Mathematics Handbook for Science and Engineering [5 ed.] 9789144031095

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LENNART RÄDE • BERTIL WESTERGREN

Machei atics 1 andbook

n

Studentlitteratur

for Science and Engineering

Lennart Räde Bertil Westergren

Mathematics Handbook for Science and Engineering

A Studentlitteratur

4n6 Copying prohibited Every reasonable effort has been made to give reliable data and information, but the authors and the publisher cannot assume reponsibility for the validity of all materials or for the consequences of their use. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.

The papers and inks used in this product are eco-friendly.

Art. No 2505 ISBN 978-91-44-03109-5 Edition 5:8 © Lennart Rade, Bertil Westergren and Studentlitteratur 1988, 2004 www.studentlitteratur.se Studentlitteratur AB, Lund Cover design: Tomas Dahlgren Printed by Elanders Beijing Printing Co. Ltd, China 2010

Contents

Preface 7 1

Fundamentals. Discrete Mathematics 9 1.1 Logic 9 1.2 Set Theory 14 1.3 Binary Relations and Functions 17 1.4 Algebraic Structures 21 1.5 Graph Theory 33 1.6 Codes 37

2 Algebra 43 2.1 Basic Algebra of Real Numbers 43 2.2 Number Theory 49 2.3 Complex Numbers 61 2.4 Algebraic Equations 63 3

Geometry and Trigonometry 66 3.1 Plane Figures 66 Solids 71 3.2 3.3 Spherical Trigonometry 75 3.4 Geometrical Vectors 77 3.5 Plane Analytic Geometry 79 3.6 Analytic Geometry in Space 83 3.7 Fractals 87

4

Linear Algebra 90 4.1 Matrices 90 4.2 Determinants 93 4.3 Systems of Linear Equations 95 4.4 Linear Coordinate Transformations 97 Eigenvalues. Diagonalization 98 4.5 4.6 Quadratic Forms 103 4.7 Linear Spaces 106 4.8 Linear Mappings 108 4.9 Tensors 114 4.10 Complex matrices 114 3

5

The Elementary Functions 118 5.1 A Survey of the Elementary Functions 118 5.2 Polynomials and Rational Functions 119 5.3 Logarithmic, Exponential, Power and Hyperbolic Functions 121 5.4 Trigonometric and Inverse Trigonometric Functions 125

6

Differential Calculus (one variable) 132 6.1 Some Basic Concepts 132 6.2 Limits and Continuity 133 6.3 Derivatives 136 6.4 Monotonicity. Extremes of Functions 139

7

Integral Calculus 141 7.1 Indefinite Integrals 141 7.2 Definite Integrals 146 7.3 Applications of Differential and Integral Calculus 148 7.4 Table of Indefinite Integral 153 7.5 Tables of Definite Integrals 178

8

Sequences and Series 183 8.1 Sequences of Numbers 183 8.2 Sequences of Functions 184 8.3 Series of Constant Terms 185 8.4 Series of Functions 187 8.5 Taylor Series 189 8.6 Special Sums and Series 192

9

Ordinary Differential Equations (ODE) 200 9.1 Differential Equations of the First Order 200 9.2 Differential Equations of the Second Order 202 9.3 Linear Differential Equations 205 9.4 Autonomous systems 213 9.5 General Concepts and Results 216 9.6 Linear Difference Equations 218

10 Multidimensional Calculus 221 10.1 The Space IV 221 10.2 Surfaces. Tangent Planes 222 10.3 Limits and Continuity 223 10.4 Partial Derivatives 224 10.5 Extremes of Functions 227 10.6 Functions f: IV -> It' (11:1n -an 229 10.7 Double Integrals 231 4

10.8 Triple Integrals 234 10.9 Partial Differential Equations 239 11 Vector Analysis 246 11.1 Curves 246 11.2 Vector Fields 248 11.3 Line Integrals 253 11.4 Surface Integrals 256 12 Orthogonal Series and Special Functions 259 12.1 Orthogonal Systems 259 12.2 Orthogonal Polynomials 263 12.3 Bernoulli and Euler Polynomials 269 12.4 Bessel Functions 270 12.5 Functions Defined by Transcendental Integrals 287 12.6 Step and Impulse Functions 297 12.7 Functional Analysis 298 12.8 Lebesgue Integrals 303 12.9 Generalized functions (Distributions) 308 13 Transforms 310 13.1 Trigonometric Fourier Series 310 13.2 Fourier Transforms 315 13.3 Discrete Fourier Transforms 325 13.4 The z-transform 327 13.5 Laplace Transforms 330 13.6 Dynamical Systems (Filters) 338 13.7 Hankel and Hilbert transforms 341 13.8 Wavelets 344 14 Complex Analysis 349 14.1 Functions of a Complex Variable 349 14.2 Complex Integration 352 14.3 Power Series Expansions 354 14.4 Zeros and Singularities 355 14.5 Conformal Mappings 356 15 Optimization 365 15.1 Calculus of Variations 365 15.2 Linear Optimization 371 15.3 Integer and Combinatorial Optimization 379 15.4 Nonlinear Optimization 383 15.5 Dynamic Optimization 389

16 Numerical Analysis 391 16.1 Approximations and Errors 391 16.2 Numerical Solution of Equations 392 16.3 Perturbation analysis 397 16.4 Interpolation 398 16.5 Numerical Integration and Differentiation 404 16.6 Numerical Solutions of Differential Equations 412 16.7 Numerical summation 421 17 Probability Theory 424 17.1 Basic Probability Theory 424 17.2 Probability Distributions 434 17.3 Stochastic Processes 439 17.4 Algorithms for Calculation of Probability Distributions 443 17.5 Simulation 445 17.6 Queueing Systems 449 17.7 Reliability 452 17.8 Tables 459 18 Statistics 479 18.1 Descriptive Statistics 479 18.2 Point Estimation 488 18.3 Confidence Intervals 491 18.4 Tables for Confidence Intervals 495 18.5 Tests of Significance 501 18.6 Linear Models 507 18.7 Distribution-free Methods 512 18.8 Statistical Quality Control 518 18.9 Factorial Experiments 522 18.10 Analysis of life time (failure time) data 525 18.11 Statistical glossary 526 19 Miscellaneous 530 Glossary of functions 544 Glossary of symbols 545 Index 547 6

Preface

This is the fifth edition of the Mathematics handbook for science and engineering (BETA). Compared to the previous editions a number of additions and corrections have been made. The Mathematics handbook covers basic areas of mathematics, numerical analysis, probability and statistics and various applications. The handbook is intended for students and teachers of mathematics, science and engineering and for professionals working in these areas. The aim of the handbook is to provide useful information in a lucid and accessible form in a moderately large volume. The handbook concentrates on definitions, results, formulas, graphs, figures and tables and emphasizes concepts and methods with applications in technology and science. The Mathematics handbook is organised in 19 chapters starting with basic concepts in discrete mathematics and ending with chapters on probability and statistics and a miscellaneous chapter. Crossreferences and an extensive index help the user to find required information. We have not included numerical tables of functions which are available on most scientific calculators and pocket computers. We have treated one variable and multivariable calculus in different chapters, because students, usually, meet these areas in different courses. In formulating theorems and results sometimes all assumptions are not explicitely stated. We are happy to have been able to draw on the expertise of several of our colleagues. Our thanks are especially due to Johan Karlsson, Jan Petersson, Rolf Pettersson and Thomas Weibull. We also want to thank Magnus Bondesson, Christer Bore11, Juliusz Brzezinski, Kenneth Eriksson, Carl-Henrik Fant, Kjell HolmAker, Lars Hornstrom, Eskil Johnson, Martin Lindberg, Jacques de Mare, Bo Nilsson and Jeffrey Steif for their helpful assistance. Furthermore we want to thank Jan Enger of the Royal Institute of Technology in Stockholm for providing new more exact tables of median ranks (section 18.1) and two-sided tolerance limits for the normal distribution (section 18.4). We also want to thank Seppo Mustonen of Helsinki University in Finland for providing us with an algorithm for the simulation of bivariate normal distributions (section 17.5) and Max Nielsen of Odense Teknikum in Denmark for an improved formula for approximation of the normal distribution function. 7

