Gives an account of Liouville's theory of integration in finite terms -- his determination of the form which the in

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- Joseph Fels Ritt

*Table of contents : PREFACECONTENTSChapter I. ELEMENTARY FUNCTIONS OF ONE VARIABLEChapter II. ALGEBRAIC FUNCTIONS WITH ELEMENTARY INTEGRALSChapter III. INTEGRATION OP TRANSCENDENTAL FUNCTIONSChapter IV. FURTHER QUESTIONS ON THE ELEMENTARY FUNCTIONSChapter V. SERIES OF FRACTIONAL POWERSChapter VI. INTEGRATION OF DIFFERENTIAL EQUATIONS BY QUADRATURESChapter VII. IMPLICIT AND EXPLICIT ELEMENTARY SOLUTIONS OP DIFFERENTIAL EQUATIONS OF THE FIRST ORDERChapter VIII. FURTHER IMPLICIT PROBLEMSREFERENCES*

INTEGRATION IN F I N I T E

TERMS

INTEGRATION IN FINITE TERMS Liouville's Theory of Elementary Methods Joseph Fels Ritt Davies Professor of Mathematics Columbia University

New

York

Columbia University Press 1 9

4

8

COPYRIGHT 1948 COLUMBIA UNIVERSITY PRESS, NEW YORK

Published in Great Britain and India by Geoffrey Cumberlege Oxford University Press, London and Bombay MANUFACTURED IN THE UNITED STATES OF AMERICA

PREFACE During the period between 1833 and 1841, J. liouville presented a theory of integration in finite terms. He determined the form which the integral of an algebraic function must have when the integral can be expressed with the operations of elementary mathematical analysis, carried out a finite number of times. He showed that the e l l i p t i c integrals of the f i r s t and second kinds have no elementary expressions. He proved that certain simple differential equations cannot be solved by elementary procedures. His papers contain other remarkable applications of his theory. The questions treated by Liouville are questions which occur to every strong undergraduate student of mathematics. Nevertheless Liouville's work never received very wide attention. It has always been something which everyone would like very much to know about but which very few undertake to study. During the nineteenth century, extremely l i t t l e was done in direct continuation of Liouville's work.* About forty years ago, the Russian mathematician Mordukhai-Boltovskoi began to write on Liouv i l l e 's theory and contributed extensively to i t . In particular, he published a book on the integration of transcendental functions and one on the integration in finite terms of linear differential equations. Through his influence, the subject seems to have been more widely studied in Russia than elsewhere. The present writer published some work on these questions between 1923 and 1927. At the present time, Ostrowski is writing on the subject. This monograph gives an account of Liouville's work and of some of that of his few followers. On the basis of what has already been said, a glance through the chapters, or even over the table of contents, will give a sufficient idea of the topics covered. I should like, however, to say something in regard to the treatment given here of Liouville's work. Liouville's methods are ingenious and beautiful. From the formal standpoint, they are entirely sound. There are, however, certain questions connected with the * T o be s u r e , equations, Galois

t h e r e appeared the P i c a r d - V e s s i o t

which f u r n i s h e s ,

f o r such equations,

t h e o r y of

linear

differential

r e s u l t s analogous t o those

f o r a l g e b r a i c e q u a t i o n s . R e c e n t work of E . R. K o l c h i n has b r o u g h t

and s i m p l i c i t y t o t h e P i c a r d - V e s s i o t

theory.

Che s h o u l d p e r h a p s m e n t i o n a l s o t h e r e m a r k a b l e work o f Bruns on t h e s o l u t i o n s of

the equations

of

rigor

of c e l e s t i a l m e c h a n i c s .

(Acta

Mathematica,

algebraic Vol.

XI.)

vi

PREFACE

winy-valued character of the elementary functions which could he pressed hack behind the symbols in L i o u v i l l e ' s tine but which have since learned to assert their r i g h t s . Such natters are mulled over in the f i r s t chapter. The mulling is inescapable. I t might be great fun to talk just as i f the elementary functions were one-valued. I might even sound convincing to some readers; I certainly could not f o o l the functions. However, i f one i s c h i e f l y interested in formal ideas, one may read the "First Survey of the Elementary Functions" in Chapter I and then pass to the summary at the end of that chapt e r . I t ought not to be hard, a f t e r that, to follow the formal processes of Chapter I I . As regards the theory of functions, I have assumed, in Chapter I , an acquaintance with the simpler facts concerning analytic continuation. Riemann surfaces of algebraic functions are mentioned in Chapter I ; their simpler properties permit, in Chapter I I , the s w i f t liquidation of questions on the i n t e g r a b i l i t y of special a l gebraic functions. In algebra, I use in Chapter I the discriminant of a polynomial and the resultant of a pair of polynomials. For the r e s t , special material of algebra or of analysis i s developed wherever i t is needed. I should l i k e , in conclusion, to say something concerning Joseph L i o u v i l l e (1809-1882). He originated the notion of derivative of f r a c t i o n a l order. The f i r s t examples of transcendental numbers were due to him; his work on this question was the starting point of the modern researches of Thue and S i e g e l . He was one of the founders of the theory of boundary value problems. As Abel had also done, earl i e r , he solved a problem involving an integral equation; this was decades before the general theory of such equations came into being. His work on doubly periodic meromorphic functions was procursive to Weierstrass's theory of the e l l i p t i c functions, just as his work in the theory of elimination anticipated, to some extent, ideas of Kronecker. He presented a method f o r treating classes of Diophantine equations which has been developed extensively, in recent times, by E. T. B e l l . In geometry, he determined the group of conformal transformations in three dimensions. He was the founder of the Journal de mathématiques pures et appliquées. Considering his achievements, one may question whether he has been adequately appreciated, even by the mathematicians of his own country. I t i s surprising, f o r instance, that his collected works were never published .

PREFACE

vi

If this monograph should promote the study of what is probably one of the most interesting portions of Liouville's work, the writer will feel amply rewarded. J. F. Ritt Columbia July,

1947

University

CONTENTS I. II. III. IV. V. VI. VII.

Elementary Functions of One Variable Algebraic Functions with Elementary Integrals

20

Integration of Transcendental Functions

40

Further Questions on the Elementary Functions

53

Series of Fractional Powers

61

Integration of Differential Equations by Quadratures

69

Implicit and Explicit Elementary Solutions of Differential Equations of the F i r s t Order

VIII.

3

78

Further Implicit Problems

87

References

99

INTEGRATION IN F I N I T E

TERMS

Chapter I ELEMENTARY FUNCTIONS OF ONE VARIABLE FIRST SURVEY OF THE ELEMENTARY

FUNCTIONS

1. The functions which we shall study in the present chapter are essentially those which make up the functional world of a student of the integral calculus. Such a student, if not familiar with the concept of algebraic function in its most general form, knows the polynomials and fractional rational functions, has seen functions involving radicals, and can imagine quite well the most general algebraic function which can be expressed in terms of radicals. He knows e x , log x, sin x, cos x, and the inverses of the latter two functions. After compounding functions of th| foregoing types in various ways to produce combinations like e*

or log (secx + tanx),

he is in possession of an extensive class of functions, each constructed with a finite number of operations. A typical example would be tan [ex The expression x

- logx (1 + x^)] + [xx + log arc sin x]*. 1

is to be interpreted, of course, as e*

x

. As

to the logarithm to the base x, it is nothing more than the natural logarithm divided by log x. The use of only a finite number of operations needs particular emphasis. One meets infinite series in a calculus course. The representation of functions by means of such series is a question foreign to our present study. We shall be concerned only with what can be obtained from the basic functions with calculations involving only a finite number of operations. 2. An inspection of our functions will permit us to describe them more closely and to classify them. We shall look twice at our material, first casually, just to see how matters stand, then fully and squarely, with no turning away from hard realities. We notice first that, if complex numbers are employed, the trigonometric functions and their inverses become redundant. For instance

and arc sin x = 1 log (ix + s/l - x").

