r; w/ D O.r2 /; respectively. The analogue to Theorem 6.15 and the main result of this section will be Theorem 6.17 below. To be prepared for the proof we will start with some exercises. Here .x; y/ is any solution to System (IV) and .Qx; yQ / D B1 .x; y/ (see (6.6)) denotes its Bäcklund transform. Exercise 6.49 Prove that x  z C    and y  N z C    ( 3 D 1/ on some sector S implies xQ  z N C    and yQ  z C    on S. Exercise 6.50 Prove that the poles of .x; y/ with residues .e˙2i=3 ; e2i=3 / are also poles of .Qx; yQ / with the same residues, while .Qx; yQ / is regular at poles of .x; y/ with residues .1; 1/. It is known that v D x C y  z vanishes at poles of .x; y/ with residues .1; 1/. Prove that the zeros of v different from those coincide with the poles of .Qx; yQ / with residues .1; 1/. Exercise 6.51 Prove that solutions having no poles with residues .!; !/ N for some ! with ! 3 D 1 are Weber–Hermite solutions of order ord.x; y/  1. Theorem 6.17 For every transcendental solution .x; y/ to System (IV) the following statements are equivalent. – – – – – –

.x; y/ is sub-normal; .x; y/ is a Weber–Hermite solution; x and y are algebraically dependent over C.z/; x and y separately satisfy first-order algebraic differential equations; V D H  13 z3 2 Y1;1 ; V satisfies some first-order algebraic differential equation.

6.6 Sub-normal Solutions

219

Proof Weber–Hermite solutions are sub-normal, and the subsequent statements in Theorem 6.17 hold. Conversely, each of the above conditions implies that .x; y/ is sub-normal. Again we just have to prove that sub-normal solutions are also Weber– Hermite solutions. Sub-normal solutions have cluster set C.x; y/ f0; 1g, and the set of poles P consists of finitely many strings. Although all examples suggest that C.x; y/ D f1g, we also have to discuss the case a) C.x; y/ D f0g. The strings of poles (if any) are distributed along the rays arg z D .2 C 1/ 4 , and x, y, and particularly H  ˛ˇ=z C R  have asymptotic expansions on the sectors between these rays. Again F.z/ D e H.z/ dz is an entire function of finite order, and the Phragmén–Lindelöf Principle applied to F.z/z˛ˇ on the small sectors j arg z  .2 C 1/ 4 j <  shows that F is a polynomial (of degree ˛ˇ), and H D F 0 =F, x and y are rational functions. b) C.x; y/ D f1g. The idea of the proof is simple, while the technical part itself is involved. Third roots of unity !, , and  will occur in different contexts: ! D e2i=3 is definitely fixed,  and N represent residues, and z and z N are principal  terms 2 in asymptotic expansions. The poles of .x; y/ are arranged in ˙ p ; 1 -strings 3   which now are asymptotic to the rays arg z D  2 . We denote by P the set of nonzero poles with resp .x; y/ D .; /. N On each sector ˙ W .  1/ 2 < arg z <  2 , 1 x   z C  z C    and y  N z C  z1 C    .3 D 1/ have asymptotic expansions, cf. (6.33). This will be expressed by the symbol

2 1 . In combination 3 4

with the asymptotic expansions, the Residue Theorem yields X 3 D1

n.r; P / D

1 2i

Z

4

x.z/ dz D  r

r2 X .1/  C o.r2 / 2i D1

(6.40)

(for the construction of r , see Exercise 4.17). Let sx ./ denote the number of strings of x with residue . Taking the real- and imaginary part we obtain 4 P

sx .1/  12 .sx .!/ C sx .!// N D  p23 sx .!/  sx .!/ N D

4 3

4 P

.1/ Im 

D1

(6.41)

.1/ Re  :

D1

To proceed we need two technical results. N with ! D e2i=3 and ˇ D ˛ C 1 Lemma 6.6 Suppose sx .1/ > 12 .sx .!/ C sx .!// holds. Then x C y  z vanishes identically. Proof Since ˛  ˇ C 1 D 0, the function v D x C y  z vanishes at least twice at poles of .x; y/ with residues .1; 1/. If v does not vanish identically, it follows from m.r; v/ D O.log r/ and Nevanlinna’s First Main Theorem that 2N.r; P1 /  N.r; 1=v/  T.r; v/ C O.1/ D N.r; P! / C N.r; P!N / C O.log r/; N against our hypothesis. holds, hence also 2sx .1/  sx .!/ C sx .!/,

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6 Higher-Order Algebraic Differential Equations