Some tables and graphs have been copied with permission from publishers, whose courtesy is here acknowledged. We are thus indebted to the American Statistical Association for permission to use the table of Gurland-Tripathis correction factors in section 18.2, the table of the Kolmogorov-Smirnov test in section 18.7 and the tables for Bartlett's test and the use of Studentized range in section 18.5. For the last two tables we also have permission from Biometrika Trustees. Furthermore we are indebted to the American Society for Quality Control for permission to use the table for construction of single acceptance sampling control plans in section 18.8 (copyright 1952 American Society for Quality Control) to McGraw-Hill Book Company for permission to use the table on tolerance limits for the normal distributions in section 18.4 (originally published in Eisenhart, et al: Techniques of Statistical Analysis, 1947) and to Pergamon Press for permission to use the graph of the Erlang Loss Formula in section 17.6 (orginally published in L. Kosten, Stochastic Theory of Service System, 1973). Lennart Rade, Bertil Westergren In the fifth edition of this handbook the sections 15.2 and 15.3 have been removed and replaced by three sections 15.2 — 15.4. These new sections have been written by Michael Patriksson of Chalmers University of Technology in Gothenburg, Sweden. I want to thank Michael Patriksson for his valuable contribution to the handbook. A new section on fractals has also been added to chapter 3. In other chapters some changes and corrections have been made. Since the former edition my dear friend and cowriter Lennart Rade has passed away and myself have retired from the faculty of the Mathematics Department of Chalmers University of Technology and the University of Gothenburg. I shall be grateful for any suggestions about changes, additions, or deletions, as well as corrections in the Mathematics handbook. It is finally my hope that many users will find the Mathematics handbook a useful guide to the world of mathematics. Bertil Westergren

8

1 Fundamentals. Discrete Mathematics

1.1 Logic Statement calculus Connectives Disjunction Biconditional Conditional Conjunction Negation

PvQ P ‹-> Q P --> Q PAQ — P or —P

P or Q P if and only if Q If P then Q P and Q Not P

Truth tables (F = false, T = true) P T T F F

Q T F T F

PvQ

PAQ T F F F

T T T F

P --› Q

P Q

T F T T

T F F T

P and —1P have opposite truth values.

Tautologies A tautology is true for all possible assignments of truth values to its components. A tautology is also called a universally valid formula and a logical truth. A statement formula which is false for all possible assignments of truth values to its components is called a contradiction. 9

1.1

Tautological equivalences ▪ - P P PA Q Q A P PvQ QvP (P A Q)A R PA (Q A R) (P v Q)vR P v (Qv R) PA (Q v R) (P A Q)v(P A R) Pv(Q A R) (P v Q) A (P vR) -,(PAQ) ▪ (PvQ) A PvPP PAPP Rv(P A -43) R RA(Pv-,P)R P Q.:=> -iPvQ ▪ -> Q) P A -Q P -> Q (-iQ-> -43) P -> (Q -> R) ((PA Q) -> R) (1 (P (P Q) (P -> Q)A (Q -> P) A -,Q) (P Q) (P Q)v(

(double negation)

(Distributive laws) (De Morgan laws)

Tautological implications PAQP PAQQ PPvQ Q PvQ (P -> Q) P (PQ) Q ▪ -> Q) P 3 ---> -,PA(PvQ) Q PA (P -> Q) Q ▪ (P -> Q) (P -> Q)A (Q -> R) (P R) (Pv Q)A (P -> R)A (Q -> R) R

T = any tautology

(simplification) (addition)

(disjunctive syllogism) (modus ponens) (modus tollens) (hypothetical syllogism) (dilemma)

F any condradiction

Exclusive OR, NAND and NOR The connective exclusive or is denoted "v" and is defined so that P v Q is true whenever either P or Q, but not both are true. The connective NAND (not and) is denoted by "T" and is defined so that P T Q --(PA Q) The connective NOR (not or) is denoted by ".L" and is defined so that P

Q -1(P vQ)

10

1.1

Tautological equivalences PvQ QvP (PvQ)vR Pv(QvR) PA (QvR) (PA Q)v(P A R) (PvQ) ((PA --,Q)v (=PA Q)) PvQ

Truth table P Q P\7Q PiQ P.LQ

PiQ QTP • Q1P

TT T F FT F F

P (Q R) v(Q A R) (P T Q)1 R (PA Q)v—iR P (Q R) =PA (QvR) (P Q) R (PvQ)A

The connectives terms of alone.

A) and

—P PT P —,1)P1P

F T T F

F T T T

v) can be expressed in terms of P Q —.(PT Q) P Q

F F F T

T alone or in

P v Q—,PT—Q P v Q --,(14Q)

Duality Consider formulas containing v, A and Two formulas A and A* are duals of each other if either one is obtained from the other by replacing A by v and vice versa. A generalisation of De Morgan's laws: —1A(P 1, P2 , •••> P n) A* (=1) 1 ,—P 2, • • • , Here Pi are the atomic variables in the duals A and A *. Normal forms If (for example) P, Q and R are statement variables, then the eight (in general 2n) formulas —,PAQAR, AQA —,PA—,QAR and PAQAR, PAQA—,R, PA—QAR, PA A-1QA—d? are the minterms of P, Q and R. Every statement formula A is equivalent to a disjunction of minterms, called its principal disjunctive normal form or sum-of-product form. Similarly A is equivalent to a conjunction of maxterms called its principal conjunctive normal form or product-of-sum form. (Cf. Boolean algebra, sec. 1.4). Example (Cf. example of Boolean algebra sec. 1.4) If P, Q R are the atomic variables, write equivalent sum-of-products and product-ofsums of A and if A =(PAQ) v (QA-1R). Solution. (Using Sv-6 T and distributive laws). 1. A (PAQA(Rv—,R)) v ((Pv—,P)AQA =I?) (PAQAR)v(PAQA—R)v v (PAQA —42)V( --IPA QA—S) (PAQAR)V(PAQA —,R)v(=PAQA—S) 2.

[The remaining minterms] —1R)V(—IPA QAR)V(-1PA-1QAR)V (—IPA QA—ER)

(Pn—,QAR)v(Pn

3. A

(=A) [Duality, see above] (=PvQv —IR) n (--,PvQvR) (Pv—.Qv—R) n (PvQv-1R) n (PvQvR)

4.