4

ELEMENTARY FUNCTIONS

Thus the functions of elementary analysis are constructed out of the variable x by repeated use of the following operations: (a) Algebraic operations perforned on one or more expressions (b) The taking of exponentials (c) The taking of logarithms. Just what is to be understood by an algebraic operation will be explained fully later. Now let us explain how the elementary functions will be classified. The variable x will be called a monomial of order zero, and any algebraic function of x will be called a function of order zero. The exponential or the logarithm of a nonconstant algebraic function will be called a monomial of the first order. An algebraic combination of x and of monomials of the first order will, if it is not an algebraic function, be called a function of the first order. The exponential or the logarithm of a function of the first order, will, if it is not a function of order zero or a function of order unity, be called a monomial of the second order. For instance, as one will see in Chapter IV, the exponential of e x , and log (log x) are monomials of the second order. An algebraic combination of monomials of orders 0, 1, 2 which is not a function of one of the orders 0, 1, will be called a function of the second order. The classification continues in this way. All of this needs a closer examination, and to that we now turn. ALGEBRAIC

FUNCTIONS

3- Let us recall the notion of algebraic function. A function u of x is algebraic if it is defined by an irreducible relation (1) a0(x) uP + ai (x) uP-1 dp(x) = 0 with p > 0 where the a are polynomials in x with any complex numbers for coefficients, a 0 not being identically zero. In saying that (1) is irreducible, we shall mean that its first member is not the product of two nonconstant polynomials in u and x. We shall find ourselves at times employing the Riemann surface of u, which is a surface of p sheets. For our immediate purposes, it is more important to consider u as a monogenic analytic function (m.a.f.) in the sense of Weierstrass, that is, as the totality of power series which can be obtained from some given power series by analytic continuation. Each power series is called an element of the m.a.f. An element which is a series of powers of x - x 0 may be represented by P(x-x0J.

ELEMENTARY FUNCTIONS

5

l e t P(x-x 0 ) be any element of our algebraic function u. Let C be any curve with x 0 for f i r s t point. We may consider C to be given by a relation (2)

x =

-*12> •••'

"^kl' _a k2> must vanish. This gives an equation

-alk

^kk

k + c x q>k_1 + . . . + c k - 0 where the c are polynomials in monomials of orders 0, . . . , n. Thus,

0, as presented in §11, is not at a l l unique. For instance, e x and e~x e 1 are the same function of order 1. Among a l l representations of a given function u of order n, there are certain ones which employ a least number of n-monomials; in any such representation, the r n in (N + 1) of §11 i s not greater than the r n of any other representation of u. Representations which have this feature of economy will be employed throughout our work on integration, and we shall now establish for them an important principle .

16

ELEMENTARY FUNCTIONS

Under the assumption that r n is a minimum, we shall show that no aliébrate

relation

u and monomials

can exist of

order

amoné the

less

r n monomials

of

order

than n . To make t h i s statement

n in ex-

p l i c i t , we refer to §11 and use the point a there mentioned. Let . . . , £p be monomials of orders less than n. Suppose that a function (12)

x^»; y i , . . . , y vp ) , n algebraic in a l l its variables and analytic for x'"' = ©["'(a), yL = ( a ) , vanishes in the neighborhood of a when each xí¡"' is replaced by i t s 0( n ' and each by We shall show that f vanishes for

f(xin',

all

replaced

values by

of

the x , n )

close

to

the

9 ( n ) ( a ) , if

only

each

is

S,l.

The principle which we have just formulated underlies a l l of Liouville's work on the elementary functions. We shall c a l l i t Liouville 's principle. The proof amounts to nothing more than solving for one of the n-monomials in any nonidentical relation which may exist. When that monomial is replaced in u by the expression found for i t , u acquires an expression involving too few n-monomials. A question connected with the implicit theorem accounts for the details which follow. Suppose that (12) is not an identity in the x ( n ' for y.¡ * £ i , j j 3 = 1 , . . . , p. We can find a point b, as close as one pleases to a, such that (13) f(x< n >, . . . , x ^ l j M b ) , . . . , f p ( b ) ) does not vanish identically in the x ^ ' . Consider the partial derivatives of f , of a l l orders, crossderivatives included, with respect to the x ' n ' . N o t a l l of them can vanish for y A = ^ ( b ) , x [ n ) * e}n> (b). I f they did, the function in (13) would be constant when each x ' f ' varies in the neighborhood of e [ n ' ( b ) . The constant would be zero, since (13) vanishes for x [ n ) = 9in ( b ) ; t h i s would contradict what we know of (13). Working now in a neighborhood SJ> of x = a, let us suppose that a l l of the above-mentioned derivatives of f up to and including those of order j vanish throughout 51 when the variables in (12) are replaced by their monomials, but that some derivative of order j + 1 does not vanish at x = b, where b is in JS. That the required j and b exist follows from what precedes. To f i x our ideas, suppose that G(x, . . . , y p ) * In this, we consider f to be its own derivative of order zero.

17

ELEMENTARY FUNCTIONS

i s a p a r t i a l derivative which vanishes over ffl but that the derivative of G with respect to x } " ' does not vanish at b. By §13, the equation G = 0 determines x'" 1 as an algebraic function of x^n> . . . , y , analytic a t (e ( s n) (b)', . . . , £ p ( b ) ) which r e duces to 9 | n ' tor the familiar replacements. I f we substitute this algebraic function for x j n 1 in (N + 1) of §11, and have regard to §13, we find a contradiction o f the assumption that r n is a minimum. One must bear well in mind that the y in (12) must be replaced by t h e i r monomials before we get a function which i s identically zero. For instance, the function F - (3 y , - y . ) * ! , which vanishes for every x when y j i s replaced by e x , y 5 by e* + log 3 and x" by log log x , is certainly not zero identically in yt, y2 x " . I t vanishes in x and x" when yt and y2 are replaced by t h e i r monomials . DIFFERSSTIATION 15. Let u be a function of order 1 , of regular structure (§11) a t some point a . We wish to form du/dx for the neighborhood of a . Let f ( x i , . . . , x^; x) be the algebraic function appearing in ( I I ) . The derivative of u can be written ^ a f ( e t , . . . , e r ; x) de t r=l 9x[ dx

+

af(e,

e„; x) 3x

+

I f Sj^ i s an exponential, e ' 1 , i t s derivative i s 6 i v [ . I f Qj^ i s a logarithm, log v i , i t s derivative i s v [ / v 1 . In the logarithmic case, v i i s not zero at a , since Qj^ i s analytic at a . Let 0 J , . . . , 9 t be exponentials and the remaining 0 logarithms. Let *

r=i

axj

1

1

i * t + i 9x' v L

ax

Then g is algebraic in x and the x'. I f we take of §11 s u f f i c i e n t ly small, g w i l l be analytic for |x - a| < plt . . . , — G r (a)| < p , . I f now we take p correspondingly small, so as to limit the v a r i a tion of the monomials, we see that g reduces to the derivative of u for |x - a| < p when each variable i s replaced by i t s monomial. I t i s now easy to t r e a t a function u of any order n. Of the algebraic functions introduced in ( I ) , . . . , (N) of §11, there are some which are used for forming logarithmic monomials. As each monomial i s analytic a t a , such an algebraic function is not zero when x = a

18

ELEMENTARY FUNCTIONS

and each accented x i s i t s 9 ( a ) ; the function is therefore d i s t i n c t from zero i f the x are close to these values. I f now p t is taken s u f f i c i e n t l y small and i f p i s taken correspondingly small, so as to limit the variation of the monomials, we may assume that none of the algebraic functions which give logarithmic monomials vanish when x d i f f e r s from a , and each accented x from i t s 0 ( a ) , by a quantity less than p t in modulus. This understood, the formulas for the d i f f e r e n t i a t i o n of composi t e functions show that If u is a function of order n, of regular structure ot a point a, there exists an aliébrate function of the x, analytic for |x - a| < p,, . . . , |x' n ) - 8 ' " ' I < p t which reduces *n *n to the derivative of u for |x - a| < p when each variable is replaced by the monomial which corresponds to It. Thus i f u i s elementary and of order n, i t s derivative i s elementary and of order not exceeding n. 16. The matter which we s h a l l now examine takes care of i t s e l f quite well in Chapter I I . S t i l l i t s t r i k e s one of the deeper notes of L i o n v i l l e ' s theory and may well be l i f t e d out from among other details. Let u of order n be derived from g(x< n >, . . . , x ) . Suppose f i r s t that 8 { n ) i s an exponential, e v . The algebraic function which yields du/dx can be written (14)

x j n ' q> + other terms.