Lemma 6.7 For any solution with symbol

4 2 1 X , .1/ Im  is non-zero. 3 4 D1

Proof The sum vanishes if and only if 1 D 2 and 3 D 4 (or 2 D 3 and 4 D 1 ). Again we consider some entire function associated with .x; y/, this time z4

F.z/ D e 12 e

R

H.z/ dz

:

From H.z/  z3 =3 C 2az C a1 =z C    and H.z/  z3 =3 C 2bz C b1 =z C    on the upper and lower half-plane (2a D 1 ˛  N1 ˇ and 2b D 3 ˛  N3 ˇ) it follows that 2

2

F.z/ D Aeaz za1 .1 C o.1// and F.z/ D Bebz zb1 .1 C o.1// holds as z ! 1, uniformly on   arg z     and  C   arg z  , respectively. In particular, the upper and lower half-plane are ‘pole-free’. Since F p p has finite order, Corollary 1.1 applies to f .z/ D F. z/ ( z > 0 if z > 0) on j arg zj < 2 , say, and yields Re a D Re b and Im a  Im b. The same argument, however, also applies to .F.z/; b; a; b1 ; a1 / in place of .F.z/; a; b; a1 ; b1 /. The second part of Corollary 1.1 then yields a D b, a1 D b1 , A D B, and F.z/ D 2 Aeaz za1 .1 C o.1// as z ! 1 on j arg zj   and also on j arg z  j  , hence F has only finitely many zeros along the real axis. Thus H D F 0 =F C z3 =3 has only finitely many poles and is a rational function in contrast to our general assumption. This proves the lemma. We claim that to any sub-normal solution satisfying x N C y  z 6 0 for every third root of unity there exists some sub-normal solution .Qx; yQ / that has fewer strings of poles than .x; y/, and this will prove Theorem 6.17.6 To reach our claim we note that by (6.41) and Lemma 6.7 there exists a 1 such that 12 .sx .2 / C sx .3 // < sx .1 / holds. Set .Ox; yO / D .N1 x; 1 y/

and .Qx; yQ / D B1 .Ox; yO /I

the latter is well-defined since sxO .1/ > 12 .sxO .!/ C sxO .!// N and xO C yO  z 6 0: From Q O Exercise 6.28, which says H  H D xQ  xO , and n.r; x/ D n.r; xO / we obtain n.r; xQ /  n.r; x/ D

6

1 2i

Z r

Q O .H.z/  H.z// dz D

1 2i

Note that x C y  z  0 already yields x0 D ˛  z2 C zx  x2 .

Z r

.Qx.z/  xO .z// dz:

6.6 Sub-normal Solutions

221

Fig. 6.3 Asymptotics and distribution of poles with residues .1; 1/; .!; !/, N and .!; N !/ (! D e2i=3 ) along the positive real axis of generic sub-normal solutions of order one, two, and three (from left to right). How to continue?

Since .Ox; yO / and .Qx; yQ / have symbols

2 1 N N and 2 1 , xQ  xO  .N   /z C    D 3 4 N3 N4

2i.Im  /z C    holds on ˙ . This implies

N n.r; xQ /  n.r; x/ D .2sxO .1/  sxO .!/  sxO .!//

p 2 3r C o.r2 /; 4

proving our claim and also Theorem 6.17. Analysis of the Proof Starting with any sub-normal solution .x; y/ we obtain after finitely many steps some sub-normal solution .x0 ; y0 / such that x0 C y0  z  0; ˛0  ˇ0 C 1 D 0, x00 D ˛0  z2 C zx0  x20 , and y00 D ˇ0 C z2  zy0 C y20 hold (cf. Exercise 6.3). Each step requires some trivial Bäcklund transformation M W .x; y/ 7! .x; y/ N D .Ox; yO / followed by B1 W .Ox; yO / 7! .Qx; yQ /; this may be abbreviated by writing M

B1

.x; y/ ! .Ox; yO / ! .Qx; yQ /:

(6.42)

The very last step only requires some M to achieve x0 C y0  z  0. Of course the sequence (6.42) may be reversed (Fig. 6.3). Exercise 6.52 Prove that generic sub-normal solutions have – k C 1 strings with residues .!1 ; !N 1 /, and k strings with residues .!2 ; !N 2 / and .!3 ; !N 3 /, respectively in each Stokes direction if ord.x; y/ D 2k;

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6 Higher-Order Algebraic Differential Equations

– k strings with residues .!1 ; !N 1 / and .!2 ; !N 2 /, respectively, and k  1 strings with residues .!3 ; !N 3 / (not necessarily in the same order as in the first case) in each Stokes direction if ord.x; y/ D 2k  1 Exercise 6.53 Prove that generic sub-normal solutions with !N ! ! – ord.x; y/ D 1 and sx .1/ D 0 have symbol !N

– ord.x; y/ D 0 and sx .1/ D 4 have symbol

! and sx .!/ D sx .!/ N D 0; !N !N and sx .!/ D sx .!/ N D 4; !