( -,Pv—,QvR) n (Pv—,QvR) 11

1.1

Predicate calculus Quantifiers Universal quantifier

Vx

For all x,

Existential quantifier

3x

There exists an x such that

Valid formulas for quantifiers

(3x)(P(x)vQ(x)) (3x)P(x)v(3x)Q(x) (V x)(P(x) A Q(x)) (V x)P(x)A (V x)Q(x) ▪ (3x)P(x) (Vx)-' P(x) ▪ (Vx)P(x) P(x) (V x)P(x)v(V x)Q(x) (V x)(P(x)vQ(x)) (3x)(P(x) A Q(x)) (3x)P(x) n (3x)Q(x) (Vx)(PvQ(x)) Pv(Vx)Q(x) (3x)(P A Q(x)) P (3x)Q(x) (Vx)P(x) Q (3x)(P(x) --> Q) (3x)P(x) Q 4=> (V x)(P(x) --> Q) P —> (Vx)Q(x) (V x)(P Q(x)) P (3x)Q(x) (3x)(P —> Q(x))

Formulas for two quantifiers

(V x) (V y)

(Vx)(Vy)P(x, y) (V y)(V x)P(x, y) (Vx)(Vy)P(x, y) (3y)(Vx)P(x, y) (V y)(V x)P(x, y) (3x)(Vy)P(x, y) (3y)(Vx)P(x, y) (Vx)(3y)P(x, y) (3x)(Vy)P(x, y) (Vy)(3x)P(x, y) (Vx)(3y)P(x, y) (3y) (3x) P(x, y) (V y)(3x)P(x, y) (3x) (3y) P(x, y) (3x)(3y)P(x, y) (3y) (3x) P(x, y)

(Vy)(Vx)

(ay) (v

(3x)(Vy)

(Vx)(3y)

(Vy) (3x)

(3y)(3x)

12

(3x)(3y)

1.1 Methods of proof Some proof methods Statement to be proved

Proof method

Procedure

Q

Modus ponens

P p --> Q ...Q

-IP

Modus tollens

-,Q p ---> Q ...--,P

Q

Disjunctive syllogism

P(a)

Universal instantiation

PvQ --,P .Q (V x)P(x) ...P(a)

P

Q

Direct proofs

Show that Q is true if P is true

P

Q

Indirect proofs

Show that -,Q

P Q

Implication proof

Show that P= Q and Q

P Q

Equivalence proofs

Show that R S where (R S) (P Q)

P

Contradiction

Assume P is false and derive a contradiction

-,(3x)P(x)

Contradiction

Assume (3x)P(x) and derive a contradiction

(3x)P(x)

Constructive proofs

Exhibit a such that P(a) is true

(ax)P(x)

Nonconstructive proofs

Show that -,(3x)P(x) implies a contradiction

-,(V x)P(x)

Counterexample

Show that (3x)-, P(x)

(V x)P(x)

Universal generalization

Show that P(a) is true for an arbitrary a

13

-,P P

1.2 Proof by induction A proof by induction that P(n) is true for all positive integers n proceeds in two steps. 1) Prove that P(1) is true. P (n + 1)) 2) Prove that ( n)(P (n) Example. k2 = n(n + 1)(2n + 1)/6.

We want to prove that k=1

1) The formula obviously holds for n = 1. 2) We make the induction hypothesis that F e=n(n+ 1)(2n + 1)/6 for some positive k=1 integer n. This implies n+

k 2+(n+ 1)2 =n(n+ 1)(2n+ 1)/6+(n+1)2 =(n+1)(n+2)(2n+3)/6 E1 k2 = k=1 k=1 This is the formula to be proved for (n+ 1). Thus the formula holds for all positive integers n.

1.2 Set Theory Relations between sets Notation: x e A, the element x belongs to the set A x0 A, the element x does not belong to the set A Let A and B be sets and Q the universal set. Then A is a subset of B, if

A c B, (Vx)(X E A = x E B).

(Sometimes the notation "A c B" is used and then "A c B" means that A c B and A # B). The set B is a superset to A, B D A, if A c B ThesetsAandBare equal, A = B, ifAcBABcA. The empty set is denoted by O. OcAcD AcA (AcB)A(BcC)AcC The power set Pa is the set of all subsets of D. If S2 has n elements, then P(Q) has 2n elements. 14

1.2 Operations with sets. Set algebra A

B

AUB

Definition {xES-2;xEAvxEB}

-41(40110

Intersection

A 11B

Ix e 0 ; x EA A x e B}

OM

Difference

A\ B

{xES2;xEAA.x0B}

4890

Symmetric difference

AAB

{xES2;x€A7xel3}

COD

Complementation

Ac, A' or CA

{xeS2;x0A}

Operation Union

Notation

Commutative laws AUB=BUA

ACIB=BCIA

Associative laws (A1113)0C=An(B11C)

(AU B)U C = AU (BU Distributive laws

A fl (B U C)= (A (1 B)U (A 11 C)

AU (B C)=(AU B)f-1(A U C) Complementation 0` =S2

12 c= 0

A UAc=c2

A flAc= 0

(Ac)c= A

De Morgan laws (A U B)`= 11Bc

(A nB)c-_,AcUBc

Symmetric difference A A /3=-B AA (AA B)AC= A A(B AC) A A 0=A A A A= 0 AA B=(A flBc)U(BflAc)

15

1.2 Cartesian product The Cartesian product A x B of A and B is A xB= {(a, b); aeilAbEB}. Here (a, b) is the ordered pair with first component a and second component b. A x(B U C)= (AxB)U (AxC) A x(B C) = (A xB) (A xC) (A U B)xC= (AxC)U(BxC) n B)xC=(AxC)n (BxC) The set of all functions from A to B is denoted BA.

Cardinal numbers Let c(A) denote the cardinal number of a set A. Writing A — B if there is a bijection between A and B, then

c(A)=c(B)A--B c(A) H is a homomorphism, then the kernel of g is a normal subgroup of G. b. (Fundamental theorem of group homomorphisms). If g is a homomorphism from (G, *) to (H, o) then G/ker(g) is isomorphic to g(G). 9. Group multiplication tables. The binary operation * in a group can be represented by a table. Example. There are two non-isomorphic groups of order 4: *

e

a

b

c

e a b c

e a b c

a b c e

b c e a

c e a b

{e, b} is a subgroup Cyclic: a2=b, a3=c, a4=e

e

a b c

e

a

b

c

e a b c

a e c b

b c e a

c b a e

{e, a}, fe, bl, fe, c}are subgroups Non-cyclic

Group order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Number of nonisomorphic groups

1 1 1 2 1 2 1 5 2

2 1 5 1 2 1 14 1 5 1

5

10. Sylow groups. If the order of G is pkm, where p is prime and does not divide m then G has a subgroup (called Sylow p-subgroup) of order pk . The number np of Sylow p-groups (i) divides m and (ii) n =-1 (mod p). Rings 1. Definition. A ring (R, +, -) or R is an algebraic system such that (i) (R, +) is an abelian group with identity element 0 and inverse -x of x, i e x +y =y +x, (x+y)+z=x+(y+z), x+0=0+x=x, x+(-x)=(-x)+x=0. 26

1.4 (ii) The operation • is associative and distributive over +, i e (x • y) • z= x • (y • z), x • (y+z)=x • y+x • z,(x+y)• z=x• z+y • z. (Note, below • is often omitted writing xy =x • y.) The ring is commutative if • is

commutative. 2. The ring has a "1" if • has an identity element, i e x • 1=1 • x=x. 3. A commutative ring with a "1" is an integral domain if for x # 0, xy = xz = y=z. (If xy = 0 with x and y 0, then x and y are called zero divisors.) 4. (A, +, .) is a subring of (R, +, .) if A c R and if -x E A (i) x, y E A x+y E A and xy E A, (ii) 0 E A, (iii) x E A 5. A subring (I, +, -) of (R, +, -) is an ideal of R if ax E / and xa E / for all ae I and all x E R. If R is commutative, then the smallest ideal of R containing an element a is (a)= {xa: x E R}. Ideals of this form are called principal ideals. An ideal MAR is a maximal ideal if whenever / is an ideal of R such that M c / c R, then either I= M or I= R. An ideal P of a commutative ring R is a prime ideal if abEP,a,bERaePorb€P. 6. Quotient rings. R/I= {cosets of I relative +} = {x +I: x E R} is a ring itself with addition S and multiplication defined by (x + /) (y + I)= (x + y)+ / and (x+/)0.0 (y +1)= xy +L The ideal / is the zero element of the quotient ring. 7. If R is a commutative ring and P is an ideal of R then P is prime •=> R/P is an integral domain. 8. If R is a commutative ring with "1" and M is an ideal of R then M is maximal •:=> R/M is a field. 9. Fundamental homomorphism theorem for rings. If R and S are rings and g: R —> S is a homomorphism (see above) with I= ker(g), then 0: R/I-* S definied by ap(x+ I)=g(x) is an isomorphism of R/I onto g(R). 10. The characteristic of a ring R is the least positive integer n such that na = a + a +...+ a=0 for all a E R. If such a number does not exist then the characteristic is 0. If D is an integral domain (i) then the characterstic is either 0 or a prime p. (ii) of characteristic 0, then D contains a subring isomorphic to Z. (iii) of characteristic p, then D contains a subring isomorphic to Zp.