In (14),

+ other terms )

ELEMENTARY FUNCTIONS where q>, algebraic in x{

n_1

19

', ..., x, reduces to the derivative of

8jn'. Let (i be a constant close to zero and let u^ be the function, analytic at x = a, which is obtained from g on replacing x j n ' by e} n ) + n and the other variables by their monomials. The same substitutions performed in (15) will produce the derivative of u^. Furthermore u , as a function of x and n, is analytic at x = a, u = 0.

^ SUMMARY we

17• In §§ made a close examination of the elementary functions. In formal work, it suffices to bear a few facts in mind. In the structure of a function u of order n, there appear certain monomials of orders 0, 1, ..., n. The function u is algebraic in these monomials, although some of the monomials may be used only for building monomials of higher order and may not appear effectively by themselves in the final expression for u. The derivative of u is algebraic in monomials which appear in the structure of u. If an expression for u is taken which employs as few as possible n-monomials, any algebraic relation among these n-monomials and any monomials of orders less than n, must hold identically in the monomials of order n. The reason is that a nonidentical relation would furnish an expression for one of the n-monomials which would permit us to write u with fewer such monomials. If 0 is an n-exponential in u, and if 9 is replaced in u by |i6, where \i is a constant, we secure a function whose derivative is obtained from the derivative of u by the same substitution. If 9 is an n-logarithm in u, and if 9 is replaced in u by 9 + u, we secure a function whose derivative is obtained from that of u by the same substitution. We are now prepared to take up questions of integration in finite terms.

Chapter I I ALGEBRAIC FUNCTIONS WITH ELEMENTARY INTEGRALS LIOUVILLE'S

FIRST THEOREM

1 . The meaning o f the e x p r e s s i o n

integrable

in finite

de-

terms

pends on the material with which one is working. An elementary function is said to he integrable in finite terms i f its integral is elementary. The f i r s t problem considered by Liouville in the field of integration in finite terms deals with the integration of algebraic functions.* We consider an algebraic function y(x) defined by an irreducible relation (1)

a„(x) ym + ai(x) y m_1 +...+ a j x ) = 0.

In dealing with the integral of y, we shall need only the barest knowledge of the nature, as a function, of the integral. Any element P(x - x 0 ) of y can be integrated into an element Q. The m.a.f. obtained from Q can be called an Integral of y and can be written /ydx. Q is determined to within an additive constant. By using a l l constants we obtain every integral of y, that is, every function whose elements have derivatives which are elements of y. Most of the time we can work on an even simpler basis. Let us consider some branch of y which is analytic in some simply connected area a. Then /ydx, which is defined to within an additive constant, is analytic throughout H. If now we ask whether y is integrable in finite terms, our question is a perfectly clear one. We are asking whether there is an elementary function u(x), of regular structure** at a point a in 81, whose derivative coincides with y for a neighborhood of a. Any further questions which may arise can be taken care of by considerations of analytic continuation. 2. We now state Liouville's f i r s t theorem on integration. THEOBEM: Let y ( x ) be an algebraic mentary.

(2) where

function

vhose

integral

Is

ele-

Then

/ydx = v 0 (x) + c t log v , ( x ) +...+ c r log v r ( x ) r is a positive

each c a

Integer,

each v ( x ) an algebraic

function,

and

constant.

* Liouville M . A l i s t of references is given at the end of the monograph. *+ 1, l i (Chapter I, §11) . When no chapter is mentioned in a reference, the chapter is that in which one is reading.

ALGEBRAIC INTEGRANDS

21

Of course, the integral may be algebraic. This case may be considered to be covered by our statement if we allow all c to be zero. An expression like the second member of (2) has an algebraic derivative. As one approaches the problem, it appears to be a foregone conclusion that the only possibility for aij elementary integral is that described in (2). If the answer contained an exponential, the exponential ought to survive after differentiation. Similarly one cannot imagine a logarithm disappearing unless it enters linearly. Considerations of this vague type had led Laplace, before Liouville's time, to conjecture the theorem established by Liouville, and Liouville claimed for his method of proof the merit of following these intuitive ideas. The proof of Liouville's theorem will be conducted as follows. Letting u be an integral of y, we shall suppose that u is elementary but not algebraic. Let n > 0 be the order of u. We shall suppose that we have for u an expression of the type described in I, 14, which involves as few as possible n-monomials, as in I, 11. We shall show first that none of the n-monomials is an exponential. It will then be shown that u has an expression like the second member of (2) with each log v¿ an n-logarithm and v 0 a function of order less than n. The proof will be completed by showing that n =1. 3- As has just been stated, we understand r n to be small as possible. We work at a point a at which u has regular structure. Let the algebraic function of (N + 1) of I, 11, from which u is obtained be represented by g(x, ..., x). Let every variable except xj n ' be replaced by its monomial. We obtain a function of x{ n ) and x which we shall represent by f(x{n',x). This latter function produces u when x} nl is replaced by ein). On this basis, letting 6 represent 6{n', we write (3)

u = f(0, x).

We now obtain the derivative of u, having regard to I, 15- On the basis of (3), we write (4)

g .

f0(e, x) jg + f, x i n )

+

fx *)

i s an identity in x j n ' and x. In particular, we may replace 9 in (5J by any function of x which i s analytic at the poiijt a and close in value to 9(a) at a. We shall replace 9 by ^9 where u i s a constant close to unity. Vie may thus write, for x close to a, (6)

y = f ^ t u e , x) n9 g

+ f x (ne, x)

where f e i s the result of replacing x( n > by U0 i n i' x {n). iy 1, 15, and even on the basis of i t s own structure, the second member of (6) i s the derivative of f(u9, x ) . By (5) and (6), f(9, x) and f(n9, x) have the same derivative and thus d i f f e r by a constant. This constant depends on n. We write (7) f(n8, x) = f(9, x) + 0(n). As f(|i9, x) i s analytic in n and x for |i =, 1, x = a, the function 3(h) must be analytic for n - 1. We differentiate (7) with respect to \i and find (8)

f^gtue, x) 6 = 0'(n),

the accent indicating differentiation. We put n = 1 and represent P ' ( l ) by c, obtaining the relation (9)

fg(9, x) 9 = c.

Again we apply Liouviile's principle; (9) holds identically in 9. We replace 9 by an independent variable z and have, for the neighborhood of x = a , z = 9(a), (10)

z f 2 ( z , x) - o.

ALGEBRAIC INTEGRANDS

23

Now (10) is a partial differential equation for f(z, x). It gives fz(z, x) = | so that (11)

f(z, x) = c log z + y(x),

where y is analytic at x = a. We notice that, as 9(a) f 0, log z is analytic for z » 9(a). To determine y(x), we replace z in (11) by any value z 0 clos6 to 9(a). We have f(z0, x) * c log ZQ + y(x). Hence, identically in z and x, (12)

f(z, x) = c log z - c log z 0 + f(z0, x).

In particular, we may replace z by 9 in (12) for the neighborhood of x = a. Thus (13)

u - f(9, x) = c log 9 - c log z 0 + f(z0, x).

In (13), log 9 is of order n-1 while f(z0, x), which by I, 13, is an elementary function of regular structure at a, involves fewer than r n monomials of order n. This contradicts the fact that the expression (3) of u is as economical as possible in n-monomials. We have thus proved that 9 is not an exponential. 4. We know now that the 9[ n ' are logarithms. We shall prove that u is a function of order less than n plus terms of the form c i 9[ n ' with constant c. We use (4). Let 9 = log w with w of order n-1. Then (14)

y = f e (9, x) Kl

+

f x (6, x).

As w and w' are of order less than n, (14) is an identity in 9. We replace 9 by 9 + u where n is a constant close to zero. Then (15)

y - fe+|t(6+n, x) «1 • f,(e+n, x).

The second member of (15) is the derivative of f(9+n, x). Then (16)

f(9+n, x) x f(9, x)

where (3 is analytic for to gives

+

f3(n)

« 0. Differentiation of (16) with respect fe+tl(9+n, x) - p'(n).

Putting |i = 0 and writing P'(G) - c 1 ( we have (17)

f e (9, x ) - o t .

24

ALGEBRAIC INTEGRANDS

Now (17) is an identity in 0. We replace 0 by a variable z and write fz(z, x) « Oj. Then f(z, x) • ci z + y(x). Determining y by a special value z 0 of z, we have f(z, x)

c 4 z - ct z 0 + f(z0, x).

Hence (18)

u =, f(0, x) » c t 0 - Cj z 0 + f(zQ, x).