(! D e2i=3 ). For exceptional (non-generic) solutions one ! resp. !N has to be replaced with !N resp. !, and the number 4 by 2. Generic symbols have ‘diagonal structure’, which simplifies matters significantly. Obviously, the method of proof provides the ‘steepest descent’, that is, the number of non-trivial steps equals the order ord.x; y/, which now may be assigned to any sub-normal solution. For the convenience of the reader we will state the analogue to Theorem 6.17 for fourth transcendents. In this form it was proved by Classen [26]. Here Weber– Hermite from the solutions to the Weber–Hermite equations p solutions are obtained 2 w0 D 2ˇ ˙ .2zw C w / by repeated application of the Bäcklund transformations p p p w0  2ˇ2zww2 w0 C 2ˇC2zwCw2 and w 7!  (for both branches of 2ˇ) w 7! 2w 2w and the elementary transformations w 7! iw.iz/ and w 7! w.Nz/. Theorem 6.18 For every fourth transcendent the following statements are equivalent. – – – – – –

w is sub-normal; w is a Weber–Hermite solution; w satisfies some first-order algebraic differential equation; W 2 Y1;1 (first integral); w and W are algebraically dependent; W satisfies some first-order algebraic differential equation.

6.6.4 The Deficiency of Zero of Fourth Transcendents Just like for second transcendents with parameter ˛ D 0, the Wittich–Mokhon’ko method does not apply to estimate m.r; 1=w/ or even to compute the deficiency ı.0; w/ for fourth transcendents with parameter ˇ D 0. The same problem occurs for non-trivial solutions to the differential equation 2vv 00 D v 02  v 4  4zv 3  .3z2 C 4˛ C 2/v 2

(Iƒ0 )

6.6 Sub-normal Solutions

223

which is derived from our System (IV) with ˛ˇC1 D 0 via v D xCyz (note that the parameters ˛ and ˇ have different meanings in Painlevé (IV) and System (IV) resp. .Iƒ0 /). Since v D x C y  z has simple poles at the poles of .x; y/ with residues .!; !/ N and .!; N !/ .! D e2i=3 /, respectively, and vanishes at least twice at poles with residues .1; 1/, n.r; v/ D 23 n.r; P/ C O.r2 / D 2n.r; P1 / C O.r2 /  n.r; 1=v/ C O.r2 / and T.r; v/ D N.r; v/ C O.log r/  N.r; 1=v/ C O.r2 / holds. Here we used n.r; P / D 13 N.r; P/ C O.r2 / for every third root of unity, which P is true for every solution and follows from the general form of (6.40), namely 3 D1 n.r; P / D O.r2 /. If v is not sub-normal, rk2 D o.T.rk ; v// holds on some sequence rk ! 1, and so ı.0; v/ D 0. This was the first part of the proof of the following, rather unexpected Theorem 6.19 The deficiency ı.0; v/ of any non-trivial solution to .Iƒ0 / is zero, 1 except when v is sub-normal of even order ord.v/ D 2k. Then ı.0; v/ D 2kC1 holds. Proof We may restrict ourselves to sub-normal solutions v D x C y  z 6 0 with parameters satisfying ˛ˇC1 D 0. Then v has poles at poles of .x; y/ with residues .!; !/ N ¤ .1; 1/, and double zeros at poles with residues .1; 1/, while from v 0 D .x.z/  y.z//v

.v 6 0/

and the uniqueness part of Cauchy’s Theorem it follows that v is non-zero otherwise. This implies (note that sx ./ denotes the number of strings with residue ) N.r; 1=v/ 

2sx .1/ T.r; v/ sx .!/ C sx .!/ N

and ı.0; v/ D

sx .!/ C sx .!/ N  2sx .1/ sx .!/ C sx .!/ N

N hence (! D e2i=3 ), and requires 2sx .1/  sx .!/ C sx .!/; N D .2k C 1/ if ord.v/ D 2k is even, and – sx .1/ D k and sx .!/ C sx .!/ – sx .!/ D sx .!/ N D k and sx .1/ D .k  1/ if ord.v/ D 2k  1 is odd; generically the factor  equals 4, and equals 2 only in special (truncated) cases. In 1 any case, this yields ı.0; v/ D 2kC1 and ı.0; v/ D 1k , respectively. We need to check in which case ˛  ˇ C 1 D 0 can hold, and will restrict ourselves to the generic case. a) ord.v/ D 2k  2. We may assume sx .!/ D 4.kC1/ and sx .1/ D sx .!/ N D 4k. Then x has the symbol