Fields 1.

Definition. A field (F, +, .) or F is an algebraic system such that (i) (F,+, .) is a commutative ring with a "1", i e x+y=y+x, (x+y)+z=x+(y+z), x+0=x, x+(-x)=0, xY =Yx, (xy)z=x(yz), x(y+z)=xy+ xz, lx=x (ii) (F-{0}, -) is a group, i e for any x 0 in F there exists a multiplicative inverse x-1

such that xx-1=x-lx = 1. 2. The following inclusions hold: Fields c Integral domains c Commutative rings c Rings. 3. Subfield. (A, +, .) is a subfield of (F,+,.) if A c F and (i) a,bEAa+bEA and abEA (ii) 0 EA and 1 E A (iii) aEA -a EA and a EA, a#0 EA 4. A field has only the ideals F and {0}. 27

1.4

5.

(i) If F is finite and has characteristic p (p prime) then F has pn elements for some positive integer n. (ii) For any prime number p and any positive integer n there exists a field with pn

6.

(i) The field E is an extension of the field F if E contains a subfield isomorphic to F. (ii) If S is a subset of E then F(S) denotes the smallest subfield of E that contains both S and F. For example,

elements.

(a) R(i)=C. (b) Q(.12)= {a+ b : a, be Q} (c) Q(ir ) = { (a0 + al +agr2+ .+ ane)/(bo + b not all bi= 01

+ b2z2+. .+ bnir m): ai, bie Q,

Integers modulo n The set Z„ consists of the congruence classes modulo n which are [k]= [..., k —2n, k—n, k, k+ n, k +2n, ...) 1. [k] = [m] n divides k — m. Furthermore, Zn

= [0], [1],

[n — 1]1.

2. Operations: [k 1 ]+n[k2]=[k 1 + k2] [k i ] X n [k2] = [ki k2]

3. [ki ] = [mi], [k2] = [m2] 4.

(i) [ki + k2] = [m„ + m2]

(ii) [k„ k2] = [m im2]

(i) Zn is a cyclic group under +,„ (ii) Zn is a ring under +n and xn [Zn is a field n is prime.]

Polynomial rings If F is a field then F[x] denotes the set of polynomials E ak xk , ak E F, n arbitrary. k=0

1. Every ideal I of the ring (F[x], +, .) is a principal ideal of the form I= (g(x))= {f(x)g(x): flx)c F[x] } for some fixed g(x) E I. 2. In F[x] the division algorithm holds, i e if f(x), g(x) e F[x] there exist q(x) (the quotient) and r(x) (the remainder) in F[x] such that f(x)=q(x)g(x)+r(x) with degree r(x)< degree g(x). 3. A polynomial p(x) E F[x] of degree is irreducible if it is not a product of two polynomials of lower degree. 4. Factor theorem. If p(x) e F[x] and ae F, then (x— a) is a factor of p(x) p(a)= O.

Example The polynomial x2+1 is (a) reducible in C[x] because x2+1= (x— i)(x+ i) (b) irreducible in R[x] because x2+1 has no zero in R (c) reducible in Z2[x] because x2+1= (x+ 1)2. 5. The quotient ring F[x]l(p(x)) is a field p(x) is irreducible over F. 6. Let p(x) in F[x] be irreducible over F. Then F= F[x]l(p(x)) is a field extension of F and p(x) has a root in F. 7. If p(x) is any polynomial of positive degree over F, then p(x) has a root in some extension F of F. 28

1.4 8. If p(x)= up+ aix + ...+anxn is a polynomial over F and I= (p(x)), then each element of bi E F. F[x]l1 can be written uniquely as I+ (b0 + bl x +

Lattices Definition. A lattice (L, 5) or L (i) is a partially ordered set, i e reflexive, antisymmetric and transitive. (ii) has for every pair a, b E L a greatest lower bound (GLB) or meet or product and a least upper bound (LUB) or join or sum. Notation: GLB {a, b}=inf{a, b} = a*b = a b= a • b=ab LUB {a, b} =sup{a, b}=a@b=avb=a+b

A lattice

Not a lattice

Properties For any a, b, CE L: a1. a • b5a, a • b5b (Idempotent) 2. a • a= a a+a=a a+b=b+a (Commutative) 3. a • b=b • a (Associative) 4. (a • b) • c = a • (b • c) (a + b)+ c= a+(b + c) (Absorption) a +(a • b)=a 5. a • (a + b)= a (Distributive inequalities) 6. a+(b • c)(a+b) • (a+ c) a • (b + c) (a • b)+ (a • c) If equality holds, then the lattice is called distributive. 7. a5ba • b=a a+b=b a• • c and a+c5b+c 8. a5banda5c a5b • c 9. aboraca5b+c (21 andac ab+c or ac ab • c 10. A lattice is complete if every non-empty subset has a LUB and a GLB. Notation: 0 = The least element of L, 1= The greatest element of L. 11. An element b E L is a complement of a E L if a • b = 0 and a + b =1. If every element of L has a complement the lattice is called complemented. (Complemented, distributive lattices, see Boolean algebras below.)

Boolean algebras A Boolean algebra (B, +, •,', 0, 1) or B is a complemented, distributed lattice. The (unique) complement of a is denoted a'. Notation: a • b=ab=GLB{a, b}, a+ b=LUB{a, b}. (Advice: To understand the following laws, think of the Boolean algebra of a power set P(S) withA+B=AUB, A • B=A nB, A'=.5 — A, 0=0, 1=SandA /3AcB). Remark. Any finite Boolean algebra is isomorphic to the powerset of some finite set and thus has 2n elements for some n. Cf. Free Boolean algebras below. 29

1.4

Properties For any a, b, c E B: 1. a • 1,a, a • b.12 a +13, a, a +171, 2. a • a=a a+ a=a 3. a • b=b • a a+b=b+a (a+b)+c=a+(b+c) 4. (a • b) • c=a • (b • c) a+ (a • b)=a 5. a • (a + b)= a 6. a + (b • c) = (a + b) • (a + c) a • (b + c) = (a • b) + (a • c) 7. (a • b) + (b • c)+ (c • a)= (a + b) • (b +c) • (c + a) 8. a • b=a • c anda+b=a+cb=c 9. 0 5a5 1, a • 0=0, a+1=1, a • 1=a, a+0=a 10. a • a'=0, a +d=1, 0'=1, 1'=0 11. (a • b)'= a' + b', (a + b)'= a' • b' 12. a5b a • b=a a+b=b a5b > a • b'=0b'.Ca' a'+b=1

(Idempotent) (Commutative) (Associative) (Absorption) (Distributive)

(Complement) (De Morgan)

o Boolean algebra. Distributive laws valid. a' =f, e' =b unique.

o Not Boolean algebras. Distributive laws not valid. Complement of a not unique. Smallest Boolean algebra B= {0, 1 }:

1

0

1

0

0 1

1 1

0 0 0 1 0 1

1

Minterms The Boolean algebra B constituted by minterms (atoms) at, a2, a„ is defined by (taking the example n = 5) Each element a of B is a sum of some minterms, e.g. a = al + a3+ (24, 1 = ai + a2+ a3+ a4+ a5 (sum of all minterms). The complement a'=a2+a5 (sum of the remaining minterms). Sum +: Remember a,+ ai= a,. Product Remember a, • a,=ai and a, • ai= 0, i #j.