In g(xi11', ..., x) of §3, let the variables corresponding to monomials of order less than n be replaced by their monomials. There results a function h(xin>, ..., x; x) where r represents r n . We have u = h(0 * C l ' There exist similarly, for i = 2, ..., r, constants c A such that 9h 3x[ n) Thus (20)

h = C l xj n ) +...+ c r x vrt For instance, in the second column of (31), v 01 , ..., v o t are various branches of v 0 . They need not be distinct branches; it is only the sets as a whole which are distinct. Let C^ be a curve which converts the first set into ylt v oi , ..., v rl . Now let * In Chapter III a more general question will be treated by algebraic methods.

AIX5EBBAIC INTEGRANDS

(32)

w0 = v 0 1 + v 0 2 + . . . +

29

V0t.

We are going to prove that w0 is unchanged when i t is continued along a curve C, starting and ending at §,, which returns to i t s e l f . For this we prove that C permutes the sets of (31) among themselves. To continue the i t h set along C i s to continue the f i r s t set along Ci and then along fi. Thus every set goes into some set when i t i s continued along C. Two distinct sets cannot go into the same s e t , since the continuation along C i s a reversible process. On this basis, when w0 is continued along C, the second subscripts of the v 0 i in (32) are permuted among themselves. This shows that w0 is unchanged. By 1, 12, w0 i s a branch of an algebraic function of x . We are going to show that this algebraic function is rational in y and x. Consider a curve Dt which s t a r t s at a and ends at a point b, avoiding singular points of y and v 0 . Along i t , y t and w0 can be continued. Let P(x - b) be the element secured for y t at b and Q(x - b ) the element secured for w0- I f a second path D, continues y t from a into P(x - b), then D2 must continue w0 into Q(x - b). Otherwise, using Di, and Ds reversed, we would have a path which leaves y t unchanged while ¿hanging w 0 . Thus the continuation of w0 furnishes an algebraic function which i s one-valued on the Riemann surface of y and i s therefore a rational combination of y and x. Let us explain the point just made in greater d e t a i l . Let the branches of y, analytic in H, be ylr . . . , ym. We denote w0 by w 0 1 . All paths which continue y! into yi continue w 0 1 , as was seen above, into a single definite function analytic in 81, which we denote by w 0 i. Now l e t functions A be determined by equations w01 = A0 + A4

yi

w02 = A0 +

y 2 + . . . + Am_1 y!?-1

Worn = A 0

+

A,

y

B

+ . . . + Am_1 y™"1

+

+ Am_1

y j "

1

.

The determinant D of this system i s not zero, for i t is a Vanderraonde determinant and the y i are distinct, we have Aj - D-i

Dj

where Dj is a determinant with the wQi in one column. By I , 12, Aj i s an algebraic function. We wish to see that Aj is unchanged when continued along any path which s t a r t s and ends at a. For this we

30

ALGEBRAIC INTEGRANDS

notice that such a continuation permutes the rows of D among themselves and performs the same permutation on Dj. Thus the continuation either leaves D and Dj unchanged or replaces them both by their negatives, so that Aj is unchanged. An algebraic function which is unchanged in this manner can have but one branch and so is rational. To sum up, w 0 = A 0 + Aj y +...+ A m _ x y1"-1 with each A a rational function of x. For i > 0, we use a product rather than a sum. If w ± = v lt v i 2 ... v i t , i * l , ..., r, vtj is a rational combination of y and x. We return now to (2). Let us show that, for i = 1, ..., t, we have a relation (33)

d.

+

/ y dx

=

Voi + ct log v t i +...+ c r log v r i

where the d are constants. We work in the neighborhood of a point a in a at which no v ^ with j > 0 is zero. Each log v ^ will have an element at a, secured in a definite manner. For i = 1, dt = 0 and (33) is merely (2). For any other i, we use a curve which continues the first set of (31) into ylt v Q l , v r i , avoiding singularities and values of x at which one or more Vji have zero values. Then each log v ^ is continued into a definite function log v ^ . Since yi returns to itself, its integral is changed through the addition of a constant, which we call Summing the equations (33) for i = 1, ..., t, we find (34)

h + t / y d x = w 0 + c1 log wt +...+ c r log w r

where h is the sum of the d. Thus • ,r-> (35)

r . we - h / y dx = — - —

c, , c log w, +...+

log W r .

In the second member of ($5), the algebraic tunctions are all rational in y and x. Equation (35 ^ has been established for the neighborhood of a. It is preserved under analytic continuation. Perhaps the simplest way to look at the situation is as follows. In (2), y is a function with a definite Riemann surface. It has been shown that the v may be selected so as to have the same surface. Each logarithmic derivative v|/vi is a function rational on this surface, and we have (36)

y = Vo + Ci V. — + . . . + cr Vh..

31

ALGEBRAIC INTEGRANDS

Every idea contained in (2) is contained in the simpler relation (36). ALGEBRAIC

FUNCTIONS

WITH ALGEBRAIC

INTEGRALS

9- As a special case of Abel's theorem, we have the result that if y is an aliébrate function of x and if the inteiral of y Is algebraic, the Integral is rational in y and x. For this special case we shall present a separate proof, almost entirely algebraic. Let u, the integral, be analytic in a. Let the irreducible equation satisfied by u be (37)

B c up

Bp = 0,

the B being polynomials in x with B0 f C. The existence of (37) is enough to show that u s a t i s f i e s in {I various equations of the form (38)

D0 u« +...+ Dq - 0

with each D a polynomial in y and x with constant coefficients, D0 not vanishing identically in x in a. We have in (37) a t r i v i a l instance of (38). From among a l l equations of type (38) satisfied by u, we select one which is of a least degree in u. We shall suppose that (38) is such an equation of least degree and proceed to prove that q = 1. Suppose that q > 1. We write (38) (39)

u" + M x ) u q _ 1 + . . . + pQ(x) = 0

with each (3 rational in y and x . Differentiating (39), we have (40)

««"I [qy + g i ]

[(q-1) y P l • g t ] •...+ [ y P q . 1 + g l ] - 0.

Now dx

_ 9£i 9x

+

aPi dy 3y dx

The derivative of y is found from (1) to be rational in y and x . Thus the coefficients in (4Ü) are rational in y and x. I f those coefficients were not zero identically in x, we would, clearing fractions, have an equation like (38) for u, of degree less than q. Thus, throughout a, Ifiii y + 5T and

r,

= 0

ALGEBRAIC INTEGRANDS

32 (41)

u = / y dx = 1 M x , y) + c. q

Now (41) is an equation (38) for u with q = 1. We have thus a contradiction of the assumption that q > 1. Then q = 1 and (38) expresses u rationally in y and x. As to the nature of the proof, the selection of an equation (38) of least degree is a device which replaces a reduction algorithm. We could instead start by differentiating (37) after dividing by B„ and replace dy/dx by its expression in terms of y and x. We would either secure an equation (38) of degree less than p or else obtain an expression for the integral as in (41). In the former case, the process would be repeated until a rational expression for u is secured. Prom the standpoint of the theory of abelian integrals, the result on algebraic integrals is a very obvious one. If the integral of y is algebraic, it can have no periods, either cyclic or polar. It is thus uniform on the surface of y and so is rational in y and x. 10. Liouville t31 furnished a method for determining whether an algebraic function has an algebraic integral and for obtaining the integral when it is algebraic. Without following the matter through to the very end, let us see what is involved in it. If y(x), defined by (1), has an algebraic integral, we have, by §8, / y dx = A 0 + A t y + ...+ A ^

y"1"1

with each A a rational function of x. Then

By (1) each y 1 " 1 dy/dx is a rational combination of x and y. Every such rational combination is a linear combination of 1, y, ..., y m - 1 with rational functions of x for coefficients. Let such linear combinations be substituted for each y 1 dy/dx in (42). Then (42) becomes (43)

with each (3 a linear combination of A 1 ( ..., A m _ 1 with rational functions of x for coefficients. The expression of a function as a linear combination of 1, ..., y m _ 1 is unique; two such expressions for a single function would furnish an equation for y of degree