1 !N , and the step !N 1

1 !N M! !N ! B1 ! !N M! 1 ! B1 1 !N ! ! ! ! !N 1 ! !N !N ! ! 1 !N 1

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6 Higher-Order Algebraic Differential Equations

.˛; ˇ/ 7! .!˛; N !ˇ/ 7! .!ˇ  1; !˛ N C 1/ 7! .ˇ  !; N ˛ C !/ 7! .˛ C !  1; ˇ  !N C 1/ has to be applied k times to obtain the parameters ˛ C k.!  1/ and ˇ  k.!N  1/. In the final step

1 !N M! !N ! ! we get the parameters ˛0 D !˛ N C k.1  !/ N and !N 1 ! !N

ˇ0 D !ˇ  k.1  !/; they have to satisfy ˛0  ˇ0 C 1 D 0. In terms of ˛ and ˇ this means !˛ N  !ˇ p C 3k C 1 D 0, which p in combination with ˛  ˇ C 1 D 0 yields ˛ D !N  ik 3 and ˇ D !  ik 3. Since the procedure may be reversed, the sub-normal parameters p solutions of even order p ord.v/ D 2k with corresponding 1 ˛ D !N  ik 3 and ˇ D !  ik 3 have deficiency ı.0; v/ D 2kC1 : b) ord.v/ D 2k  1  1. Then v has a predecessor vQ of even order ord.v/ Q D 2k and parameters ˛Q D !ˇ C ! and ˇQ D !˛ N  !. N After 2k steps we arrive at some solution of order 0 with parameters ˛0 D !N ˛Q C k.1  !/ N D ˇ C 1 C k.1  !/ N and Q ˇ0 D ! ˇk.1!/ D ˛1k.1!/. Again ˛0 ˇ0 C1 D 0, thus ˇ˛C3kC3 D 0 is necessary, which, however, is incompatible with ˛  ˇ C 1 D 0. p p Remark 6.14 The parameters .˛; ˇ/ D .!N  ik 3; !  ik 3/ in System (IV) and .Iƒ0 / correspond to .˛; ˇ/ D .2k C 1; 0/ in Painlevé (IV). Example 6.9 ([60], p. 127) Sub-normal solutions to 2ww00 D w02 C 3w4 C 8zw3 C 4.z2  3/w2

.˛ D 3; ˇ D 0/

also satisfy w03 C w.w C 2z/w02  w.w.w C 2z/2  16w C 32/w0 C P0 .z; w/ D 0 with P0 .z; w/ D w2 .w4 C 6zw3 C .12z2  16/w2 C 8z.z2  6/w  32.z2  1// and deficiency ı.0; w/ D 13 .

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Index

Abel function, 117 Abel’s functional equation, 118 Ahlfors–Shimizu formula, 39, 59 Ahlfors–Shimizu formula modified, 131 Airy equation, 188 Airy function, 212 Airy solution, 188 algebraic curve, 5 algebraic degree, 5 algebraic differential equation, 78 algebraic function, 2 algebraic pole, 3 algebraic singularity, 3 algebroid function, 63 almost entire, 88 analytic dependence, 16 Arzelà–Ascoli Theorem, 6 asymptotic expansion, 19 asymptotic series, 19

Bäcklund transformation (II), 188 Bäcklund transformation (IV), 222 Bäcklund transformation (IV), 178 Bernoulli number, 20 binomial differential equation, 160 Bloch’s principle, 122 Borel exceptional value, 53 Borel Identity, 61 Borel’s Theorem, 53 Borel–Carathéodory inequality, 69 Briot–Bouquet equation, 161

calcul des limites, 14 canonical form of Riccati equations, 148 canonical product, 41 Cartan characteristic, 58 Cartan’s First Main Theorem, 59 Cartan’s Identity, 38 Cartan’s Second Main Theorem, 60 Cauchy’s Existence Theorem, 11 central index, 29 chordal metric, 6 Clarkson–McLeod solution, 212 Clunie’s Lemma, 79 cluster set, 149 cluster set of (IV), 197 cluster set of Painleve (I), (II), (IV), 202 continuous dependence, 16 counting function, 35 cross-ratio, 31

deficiency, 47 deficiency relation, 47 deficient value, 47 degree of a differential polynomial, 79 differential equation, 9 differential polynomial, 79 discriminant, 1 doubly periodic function, 24

elliptic curve, 5 elliptic function, 24

© Springer International Publishing AG 2017 N. Steinmetz, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations, Universitext, DOI 10.1007/978-3-319-59800-0