•

• •

+a2

ai+a3

a4+a5

• • •

a3

a4

0 Remark. The above Boolean algebra is isomorphic to a free Boolean algebra if n=2k, k integer. 30

1.4 Free Boolean algebras In a free Boolean algebra generated by n variables x1, x2, ..., xn, the elements are (finite) combinations of the xi, xi' and + and The elements can be written uniquely as a sum of minterms xi al x2a2 xnan, its disjunctive normal form (or similarly as a product of maxterms), where ai= 0 or 1 and xi° =xi, xil =xi . Sum of all minterms = 1. Remark. The number of minterms is 2' and the number of free Boolean expressions is 22". Duals The dual /3(x1, x2, ..., xn) of a Boolean expression a(x1, x2, ..., xn) is obtained by interchanging the operations • and +. Note: [a(xi, x2, ..., xn)]' = p(xi' , x2' , xn) (Cf. 11. above). Example (Cf. Example of sec. 1.1) Consider the free Boolean algebra generated by x, y and z. Find the sum of minterms and product of maxterms of b and b' if b = xy + yz'. Solution b = xy(z+ z')+ yz'(x+ x')= xyz+ xyz' +xyz +x'yz =xyz+ xyz' + x'yz' (= min7 + min6 + min2 = • 2, 6, 7) b' = (. 0, 1, 3, 4, 5) = +x'yz+xy'z'+xy'z b =(x')'=+0, 1, 3, 4, 5 = (x + y+ z)(x+y+z')(x+y'+z')(x'+y+z)(x'+y+z') =+ 2, 6, 7 = (x'+y'+ z')(x'+y'+ z)(x + y'+ z) If B= {0, 1}, the values of the above Boolean function are given in the table x

y

1 1 0 0

z 1 0 1 0

AY 1 1 0 0

yz' 0 1 0 0

b =.ry + yz' 1 1 0 0

b'

1 1 1 1 0 0 0 0

1 1 0 0

1 0 1 0

0 0 0 0

0 1 0 0

0 1 0 0

1 0 1 1

0 0 1 1

Minimization of Boolean polynomials Given a Boolean polynomial (expression, function), how to reduce it to its simplest form? This problem may be solved using succesively the reduction idea xyz +xyz'=xy(z + z')=xy. A systematic way of simplifying a Boolean polynomial (McCluskey's method) is illustrated by the example (written in disjunctive normal form): b= xyzw + xy'z'w + x'yzw + x'yz'w'+ xyzw'+ xyz'w'+ x'yzw' Enumerate the atoms (minterms) in a first column (a). Beginning from the top, compare the terms. 31

1.4 Those which differ by only one variable and its complement are reduced, and the reduced new terms are listed in the next column (b). This procedure is repeated until no more reductions can be made. Finally go backwards in the columns and pick up terms until all atoms are covered. (c)

(a)

(b)

1 xyzw 2 xy' z'w 3 x'yzw 4 x'yz'w' 5 xyzw' 6 xyz'w' 7 x'yzw'

1,3 yzw 1,5 xyz 3,7 x'yz 4,6 yz'w' 4,7 x'yw' 5,7 yzw'

1,3,5,7 yz 1,3,5,7 yz 4,5,6,7 yw' (No 2 is missing)

Thus, b= yz+ yw'+ xy'z'w

Logic design Input-output table Gate

Input a

Output

Logic gate symbols (IEC 612-12)

b a

1

0 0 0 1

a b

&

0 1 0 1

0 1 1 1

a b

>1

a+b

0 0 1 1

0 1 0 1

0 1 1 0

a b

=1

a(Db

NAND gate 1'

0 0 1 1

0 1 0 1

I 1 1 0

a b

& •

(ab)..a.4- b.

NOR gate 1

0 0 1 1

0 I 0 1

1 0 0 0

a b

1 0

Inverter —1, —

0 1

AND gate

0 0 1 1

0 1 0 1

OR gate v, +

0 0 1 1

Exclusive Or gate v,0

32

•

>1 •

a'

a b-ab

(a+b)..a.b.

1.5

X•

• z •

y Z•

Realization of the Boolean expression x (y+z) + y'z

1.5 Graph Theory A graph G is an ordered tripple (V, E, q)), where V is the set of nodes or vertices, E is the set of edges and cp is a mapping from E to ordered or unordered pairs of V.

F7iD Directed graph Undirected graph or digraph

Multigraphs

Simple graphs

An edge of a graph which is associated with an ordered pair of nodes is directed. An edge which is associated with an unordered pair of nodes is undirected. A graph with every edge directed (undirected) is called a directed (undirected) graph. A graph is simple if there is at most one edge between all pairs of nodes. If the graph has parallel edges it is called a multigraph. The converse G-1 =(v, E-1) of a digraph G= (V, E) is a digraph in which E-1 is the converse of the relation E (i e the arrows have opposite direction).

33

1.5

11* Outdegree = 3 Total degree = 4 Indegree = 2 Total degree = 3

V

V1 V2 V3

node simple path of length 2

V1 V3 v1 V2

edge simple path of length 3

V1 V2 V3 V1

simple cycle of length 3

For a directed graph the outdegree of a node v is the number of edges which have v as their initial node and the indegree of v is the number of edges which have v as their terminal node. The total degree of v is the sum of the outdegree and the indegree of v. For an undirected graph the degree of a node v is the numbers of edges, which are incident with v. For a digraph a path is a sequence of edges such that the terminal node of every edge is the initial node of the next. The length of a path is the number of edges in the sequence. A path is node (edge) simple if all nodes (edges) in the path are different. A path with origin and end in the same node is called a cycle. V2

V3

Node v4 is reachable from v1. Node v5 is not reachable from any other node. V5

A node u of a simple digraph is reachable from a node v if there exists a path from v to u. If for any pair of nodes of such a graph each node is reachable from the other, the graph is (strongly) connected. For a simple digraph a maximal strongly connected subgraph is called a strong component, and every node lies in exactly one strong component. Thus the strong components constitute a partition of the digraph. Matrix representations A: 0 1 1 0 0

1 0 1 1 0

0 0 1 0 0

0 0 0 1 1

0 1 0 1 0

For a simple digraph (V, E, (p) with V= { v1, v2, ..., } the adjacency matrix A = [au] is the n x n matrix such that 34

1.5

a ii =

{ 1 if (vi, vi )E E 0 otherwise

Let Ak be the kth power of A. The element in row i and column j of Ak is the number of paths of length k from node vi to node The path (connection, reachability) matrix P= [pa] of a simple digraph with nodes { v1, v2, ..., vn} is defined such that 1 if there is a path from vi to v./ or i =j 0 otherwise {

Then

P = (1+ AP-1) = 1+ A+ A(2) +

A(n-1)

where A(k) denotes the Boolean power matrix (with entries 0 or 1) calculated with Boolean matrix addition and mutiplication. Properties Converse digraph. If G is represented by A then G-1 is represented by AT (the transpose of A). For a symmetric (or for an undirected) graph, A =AT. 1. Two nodes vi and vj are in the same strong component .4=. [PAPT]y=pu pii= 1. 2. [AA T]ij = number of nodes which are terminal nodes of edges from both vi and vi 3. [ATA]; =number of nodes which are initial nodes, whose edges terminate in both vi and v./ 4. [A% = number of paths of length k from vi to v./. 5. [A(k)]ij = 1

There is a path of length k from vi to v./.