33

ALGEBRAIC INTEGRANDS

l e s s than m. Thus we may equate c o e f f i c i e n t s of l i k e powers of y in both members of (43). This f u r n i s h e s f o r the A a system of l i n e a r d i f f e r e n t i a l equations of the f i r s t order, with known r a t i o n a l funct i o n s of x f o r c o e f f i c i e n t s . Oui problem i s t o determine whether t h i s system has a s o l u t i o n with each A r a t i o n a l . Liouville c a r r i e s t h i s question through, lie s h a l l not go into the d e t a i l s s i n c e , from the standpoint of the theory of l i n e a r d i f f e r e n t i a l equations as i t e x i s t s a t p r e s e n t , the question i s an e s s e n t i a l l y routine one. In I I I , 11 we work out a simple problem of t h i s type. No general method e x i s t s f o r determining whether an algebraic function whose i n t e g r a l i s not algebraic can be integrated with logarithms. This problem, which depends on d e l i c a t e questions in the theory of numbers, appears t o be a d i f f i c u l t one. RESIDUES

AND

INTEGRATION

11. The methods which were a t L i o u v i l l e ' s disposal f o r the examination of s p e c i a l algebraic functions were of a somewhat laborious type. Great s i m p l i c i t y can be secured by the use of the expansions of an algebraic function a t places on i t s Riemann s u r f a c e . We s h a l l r e c a l l the f a c t s . Let y(x) be a l g e b r a i c aDd l e t u(x) be an algebraic f u n c t i o n , not i d e n t i c a l l y zero, which i s uniform on the surface of y ; u thus i s r a t i o n a l in x and y . At a point on the surface of y which i s n e i t h e r a branch point nor a point a t u e i t h e r i s a n a l y t i c or has a pole; i t w i l l have a Taylor development a t the point in the f i r s t case and a Laurent development in the second. Now consider a branch p o i n t , corresponding t o a f i n i t e value x 0 of x . There u has an expansion (44)

a 0 ( x - x 0 ) P / r + a , (x - x 0 ) ' P + 1 » / r

where p i s some integer and r the number of sheets which c i r c u l a t e a t the p o i n t . We suppose t h a t a 0 f 0. If p > €, u i s said t o have a zero of order p a t the p o i n t , and if p < 0, u i s said t o have there a pole of order - p . At each point on the surface a t u has an expansion (45)

a0 xP/r +

x ' P " 1 ' ^ +...

with a 0 f 0. If the point i s not a branch p o i n t , pole of order p i f p > 0 and a zero of order - p Consider a f i n i t e point on the surface of' y a t We write the expansion of u a t t h i s point in the

r =• 1. There i s a if p < 0. which u has a pole. form (44), with

34

ALGEBRAIC INTEGRANDS

r = 1 if the point is not a branch point. The product by r of the coefficient of (x - x 0 ) - 1 in (44) is called the residue of u at the pole. At a point at we call the product by -r of the coefficient of x _ 1 in the expansion (45) of' u, the residue of u at the point. Thus there are two types of places at which we speak of residues, the poles at finite points and all points at We now consider u'/u, the logarithmic derivative of u, which is uniform on the surface of y. We examine a finite point on the surface, at which u has an expansion (44). If p = 0, the expansion of u'/u, found formally from (44), will contain no negative powers if r = 1. If r>l, the expansion may contain negative powers, but the exponents of x - x 0 will all exceed -1. Thus u'/u may have a pole at the point if r > 1, but, if so, the residue is zero. If p i 0, the expansion of u'/u starts with r-1 p(x - x j " 1 so that u'/u has a pole with p for residue. For a point at «=, we find from (45) that the residue of u'/u is -p. Consider now u'. None of its expansions can contain a term of exponent -1. Thus the residues of u' are all zero. 12. We return now to Liouville's theorem of §2, supposing that the integral of y is elementary but not algebraic. A set of r complex nunbers c 1 ( ..., c r will be called Independent if there does not exist a set of rational nunbers q 4 , ..., q r , not all zero, such that 1i Ci +•••+ q r c r » 0. Let an expression (2) of the integral of y be given which contains tains a minimum number of logarithms. We shall prove that c1, ..., c r are independent. Suppose, for instance, that Ci = s 4 c a +...+ s r c r with rational s. We can write the second member of (2) as v 0 + c s log v®2 vs +...+ c r log v / v r . In this expression, the functions of which logarithms are taken are algebraic. Thus r is not a minimum. This proves the independence of the c. When we have an expression for / y dx which contains a least nunber of logarithms, we can, as in §8, replace it by an expression with the same number of logarithms and with each algebraic function rational in y and x.

35

ALGEBRAIC INTEGRANDS

We s h a l l use, f o r / y dx, such an expression as has j u s t been des c r i b e d . We r e f e r t o (2). No v L with i > 0 i s a c o n s t a n t . An a l g e braic function which i s not a constant has a t l e a s t one zero and a t l e a s t one pole. Suppose, f o r instance, t h a t v, has a zero a t a f i n i t e point on the s u r f a c e of y , f o r which x * x 0 . At t h i s point v j / v , w i l l have a residue d i s t i n c t from z e r o . Let the c o e f f i c i e n t of (x - x 0 ) - 1 in the expansion of v j / v ^ a t the point under cons i d e r a t i o n be q 1 , i t l , . . . , r . The q are a l l r a t i o n a l and q t t 0. As the residues of v¿ are a l l zero, we see from (2) t h a t q j c t + . . . + q r c r i s not zero so t h a t y has a nonzero residue a t the p o i n t . S i m i l a r l y , i f v t has a zero a t a point a t y has a nonzero residue a t the p o i n t . Thus, if y(x) is an algebraic function whose integral is elementary but not algebraic, there must exist points on the surface of y at uhich y has resiaues distinct from zero.

ELLIPTIC

INTEGRALS

13- We s h a l l now demonstrate the nonelementary character of Legendre's e l l i p t i c i n t e g r a l s of the f i r s t and second kinds, r II

' J

dx s

r s

2

- X ) (1 - k X ) '

'J

x" dx s

n/(1 - x ) (1 - k* X s ) '

where k i s a constant with k 5 f 0, 1. We must show f i r s t t h a t there are no nonzero residues and then t h a t the i n t e g r a l s a r e not a l g e braic. For those versed in the theory of abelian i n t e g r a l s , the nonelementary c h a r a c t e r of follows immediately from §12. An i n t e g r a l of the f i r s t kind, 'on a surface of any p o s i t i v e genus, has an i n f i n i t e number of branches and has no logarithmic branch p o i n t s . Thus no i n t e g r a l of the f i r s t kind can be elementary. An i n t e g r a l of the second kind cannot be elementary if i t i s not a l g e b r a i c . We p r e f e r , however, t o t r e a t the question by more elementary methods. Consider . Let s s 5 s / ( l - x ) (1 - k x ) : 1, i k. Writing

(46)

1

(x - 1 ) " 1 / 2

y

x/(- x - 1 ) a - k s x a ) '

we see t h a t the function under the r a d i c a l in (46) i s not zero a t x = 1 , so t h a t the r e c i p r o c a l of i t s square root i s a n a l y t i c f o r

36

ALGEBRAIC INTEGRANDS

x = 1. Thus the residue of 1/y at x = 1 is zero. The same is true at the other three branch points. At 0 over 3 , and l e t i t be expressed with a least number of n-monomials. Let the function, algebraic over 3 , associated with u, be g(x{n),

. . . , x)

Let each accented variable except x| n 1 be replaced by i t s monomial. There results a function f ( x j n ) , x ) . Let 9 represent 9 { n 1 . We have u = f(9, x). Then (5)

y = f e ( e , x) e' + f x ( 9 , x ) .