233

234 elliptic order, 24 elliptic parametrisation, 5 entire curve, 58 Eremenko’s Lemma, 77 exceptional fundamental set, 143 exponent of convergence, 41 exponential polynomial, 143

Five-Value Theorem, 94 Four-Value Theorem, 97 Frei’s Theorem, 138 fundamental set, 12 fundamental system, 12

Gamma function, 37 genus of a canonical product, 41 genus of a meromorphic function, 53 genus of an algebraic curve, 5 Gol’dberg Conjecture, 52 Green’s formula, 39 Green’s function, 33 Gronwall’s Lemma, 12 Gundersen’s example, 95

Hadamard’s Theorem, 52 Hamiltonian, 178 Hamiltonian system, 177 harmonic measure, 25 Hartogs’ Theorem, 10 Hastings–McLeod solution, 212 Hayman’s alternative, 91 Hurwitz’ Theorem, 6

Implicit Function Theorem, 15 irreducible, 1 isomonodromic deformation, 176

Jacobi’s elliptic functions, 25 Jensen’s formula, 34 Jensen’s inequality, 45 Julia set, 118

Laplace transformation, 154 lattice, 24 lattice structure, 199 Lemma on the Logarithmic Derivative, 44

Index Liouville’s Theorem, 24 locally univalent, 157 Lyapunov function, 183

Malmquist’s First Theorem, 76 Malmquist’s Second Theorem, 78, 158 Malmquist-type theorem, 180 Marty’s Criterion, 9 maximum modulus, 40 maximum term, 28 minimal polynomial, 63 Montel’s Criterion, 6 Montel’s Second Criterion, 120 movable singularity, 175 Mues Conjecture, 52

Nevanlinna characteristic, 36 Nevanlinna’s First Main Theorem, 36 Nevanlinna’s Second Main Theorem, 46 Nevanlinna’s Third Main Theorem, 62 Newton–Puiseux polygon, 4 Ngoan–Ostrovskii Lemma, 43 normal family of holomorphic functions, 6 normal family of meromorphic functions, 9 normal function, 120

order of a Weber–Hermite solution, 201 order of growth, 40

Painlevé hierarchy, 205 Painlevé property, 173 Painlevé story, 175 Painlevé test, 174 Painlevé transcendent, 176 Painlevé’s Theorem, 17 Painlevé–Yosida Theorem, 188 parabolic fixed point, 117 Phragmén–Lindelöf indicator, 141 Phragmén–Lindelöf principle, 26 Picard exceptional value, 47 Picard’s Great Theorem, 120 Picard’s Little Theorem, 119 Poincaré density, 121 Poincaré function, 115 Poisson kernel, 33 Poisson–Jensen formula, 33 pole-free sector, 152 proximity function, 35 Puiseux series, 3

Index ramification index, 56 ramified value, 55 rational curve, 5 rational parametrisation, 5 repelling fixed point, 115 resonance condition, 178 Riccati differential equation, 18 Riemann–Hilbert method, 176 Schröder’s functional equation, 116 Schwarzian derivative, 30 Selberg characteristic, 64 Selberg–Valiron First Main Theorem, 66 Selberg–Valiron Second Main Theorem, 70 singular solution, 159 small function, 48 spherical characteristic, 40 spherical derivative, 9 spherical derivative modified, 130 spherical First Main Theorem, 40 stereographic projection, 7 Stirling’s formula, 20 Stokes ray, 149 Stokes sector, 149 string, 136 string of poles, 199 string structure, 199 sub-normal solution, 141 sub-normal solution (II), 214 sub-normal solution (linear), 213 sub-normal solution (IV), (IV), 218 subdominant solution, 154

235 triply truncated solution, 212 truncated solution, 153 Tumura–Clunie Theorem, 54 Two-Constants Theorem, 26

Valiron characteristic, 65 Valiron’s Lemma, 37, 75 Vitali’s Theorem, 6

Weber–Hermite equation, 189 Weber–Hermite solution, 189 Weierstraß P-function, 24 Weierstraß prime factor, 41 Weierstraß’ Theorem, 6 weight of a differential polynomial, 79 Wittich–Mokhon’ko Lemma, 79 Wronskian determinant, 12

Yosida class, 130 Yosida function, 129 Yosida property, 174 Yosida re-scaling, 195 Yosida test, 189 Yosida–Zalcman–Pang re-scaling, 190

Zalcman function, 118 Zalcman’s Lemma, 114 Zalcman–Pang Lemma, 114