[A(k)]ij= 0 = There is no path of length k from vi to vi.

'frees

branch node at level 2

• E-- leaf at level 3

A directed tree is a digraph without cycles such that exactly one node called the root has indegree 0 while all other nodes have indegree 1. Nodes with outdegree 0 are called leaves or terminal nodes. All other nodes are called branch nodes. The level of a node is the length of its path from the root. 35

1.5

Weighted digraphs A weighted digraph is a digraph in which each directed edge (vi, vi) is assigned a positive number (the weight) wii = w(vi, 9. If there is no edge from vi to v./ then wii= 00. The graph can be represented by a weighted adjacency matrix W = (wii). The weight of a path is the sum of the weights of the edges occuring in the path. b

00 2 6 5 co 4 oo 7 W = 0. 00 00 1 2 00 .0.0 co 1 00 00 00 00 00

a= b= c= d=

v1 v2 v3

v4

z= V5

Dijkstra's algorithm for finding the shortest path To find the shortest path from a to z. The instructions in the brackets [...] are not needed if only the minimum weight of a path from a to z is sought. Notation: TL= Temporary Label, PL= Permanent Label, SP= Shortest Path. Step 0. Set PL(a)=0 and V= a, TL(x)= 00, x# a. Here and below the node which most recently is assigned a PL is denoted V. [SP(a)= {a}, SP(x)= 0, all x#a]. Step 1. (Assignments of new TL.) Set for all x without PL new TLs by TL(x) = min(old TL(x), PL(V)+w(V, x)) Let y be that node with the smallest TL. Set V=y and change TL(y) to PL(y). [(i) If TL(x) is not changed, do not change SP(x). (ii) If TL(x) is changed, set SP(x)= {SP(V), x}] Step 2. (i) If TL(V)= 00, then there is no path fram a to z. Stop. (ii) If V= z then PL(z) is the weight of the shortest path from a to z. Stop. [The shortest path is SP(z).] (iii) Return to step 1.

Example Consider the weighted graph above. Let a= v1 and z = v5. Step 0.

PL(a)= 0 V= a SP(a)= {a} TL(b)= .0 SP(b)= 0 TL(c)= .0 SP(c)= 0 TL(d)= 00 SP(d)= 0 TL(z)= oo SP(z)= 0 V=a PL(V)={a}

Iteration 1. Step 1. PL(a)= 0 TL(b)= 2 TL(c)= 6 TL(d)= 5 TL(z)= oo .•. V = b, PL(V)= 2

Iteration 2. Step 1. PL(a)= 0 SP(a)= {a) PL(b)= 2 SP(b)= {a, b} TL(c)= 6 SP(c)= {a, c} TL(d)= 5 SP(d)= {a, cl} TL(z)= 9 SP(z)= 0 V = d, PL(V)= 5 SP(V)= {a, Iteration 3. Step 1. PL(a)= 0 SP(a)= {a} PL(b)= 2 SP(b)={a, b} TL(c)= 6 SP(c)= {a, c} PL(d)= 5 SP(d)= {a, d} TL(z) = 6 SP(z) = {a, d, z} V= z , PL(V)= 6. Stop. Shortest path= {a, d, z} with weight 6.

SP(a)= {a) SP(b)-= {a, b) SP(c)={a, c} SP(d)= {a, d} SP(z)= 0 SP(V)= {a, b}

36

1.6

1.6 Codes Matrix group codes Below, arithmetic modulo 2 is used, i.e. 0+0=0, 0+ 1 = 1 + 0 = 1, 1 + 1 = 0 and 0.0=0.1=1.0=0, 1 • 1=1. Notation: Z2= {binary n-tuples} = {a = (al, a2...., an): a,=0 or 1, all 1. The Hamming distance H(a, b) between a, b in ZZ is defined by H(a,b) = number of coordinates for which ai and bi are different. The weight of a is H(a, 0) = number of coordinates # 0 in a. m 2. An (m, n)-code K is a one-to-one function K: X c Z7 ---> Z2, 3. A code K: Z7 —> Zn2 is a matrix code if x = x'A, where the matrix A is of type (m, n). 4. The range of K is the set of code words. , 5. A code K: Z2 —> Zn2 is a group code if the code words in Z2n form an additive group. 6. The weight of a code is the minimum distance between the code words = the minimal number of rows in the control (decoding) matrix C whose sum is the zero vector. 7. A code detects k errors if the minimum distance between the code words (the weight) is at least k+ 1. A code corrects k errors if the minimum distance between the code words (the weight) is at least 2k+ 1. Encoding and decoding matrices Encoding k parity checks x=x'A message X.E Z2

Control

Transmission fly

Y

received code word yE ZZ

code word XE Z2 (n=m+k)

z=yC error syndrome ZE Zk2

The encoding matrix: A = 1/m, Q] (normalized form) (mm)(m,k) (m,n)

The control matrix: C=

(m k)

(n k)

1k(Ic,k)

(Sometimes C T is called the parity check matrix). Decoding procedure: (Cf. example below.) 1. If z = 0: y is a code word. The decoded message x' is the first m coordinates of y. 2. If z#0: Find yi of minimal weight (the coset leader) with the same error syndrome z. Then y+yl is (probably) the transmitted code word and the first m coordinates the meassage x'. To find yi, write z as a sum of a minimum number of rows of the matrix C, say z rii+ ri2+ + ris . Then y = eii+ ei2+ + eis , where ek = (0, ..., 0, 1, 0, ..., 0) with the "1" on the kth place. (Alternatively, a decoding table may be used.) 37

1.6 8. Hamming codes. In a Hamming code the rows of C= Cn are formed by all binary r-tuples 0, i.e. n= 2r-1. For example. eT

3

[1 1 10 1

CT 7

=

1 1 1 01 00 1 1010 10 1011001

Example 100110 Given A = 0 1 0 1 0 1 001111

i.e. m=3, k=3, n=6.

Decode (a) y= (110011), (b) y = (011111) and (c) y = (111111) and find the message. Solution: -1 1 0 101 Control matrix C = 1 1 1 100 010 001

Weight of C=3, row(1)+ row(4)+row(5)= (000). The code detects 2 errors and corrects 1 error. (See 6. and 7. above.)

(a) z=yC= (000). Thus, y is a code word. Message x'= (110). (b) z=yC=(101)=r2 (rd row of C). yi=e2 = (010000) = yi C= (101). Thus, y+y1=(001111) and x'=(001) (c) z = yC= (011) = r1 + r2 = r5 + r6. No code word of weight 1 has this error syndrome. The error has weight 2. No "safe" correction can be made.