The p a r t i a l derivatives in (5) are obtained by replacements from functions algebraic in the accented l e t t e r s over 3 . We now follow with no e s s e n t i a l change §§3> 4, 5 ° f Chapter I I . We secure the relation ( 4 ) . In § 6 we t r e a t the r a t i o n a l i t y question. OSTROHSKI'S METHOD OF FIELD EXTENSIONS 5 . We have followed above the Liouville proof pattern. Ostrowski uses a method which contains a genuinely novel idea and constitutes a noteworthy addition to L i o u v i l l e ' s technique. We s h a l l give an account of i t . We c o n s i d e r s as above. Let there be given any f i n i t e set of functions u , , . . . , u which are meromorphic in a and algebraic over

TRANSCENDENTAL INTEGRANDS

43

3. Let 3' be the totality of rational combinations of the u and functions in 3. Then 3' is a differential field, because the derivative of each u ± is rational in that Uj and functions in 3. We call 3' an algebraic extension of 3. By I, 12, every function in 3 1 is algebraic over 3. Now let u be analytic in a and algebraic over 3. Suppose that e u is not algebraic over 3. Let 3' be the differential field consisting of all rational combinations of e u and u' with coefficients in 3.* We call 3', or any algebraic extension of 3', an exponential extension of' 3. Similarly, let u, algebraic over 3, be analytic in JI and suppose besides that log u is uniform in SI but not algebraic over 3. Let 3' be the differential field consisting of all rational combinations of log u and u'/u with coefficients in 3. We call 3', or any algebraic extension of 3', a logarithmic extension of 3. Given two distinct differential fields 3 and 3 i if there exists a finite sequence of differential fields, 3, 3',

3 ( n ) = 3,,

each after the first an exponential or logarithmic extension of its predecessor, we call 3 4 an extension of 3 of positive rank. If 3 t is such an extension of 3 and if n, as above, is the smallest number of extensions which will permit the passage from 3 to 3 t , we call n the rank of 3 a with respect to 3. Now let y (x) be algebraic over 3 and let the integral u of y be elementary, but not algebraic, over 3. From our study of the structure of u, it is seen that we can find an area 8 t and an extension Si of 3, (relative to a, which contains u. After forming an algebraic extension of 3, one brings in, in succession, monomials analytic in Hi and functions algebraic in such monomials. Having such an area JIj in which extensions containing u exist, we use an extension 3t whose rank, r, is as small as possible. For all we can say offhand, r may depend on the area Sli which is chosen; r will not inerease if Ht is shrunk. We suppose Hi to be such that r is as small as possible. Let the successive extensions which produce 3t from 3 be 3', 3", ..., 3' r ' = 3 4 . We shall show that the extensions are all logarithmic extensions and that U = V0

+ C! log v, +...+ c r log v r

with each v x algebraic over 3. * Vote that as u ' i s a l g e b r a i c over 3, the higher d e r i v a t i v e s of u are r a t i o n a l in u* and functions in 3 '

44

TRANSCENDENTAL INTEGRANDS

F i r s t , l e t r = 1. Denoting by 9 the exponential or logarithm used in building 3 ' out of 3, we have u = v(e, x) with v algebraic in 0 and functions in 3. Ely the Liouville procedure, we find that u = v 0 + c 4 log v t with v 0 and v 4 algebraic over 3. Suppose now that our theorem i s proved for r = 1, . . . , s . Let r = s + 1. We simply replace 9 by 3 ' . 3 ( r l is of rank s over 3 ' . Then U = V0 + C! log v, + . . . + c s log v s with each v algebraic over 3' and hence algebraic in the 0 used above and functions of' 3. The procedure of Liouville then gives our theorem. Ostrowski's method of field extensions lends an intriguing Galoisian aspect to the theory of Liouville. As there are no groups in the Liouville theory, the resemblance i s not too deep. Ostrows k i ' s method consists, broadly speaking, in expressing u with a minimum t o t a l nunber of monomials of a l l orders. GENERALIZATION

OF ABEL'S

THEOREM

6. We shall now show that the v in (4) may be taken rational in y and functions in 3. Our method will be algebraic, rather than analytic as in I I , 8. F i r s t we secure a function t ( x ) , linear in the v with constant coefficients such that the v are rational in t and functions in 3. Rather than the r + 1 functions v, we consider two functions cp and y, analytic at a point a in H and algebraic over 3. One will see that our conclusion holds for any number of functions. Let

. We shall make, in F of (2), the substitution z = c y P l where c is an indeterminate. F becomes a collection of terms of the type c 1 a(x) y b , where b is rational. We shall show that the least b is a 0 and we shall find the terms with b = o 0 . From B 1 z 1 with Bj^ f 0, the lowest term secured is a

o l

W

ci y°i + l p i

If i = 0, or if i > 0 and (5) equals p t , o^ + i pt equals o 0 . If (5) is less than a 1 + i pt exceeds o 0 . Thus the least b is a 0 , and the sum of the terms with b = o 0 may be written y°° [k«(x) + kj(x) c +...+ k^(x) c g l ] . The bracket in this expression will be designated by c). If now t is the least value of b which exceeds o 0 , we may write (10)

F(x, y, c y P l ) . Vl (x, c) y°°

+ ¥l

(x, y, c) y*

where y, is a polynomial in c whose coefficients are series of terms of the form a(x) y d with d rational and nonnegative. In (10), we put c = c 1 (x) + zl

y

Pl

.

The first member of (10) becomes F(x, y, C! y P l + z t ), which equals F!(x, y, Zj). Hence (11) F, (x, y, z,) - (Pi (x, c, + z, y - P l ) y°° +

(x, y, c t + z, y" P l ) y T .

64

FRACTIONAL POWER SERIES Expanding in powers of z,, we find

, 1CM (12)

n,

=y

Oo-iPi «p^Ux, c j t-iPl ^ ' ( x , y, c t ) +y

where the superscript (i) denotes i differentiations with respect to c. Let us consider a B[ which is not zero. As x > o 0 ) we find that if ipj^ix, c1) does not vanish identically in x, we have - o 0 - i pt

(13)

hut that, if the vanishing does occur, o{ > c„ - i p t .

(14)

In particular, (14) holds for i = 0, so that a'0 > o 0 . On the other hand, s, being, as at the end of §3» the multiplicity of the solution ct(x) of (6), we find (13) to hold for i = s ± . We have (15)

°° - g l _ °o - gp

+

Op - oi

For i - s u the second term in the second member of (15) is p t . The first term in the second member is positive, since o'0 > a 0 . Thus p 2 , the greatest value of (9), exceeds p t . 6. Let g 2 be the greatest value of i for which (9) equals p 2 . We denote by k'^(x) the coefficient of y°° in B£. For i > 0, if B{ f 0 and if (9) equals p,, we let kj(x) denote the coefficient of y" 1 in B|. In the other cases with i > 0, we let kj(x) denote 0. We consider the equation (16)

kg(x) + k»(x) c +...+ k" (x) c g s = 0. e2

The coefficients in (16) are analytic in S4 of §3- I n some area in Bi, (16) has g 2 analytic solutions. Let c 2 (x) be one of these. We assign an area S 2 to c 2 in such a way that if c 2 is of multiplicity s 2 , the s 2 t h derivative with respect to c of the first member of (16) is distinct from zero throughout 8 2 for c = c2(x). We understand that S 2 lies with its boundary inside of B 4 . It may be that c 2 y P a annuls F t when substituted for z,. In that case Ps „ , P» Ci y + c2 y

FRACTIONAL POWER SERIES

65

is a series (4) which annuls F. If c 2 yP* does not annul F ^ we put z

i - o 2 y P s + z 2 , we write Fi(x, y, z j = F 2 (x, y, z 2 ),

and we give F 2 the treatment accorded to F and F t . 7. It nay be that at some stage in our process we are led to a finite series. C l (x)

y P l + ...• c r U ) y P r

which annuls F. In what follows, we assume that this does not happen, so that we are led to consider an infinite sequence of terms (17)

c, y P l , ..., c r y P r , ...

where the p increase with their subscripts and the c are all analytic at some point a contained in S 1 ( 8 2 , .... 8. We shall be able to build a series (4) out of (17) through the proof of various facts. We shall show that the p have a common denominator. The c(x), analytic at a, will be seen to be continuable over an area £ containing a. The series (18)

c,(x) y P l +...+ c r (x) y P r +...

will be found to converge for x in S and for |y| small and not zero. If s is the common denominator of the Pj, the replacement of y 1 ' 3 by v in (18) will produce a function of x and v, analytic for x in £ and for |v| small and not zero. 9. Towards proving the above statements, we shall show that gy, the degree of (16), does not exceed the multiplicity s, of c t . This will show that g 2 s g 4 . We inspect (15). The term (o0 - o|)/i is a maximum for i * s t . The term - o0)/i is less for i > s, than for i £ s,. Hence the greatest value of i for which the first member of (15) is a maximum cannot exceed s 4 . Then g 2 s s t s g t . 10. We can now prove that the p have a common denominator. From §9 we see that for j large, say for j > e, where e is some integer, the gj have a common value, say q. Let j have any fixed value greater than e. The equation (19)

k^'

k ^ 1 cq . 0

which determines Cj, must have q equal solutions. The first member of (19) is therefore of the form

FRACTIONAL POWER SERIES

66

k^J

(20)

1

(c - o p 9 .