Polynomial codes Consider the polynomial ring Z2[x] with coefficients in the field Z2. In taking sums and products of polynomials in Z2[x] the laws of Z2 have to be respected. [E.g. (1 +x+x2)+(l+x+x3)=x2+x3 and (1 +x+x2)(1 +x+x3)= 1 +x4+x5.1 Identifying f(x)E Z2[x] with the binary vector (sequence) of the coefficients off [e.g. 1+x+x3 (1101)], the following definition can be made: Definition. Given the generator polynomial g(x)=a0+a1x+ ...+akxk H (opal ak), the polynomial code K: Z2 --> Zn2 (n=m+k) maps x=p(x)E Zn2 to y=q(x)E Zn2 by q(x)=g(x)p(x). The last equality can also be written (q0 q1 G=

ito a l O opal O

qn-1)= (POI • • • pm_i)G, where ak 0 01 ak 0..0 of type (m, n)

0 a0

a

Thus, a polynomial code is a matrix code (not necessarily normalized). 38

1.6 Example K: 4 —> ZS , g(x)= 1 + x2+ x3 . X = (X0X1) H X0 -1- X I X,-"A (X0-1- XIX)( 1 -I- X2 + X3) = x0 + XIX+ X0X2 + (X0-1- ).X3+ XiX4 (X0, Xi, X0, X0+ Xi, Xi) = y

Thus, (00)

(00000), (01) -(01011),(10) -.(10110),(11) -(11101)

BCH-codes (BCH =Bose, Ray-Chaudhuri, Hocquenghem) Algebraic concepts and notation, see sec. 1.4. Let Z2 denote a field which is an extension of Z2 such that any polynomial with coefficients in Z2 has all its zeros in Z2 . 9. (xi + x2+ ... + xn)2 =x12 +x22++xn2 since 1 + 1 = 0. 10. Two irreducible polynomials in Z2[x] with a common zero in Z2 are equal. 11. Let an Z2 be a zero of an irreducible polynomial g(x)E Z2[X] of degree r. Then for any positive j (i) oti is a zero of some irreducible polynomial gi(x)e Z2[x] of degree (ii) gi(x) divides 1 +x2 -1. (iii) degree gf (x) divides r. 12. The exponent of an irreducible polynomial g(x)e Z2[x] is the least positive integer e such that g(x) divides 1 +xe. Note: e 5 2r-1, where r=degee g(x) and e divides 2r-1. 13. For any positive integer r there exists an irreducible polynomial of degree r and exponent 2r-1. Such a polynomial is called primitive. 14. If g(x) is a yrimitive polynomial of degree r and a E Z2 is a zero of g(x), then a° =1, a, a2, a2 -2 are different (and a2r-1 = 1). 15.

A0=0 g(a2)=g(a4)=g(a8) =... =0 (Cf. 9. above). Definition and construction of BCH-codes 1° Decide a minimal distance 2t + 1 between the code words and an integer r such that 2r> 2t+ 1. 2° Choose a primitive polynomial g1(x) of degree r (see table) and denote by an i2 a zero of gi (x). 82t(x) of degree 3° Construct (cf example below) irreducible polynomials g2(x), with zeros a2, ..., a 2t, respectively. 4° Let g(x) of degree k (k is always tr) be the least common multiple of the polynomials gi (x), 82t(x) (i.e. product of all different of the polynomials gi (x), g2r(x))5° The BCH-code is that polynomial code K: Z2 ---> Z; generated by g(x). Here, n = 2r-1, m =n - k and the weight of the code is at least 2t + 1.

39

1.6

Example 1° According to the above notation, take t = 2, r = 4 (and thus n=15). 2° Choose gi (x)= 1 + x3+ x4. (Cf the table of irreducible polynomials below.) 3° Construction of the polynomials g2(x), g3(x), g4(x): g1(x)=g2(x)=g4(x) since gi(a= gi(a2)= gi(a4)= 0 (see 10 and 15 above). It remains to construct g3(x) with g3(a')= 0. One way to do that is the following (another way is to use the table below): Set g3(x)=1+Ax+Bx2+Cx3+Dx4. Now, 81(a)=0= a4+ a3+1=0 = a4 =1 + a3. Therefore, recursively, a5=aa4=a(l+a3)=a+a4=l+a+a3 a6 =aa5=a(l+a+a3)=a+a2+a4=1+a+a2+a3 a9=a3a6 =a3(1+a+a2+a3)-=a3+a4+a5+a6=1+a2 a12 = a3a9 =a3(1 + a2 )= a3 + a5 =l+a g3(a3)=0 1 +Aa3 +B(1 +a+a2+a3)+C(1 +a2)+D(1 +a)=0. Identifying coefficients yields 1+B+C+D=0,B+D=0,B+C=0,A+B=0A=B=C=D=1. Thus, g3(x)= 1 +x +x2+x3+x4. 4° g(x)= gi (x)g3(x)=(1+ x3+ x4)(1+ x+ x2+ x3+ x4)=1+ x + x2+ x4 + x8. 5° Degree g(x)= 8 = k, m =7. E.g. (1100100) 1 +x+x4 (1+x+x4)(1+x+x2 +x3+x4 +x8)= = 1 +x3 +x°+x9 +x12 (100100100100100).

Examples of known BCH-codes m

n

t

m

n

t

m

n

4 5 6 7 11 11

7 15 31 15 15 31

1 3 7 2 1 5

16 18 21 24 26 30

31 63 31 63 31 63

3 10 2 7 1 6

36 64 92 139 215 231

63 127 127 255 255 255

5 10 5 15 5 3

Table of irreducible polynomials in Z2[x] Explanations 1. Given a polynomial p(x) =akxk Z2[x] the reciprocal polynomial p*(x) is defined k=0 by p*(x)= y, akin-k [e.g. p(x)= a+ bx+ cx2 H (a,b,c) = p*(x)*--> (a,b,c)*=(c,b,a) k=0 c+bx+ax2 2. If p(x) is irreducible, [primitive] and has a zero a, then p*(x) is irreducible, [primitive] and has zero a-1. Therefore, in the table below only one in the pair p(x) and p*(x) is listed, i.e. the sequence of coefficients may be read either from the left or from the right to obtain an irreducible polynomial. 40

1.6 3. For each degree, let a be a zero of the first listed polynomial. The entry following the bold j is the minimum polynomial of af. How to find the minimum polynomial not listed, use 10, and 15, above and a2-1 = 1. [Example: Let a denote a zero of theyolynomial (100101) 1 +x3 +x5 of degree 5 (see table). By the table it is clear that a-1 is a zero of (111101) and a5 a zero of (110111). Then the minimum polynomial of (remember a31 = 1) a, a2, a4, a8, a16, a32 = a is pi(x)= (100101) a3, a6, a12, a24, (a48= 0148-31 =)a17 is p3(x)= (111101) a5, a10 WO, (a4043,9,a18isp5(x)=(110111) Using the zeros of the reciprocal polynomials, the minimum polynomial of (a-t =)a.30, (a 2=)a29 (a8=)°,27, (a 8=)a23 (a-16 =)a15 is p i *(x)= (101001) (a-3=)a28, (a-6=)a25, (06-12 =)0(19, (a38=)a7, a14 iS p3*(x)= (101111) (a-5=)0126, (a-10=)0121, (a-20=)al

a22, (a44=)«13 iS p5*(x)= (1 11011)]

4. The exponent e of the irreducible p(x) of degree m can be calculated by e = (2m- 1)1(GCD(r - 1, j)). (In factorization of 2m-1 the table of factorization of Mersenne numbers in section 2.2 may be used.) [E.g. The exponent of (1001001) is e = (26-1)1 (GCD(26 1,7) = 63/GCD(63,7) = 9].

TABLE OF ALL IRREDUCIBLE POLYNOMIALS IN Z2[x] OF DEGREE (P = Primitive, NP = Non-primitive) Degree 2 (a3 = 1) 1 111 (P) Degree 3 (a7 = 1) 1 1011 (P)

[Note: (1011)* = 1101 (P) is also irreducible etc. below. See 2. above.]