This means, because Cj / 0, that k ^ ' in (19) is not zero. It follows that the ratio C J —l) (i_n o0 - a J i)

attains its maximum value pj for i = 1. This means that we can use, for the denominator of p^, the common denominator of the exponents of y in Fj_ 1 (x, y, Z j ^ ) . The same denominator can be used for the p i with i > j . It follows that the p i increase towards + =•= with i. 11. We shall show that the series (18) is a formal solution of F(x, y, z) = 0 . This will help us to establish analyticity properties of (18). By § 5» the quantity o 0 J

increases with j. As the o 0 J

all have

a common denominator, namely, that of the p, it must be that o ^ ' increases towards + 1. For j large we have, by §10,

(j ) (j ) (J I (j ) o0 - aJ P J + i - "o - Oi '

Hence (23)

o|J) = a ^

+

(q - 1) p J + 1 .

It is easy to see from (2), (21), and (22) that the set of all numbers aj^ ', for all i and j, is bounded from below. Indeed, if o is the least of the c^ associated with (2), no a ^ ' is less than

FRACTIONAL POWER SERIES

67 1

a - n IpJ. From (23) it follows, because q > 1, that a'^ tends towards + = with j. Now, by (22), (24)

9F

.1 ( X ' y ' Z l' - n z'""1' az

2 B^1

Zj +

B| J) .

J

The least exponent of y in b|J 1 is oiJ Also B ^ ' is the result of replacing z in 3F(x, y, z)/3z by c t yPl+...+ C j y P j . It follows that (18) is a formal solution of 3F/9z = 0. This contradicts the fact that (2) is irreducible. Thus q = 1. 13. We have, analogously to (11), (j-1) (25)

Fj(x, y, zj) = Tj (x,

Cj +

zj y" P J) y a °

i|/j(x, y, Cj + Zj y with T ( j _ 1 ) > OoJ

lf

J

) yT

. For j > e, we have, by (20),

(26)

We put, in (25), Zj = yPJ u. Then "PJ = Cj

+

u

and the equation Fj(x, y, Zj) = 0 gives, for u, the equation (27)

k,

(x) u = - ijij (x, y, Cj + u) y T

°°

Let s be the common denominator of the p in (17). We put y = v 3 in (27), and (27) becomes an equation for u in terms of x and v, (28)

ki j ) u = D 0 (x, v)

+

D,(x, v) u

D n (x, v) u n .

The D are series of positive integral powers of v with coefficients analytic in Bj which represent functions analytic for x in 8j and for |v| small. From (26) we see that 8j is taken so that kJJ'(a) f 0, where a is as in §7We are now able to solve for u in (28) by means of the implicit function theorem. As D ^ a , 0) = 0 for every i and as kp'(&) / 0, (28) defines u as a function of x and v, analytic for x in some area £ containing a, and for |v| small. Now Fj(x, y, Zj) is annulled formally when Zj is replaced by the series

68

FRACTIONAL POWER SERIES

(29)

c J + 1 ypJ+1 • . . . .

This means that (28) is satisfied by the series obtained by dividing the series in (29) by ypJ and then replacing y 1 / s by v . But (28) is satisfied by just one series of positive powers of v with coefficients analytic at a, namely, the expansion of the solution of (28) found above. Thus the coefficients in (18) are continuable over S, and (18) converges for x in E and for |y| small but not zero. Prom this follow the properties of (18) stated in § 8 . 14. We have obtained one of the series (4); l e t i t be called P t . By division, we find F(x, y, z ) . (z - P J G(x, y , z ) where G is degree n-1 in z, with coefficients of the type of the B in (2), except that y may enter in fractional powers. I f n > 1, we can treat G as F was treated, to obtain a second series (4). The fractional powers of y constitute no d i f f i c u l t y . We cannot use i r reducibility for G as we did for F in §12. However, i f q > 1 for G, G w i l l have a multiple formal solution, so that the same w i l l be true for F. The proof of the existence of the n series (4) is thus completed. 15- Let us suppose now that the in (1) are polynomials in y . We are interested in the neighborhood of y = (t),

0 < t s 1,

with q>(0) = 0(a), the function tp(t) having a large modulus for t = 1 and assuming nowhere on (0, 1) any of the values of 9 which annul K(9, a). Then F(0, x) can be continued along C from t = 0 to t - 1. As 1 is fractional and indeed the least fractional exponent. The least fractional exponents in v' and v 2 will be found in P i a i (z

- c) P i _ 1 ,

2 ai

a i (z

- c)Pl+Pl

We must therefore have pt = -1 and pj^ = -2ai. The terms of lowest degree in v' and v 2 are respectively,

INTEGRATION BY QUADRATURES - a! (z - c) ",

75

a 2 (z - c)~2

The expansion of the second member of (11) about z - c has no negative powers, since c f 0. Then - at + a 2 = 0 so that a t = 1. Hence

= -2, and pj^ is not a fraction.

Thus an irrational algebraic solution ol' (11) could have a critical point only at z - 0. Then every algebraic solution of (11) is rational. 8. We are going to show that, if (11) has a rational solution, p is an integer. Let (11) have a solution

v = EM Q(z)

with P and Q polynomials with no zero in common. From (11), we see that v has no pole at = and that the poles of v in the finite plane are all simple poles. There is certainly a pole at z = 0, and if there is a pole at c / 0, the term in l/(z - c) has unity for coefficient. Let v(«0 = h and let the zeros of Q be o,, ..., c r . From (11) we see that h f 0. Then v

. h + li + - 1 z

z - c,

+ ...+ — 1

.

z - Cr

We have for k the equation (15)

k 2 - k = p(p • 1),

so that k = p + l o r k = - p . The development of v about «= is v = h + fe-t-E +... . z Now k + r must be zero or the term 2h (k + r) z - 1 would be present in v 2 and would not be cancelled by v" or the second member of (11). Thus k is an integer. This means, by (15), that p is an integer. 9. We show finally that when p is an integer, (11) has a rational solution. There is no generality lost in assuming that p is positive. When p is either 0 or -1, (11) is satisfied by v = +1. If p < -1, we let p' = -p -1. Then p(p + 1) = P'(P' + 1). We put v = w'/w in (11) so that

76

INTEGRATION BY QUADRATURES w«

=

(1 + PÍP t D ) w. z4

Now l e t w = e z z~P u. Ihen (16)

u" + 2 (1 - 2) u' - 2 Eä . 0. z

z

Let us determine the power series u » a 0 + a 4 z + . . . + am z™ + . . . so as to s a t i s f y (16). We find equations - 2 p a

1

- 2 p a

»s(2-4p)

o

, 0

+ a 4 ( 2 - 2 p) = 0

a B [m(m - 1) - 2 m p] + a n _ 1 (2 • - 2 - 2 p)

.0.

We are supposing that p is a positive integer. Our equations give an „ in terms of a_in—l , unless m - 1 * 2 p. * But when m is 1p + 1, which i s less than 2 p + 1, we find that a n = 0. Thus, i f we take am . 0 f o r m > p + 1, we get a polynomial u which s a t i s f i e s (16). The logarithmic derivative of the corresponding w i s rational and sati s f i e s (11). This concludes our treatment of the equations of Riccati and Bessel. The existence of a single solution of (1) of the Liouville type is easily seen to imply that a l l solutions are of that type. The same holds f o r (2) i f we disregard the solution y • 0. FURTPER

STUDIES

10. Mordukhai-Boltovskoi [12] investigated the i n t e g r a b i l i t y by quadratures of linear d i f f e r e n t i a l equations of any order. In the following chapter we shall go with Mordukhai-Boltovskoi into the question of integrating algebraic d i f f e r e n t i a l equations of the f i r s t order in terms of elementary functions. There is another type of problem on integrability by quadratures. The linear equation of the f i r s t order (17)

y ' • P(x) y = Q(x),

where P and Q are any functions, can be integrated by two quadratures. This raises the problem of finding classes of equations, involving arbitrary functions, which can be solved by f i n i t e

INTEGRATION BY QUADRATURES

77

algorithms, with integration among the permitted operations. Such a question was examined by Maximovich [11] in 1885- As the author has not been able to secure Maximovich's paper or any account of it except those given in an abstract in the Janrbuch and in one in the Paris Comptes Rendus, he is unable to make a definite statement in regard to it. It appears that, with certain assumptions, Maximovich shows that the linear equations are, essentially, the only general class of equations of the first order which can be integrated in explicit form by quadratures.

Chapter VII IMPLICIT AND EXPLICIT ELEMENTARY SOLUTIONS OP DIFFERENTIAL EQUATIONS OF THE FIRST ORDER IMPLICIT

REPRESENTATIONS

1. In the preceding chapters we have discussed the possibility of representing functions in explicit form by means of certain operations. In solving differential equations, one is perfectly happy to end up with a relation among the unknown function, the variable, and arbitrary constants. For instance, the equation p- (i - ey) •= 1 dx has for solution y - ey = x + c. For no value of c is y an elementary function of x. There arise thus questions on the possibility of solving differential equations in finite Implicit terms. We shall consider some problems of this type. ELEMENTARY FUNCTIONS

OF TVO VARIABLES

2. For our purposes, we must construct the elementary functions of two variables, x and y. We consider an m.a.f. u(x, y). An element of u s a power series P(x - x0, y - y0). % the radius of convergence r of P we shall mean the least upper bound, finite or infinite, of those positive numbers p which are such that P converges for |x - x0| < p, |y - y 0 | < p. An immediate continuation of P is an element secured by developing P in powers of x - x,, y - y,, where |xt - x 0 | < r, |y4 - y0| < r. We shall call u fluent if for each element P(x-xD, y-y 0 ) of u, for each curve (1)

x = „(X), y = f(\)

(0 i U

1)

where cp(0) = x0, 0, there exists a curve (2)

x = 0 i s c l o s e enough t o ( x i _ i > y i _ i ) , u w i l l be continuable along (1) by a chain of e l e ments P(x - x i , y - y i ) , each a f t e r the f i r s t an immediate continua t i o n of i t s predecessor. The f i r s t segment i s given by (4)

x = x0 + t(xi - x0),

y = y0 + t(y, - y 0 ) ,

0 s t s 1.

I f x and y are replaced in P(x - x 0 , y - y 0 ) by t h e i r expressions in ( 4 ) , P becomes a function Q(t) a n a l y t i c f o r |t| s 1 and not zero f o r t - 1 . Let t = 0 be joined to t - 1 by a curve x = £ ( t ) , 0 s t s l with |t - t | very small along the curve, Q being d i s t i n c t from 0 along the curve, except.perhaps a t the f i r s t p o i n t . Then P(x - x 0 , y - y 0 ) i s a n a l y t i c along (5) x = x 0 + £ ( t ) ( X l - x 0 ) ,

y = y 0 + S ( t ) (yi - y 0 )

0 s t g 1,

and w i l l not be zero on (5) except perhaps a t the f i r s t p o i n t . One makes s i m i l a r replacements f o r the other segments. 3 . The v a r i a b l e s x and y w i l l be c a l l e d complete monomials of order z e r o . An a l g e b r a i c function u ( x , y) w i l l be c a l l e d a funct i o n of order z e r o . An a l g e b r a i c u(x, y) i s f l u e n t , f o r , given any curve ( 1 ) , we can replace i t by a curve (2) c l o s e to i t along which, except perhaps a t the f i r s t p o i n t , the discriminant o f the equation d e f i n i n g u and the c o e f f i c i e n t of the highest power of u a r e d i s t i n c t from z e r o . I f u i s a l g e b r a i c and nonconstant, e u and log u are c a l l e d complete monomials of order 0 . Of course e u i s f l u e n t ; so a l s o , by § 2 , i s log u. A branch of a complete monomial, a n a l y t i c in a region in the space of x and y , i s c a l l e d a monomial. Continuing, we obtain the elementary functions of x and y . The s t r u c t u r e of such a function i s described as in I , 1 1 . One u s e s , in the

80

IMPLICIT AND EXPLICIT SOLUTIONS

description, a branch of a function u of order n, analytic in a region U. One employs, in the place of a of I, 11, a point (a, b) in H. In (I), one uses r, algebraic functions of x and y, analytic for |x - a| < p 1? |y - b| < pt. In (N + 1), one uses an algebraic function of x, y, xj, ..., x ^ ' . I f a function has the structure just described at (a, b), it is said to be of regular structure at (a, b). IMPLICIT

SOLUTIONS

OF ALGEBRAIC

DIFFERENTIAL

EQUATIONS

4. Mordukhai-Boltovskoi [14] investigated differential equations of the form (6)

y 1 = f(x, y)

with f algebraic in both variables, inquiring as to whether (1) has a solution g(x, y) = c with c an arbitrary constant and g elementary. He proved the following theorem. THEOREM: Let y' = f(x, y) with f algebraic in x and y have a solution g(x, y) = c with g elementary. Then it has a solution (7)

ip0(x> y) + a 4 log tpl (x, y) +...+ a r log 0. Suppose that 9 = e v with v of order r 0 - 1. For w and x independent variables close to (b, a) we write (6) in the form (7)

v(w, x) = log f(w, x).

Now, letting w in (7) be w(x), we find, differentiating with respect of x, io\ (8)

i x)\ y(x) i \ + v i *•>» v (w, (w, x)\ = -S " *

+ M Sy W —i f(w, x)

w

>

.

In (8) y(x) is the function of which w(x) is an integral. We suppose, as we may, that y has regular structure at a. Now (8) must be an identity in w and x for a neighborhood of (b, a). Otherwise it would give a function in A with r s r 0 and with s < s 0 if r = r 0 . The two functions (9)

v(w + n, x),

log f(w + n, x)

are analytic in w, x, n for w = b, x = a, ^ = 0Let us imagine that in (9) we replace w by w(x). The two functions of (9) become functions of x whose derivatives are given hy the two members of (8) with w replaced hy w(x) + ix. We therefore have (10)

v[w(x) + n, x] = log i'[w(x) + n, x] + p(n)

with p analytic at n = 0. Then (11)

V

(w, x) . f " ( W ' x ) + c f(w, x)

for a neighborhood of (b, a) on and therefore identically in w and x. Integrating, we have in w and x (12)

v(w, x) = log f(w, x) + c w + y(x) .

To determine y(x) we consider a ( V , a') close to (b, a) for which (7) does not hold. For w = b', (12) gives

94

(13)

FURTHER IMPLICIT PROBLEMS Y(x) - v(b', x) - log f(V, x) - cb'.

By (7) and (12) we have for w - w(x), (14)

c w = - y(x).

Now c is not zero. If it were, (13) would imply that v(b', x) - log f(b', x) vanishes for every x, whereas it is not zero for x - a'. By (14), w(x) is elementary. If 9 - log v, (6) becomes f(w, x) • log v(w, x) and we find again that w(x) is elementary. We might have used, in what precedes, a different classification of functions of w and x, letting w, and every elementary function of x alone, be monomials of order zero. This would have spared us the considerations relative to monomials involving w. Having already classified our functions in Chapter VII, we found it easier to proceed as above. A classification of the new type will be used below in connection with Bessel's equation. In the introduction to my paper, [18], I mentioned the problem of determining whether the integral of an elementary function is elementary if it is one of a set of functions which satisfy a set of elementary equations. I stated that the formal elements of the proof for one function could be carried over to answer this question affirmatively. I wish to withdraw this statement. I do not have the details now, if indeed I ever did. In particular, the question as to whether the integral of an elementary function may be represented parametrically by elementary expressions may be of interest.

LIHEAR DIFFERENTIAL

EQUATIONS

OF THE SECOND

ORDER

11. In Chapter VI, we determined when a solution of Bessel's equation is a function of Liouville. We shall now examine the question of implicit representations. We shall-construct Liouville functions of w and x, letting w and every Liouville function of x be complete monomials of order 0. A function of order 0 is an algebraic combination of complete monomials of order 0. An m.a.f. u(w, x) is a complete monomial of order 1 if it is not a function of order 0 and if either

FURTHER IMPLICIT PROBLEMS

95

(a) u • e v with v of order 0 or (b) and ux are both functions of order 0. The construction continues in the usual way. 12. The d i f f e r e n t i a l equation y " + 2 P ( x ) y ' • Q(x) y = 0 goes over, under the transformation y » e-J" P dx^ into an equation (15)

w" »