Degree 4 (a15 = 1) 1 10011 (P)

3

11111 (NP)

5

111

3

111 101 (P)

5

110 111 (P)

Degree 5 (a31 =1) 1 100 101 (P) Degree 6 (a63 = 1) 1 100 0011 (P) 9 1101

3 101 0111 (NP) 5 110 0111 (P) 11 10 1101 (P) 21 111

7

100 1001 (NP)

Degree 7 (a127 = 1) 1 1000 1001 (P) 3 1000 1111 (P) 5 1001 1101 (P) 7 1111 0111 (P) 9 1011 1111 (P) 11 1101 0101 (P) 13 1000 0011 (P) 19 1100 1011 (P) 21 1110 0101 (P) 41

1.6

Degree 8 (a255= 1) 1000 11101 (P) 7 1011 01001 (P) 13 1101 01011 (P) 19 1011 00101 (P) 25 1000 11011 (NP) 43 1110 00011 (P)

1

3 9 15

21 27 45

1011 10111 (NP) 1101 11101 (NP) 1110 10111 (NP) 1100 01011 (NP) 1001 11111 (NP) 1001 11001 (NP)

5

11 17

23 37 51

1111 10011 (NP) 1111 00111 (P) 10011 1011 00011 (P) 1010 11111 (P) 11111

85 111 ( 511 =1) Degree 9a 1 10000 10001 (P) 7 10100 11001 (NP) 13 10011 10111 (P) 19 11100 00101 (P) 25 11111 00011 (P) 35 11000 00001 (NP) 41 11011 10011 (P) 51 11110 10101 (P) 73 1011 83 11000 10101 (P)

3 9 15

21 27 37

43 53 75

85

10010 11001 (P) 11000 10011 (P) 11011 00001 (P) 10000 10111 (NP) 11100 01111 (P) 10011 01111 (P) 11110 01011 (P) 10100 10101 (P) 11111 11011 (P) 10101 10111 (P)

5

11 17

23 29 39 45 55 77

11001 10001 (P) 10001 01101 (P) 10110 11011 (P) 11111 01001 (P) 11011 01011 (P) 11110 01101 (P) 10011 11101 (P) 10101 11101 (P) 11010 01001 (NP)

Degree 10 (023 =1) 1 7 13

19 25 31 37

43 49

10000 001001 (P) 11111 111001 (P) 10001 101111 (P) 10111 111011 (P) 10100 100011 (P) 11000 100011 (NP) 11101 100011 (P) 10100 011001 (P) 11101 010101 (P) 11100 101011 (NP) 10111 000001 (NP)

3 9 15

21 27 33

39 45

51 57 69 71 75 10100 011111 (NP) 77 85 10111 000111 (P) 87 91 11010 110101 (P) 93 101 10000 101101 (P) 103 107 11001 111001 (P) 109 149 11000 010101 (P) 155 171 11011 001101 (NP) 173 55

10000 001111 (NP) 10010 101111 (NP) 10110 101011 (NP) 11111 101011 (P) 11101 111011 (NP) 111101 10001 000111 (NP) 11000 110001 (NP) 10101 100111 (NP) 11001 010001 (NP) 11011 010011 (P)

73

10100 001011 (NP) 10011 001001 (NP) 11111 111111 (NP) 11101 111101 (P) 10000 100111 (P) 10010 101001 (NP) 11011 011111 (P)

83 89 99 105 147 165 179

341 111

42

5

11 17

23 29 35

41 47 53

59

10100 001101 (P) 10000 110101 (NP) 11101 001101 (P) 10000 011011 (P) 10100 110001 (P) 11000 010011 (P) 10111 100101 (P) 11001 111111 (P) 10110 001111 (P) 11100 111001 (P) 11101 000111 (P) 11110 010011 (P) 10011 010111 (P) 110111 11110 000111 (NP) 10011 101101 (NP) 101001 11010 001001 (P)

2.1

2 Algebra

2.1 Basic Algebra of Real Numbers Sum and product symbols

n

X i = k Xk = X i + X 2 + + X n i=1

xk = x1 x2 ...x„

k=1

k=1

Algebraic laws (commutative laws) (associative laws) (distributive laws)

a+b=b+a ab=ba (a+ b)+ c = a + (b + c) (ab)c = a(bc) a(b+c)=ab+ac (a+b)(c+d)=ac+ad+bc+bd a b c d

a c ac -b' d - =bd

a + c ad+-bc b -d bd

a d = ad b c be

a+(-b)=a-(+b)=a-b (+a)(- b)=(-a)(+ b)= - ab

a+(+b)=a-(-b)=a+b (+a)(+ b)= (-a) (- b)= a b

a

+a -a = a +a -a +b -b b -b +b

b

(laws of sign)

a 0)

2.1

Table of Binomial Coefficients (Z) 0

1

2

3

4

9

8

7

6

5

10

1 2 3 4 5

1 I I 1 1

2 3 4 5

6 10

6 7 8 9 10

1 1 1 1 1

6 7 8 9 10

15 21 28 36 45

20 35 56 84 120

70 126 210

252

11 12 13 14 15

1 11 I 12 1 13 1 14 1 15

55 66 78 91 105

165 220 286 364 455

330 495 715 1001 1365

462 792 1287 2002 3003

924 1716 3003 5005

3432 6435

16 17 18 19 20

1 1 1 1 1

16 17 18 19 20

120 136 153 171 190

560 680 816 969 1140

1820 2380 3060 3876 4845

4368 6188 8568 11628 15504

8008 12376 18564 27132 38760

11440 19448 31824 50388 77520

12870 24310 43758 75582 125970

48620 92378 167960

184756

21 22 23 24 25

1 I I 1 1

21 22 23 24 25

210 231 253 276 300

1330 1540 1771 2024 2300

5985 7315 8855 10626 12650

20349 26334 33649 42504 53130

54264 74613 100947 134596 177100

116280 170544 245157 346104 480700

203490 319770 490314 735471 1 081575

293930 497420 817190 1 307504 2 042975

352716 646646 I 144066 1 961256 3 268760

26 27 28 29 30

1 1 1 1 1

26 27 28 29 30

325 351 378 406 435

2600 2925 3276 3654 4060

14950 17550 20475 23751 27405

65780 80730 98280 118755 142506

230230 296010 376740 475020 593775

657800 888030 1 184040 1 560780 2 035800

1 562275 2 220075 3 108105 4 292145 5 852925

3 124550 4 686825 6 906900 10 015005 14 307150

5 311735 8 436285 13 123110 20 030010 30 045015

31 32 33 34 35

1 1 I 1 1

31 32 33 34 35

465 496 528 561 595

4495 4960 5456 5984 6545

31465 35960 40920 46376 52360

169911 201376 237336 278256 324632

736281 906192 1 107568 1 344904 1 623160

2 629575 3 365856 4 272048 5 379616 6 724520

7 888725 10 518300 13 884156 18 156204 23 535820

20 160075 28 048800 38 567100 52 451256 70 607460

44 352165 64 512240 92 561040 131 128140 183 579396

, yc

11

12

13

14

15

22 23 24 25

705432 1 352078 2 496144 4 457400

2 704156 5 200300

26 27 28 29 30

7 726160 13 037895 21 474180 34 597290 54 627300

9 657700 17 383860 30 421755 51 895935 86 493225

10 400600 20 058300 37 442160 67 863915 119 759850

40 116600 77 558760 145 422675

155 117520

31 32 33 34 35

84 672315 129 024480 193 536720 286 097760 417 225900

141 120525 225 792840 354 817320 548 354040 834 451800

206 253075 347 373600 573 166440 927 983760 1476 337800

265 182525 471 435600 818 809200 1391 975640 2319 959400

300 540195 565 722720 1037 158320 1855 967520 3247 943160

6 6 For k> - use (nk ) —— (n n k) ' e'g' (112 ) = (14 ) 182° 2 47

16

601 080390 1166 803110 2203 961430 4059 928950

17

2333 606220 4537 567650

2.1 Some inequalities 1. ixyl

IXY1 < ,*X2 + Ely2) , any E>0

(x2 + y2)

2. The triangle inequality. n k=1

xk

I 1.Xkl k=1

1 1 3. Holder's inequality. If - + - = 1, p, q> 1 then P q 1

1 (n

k =1 IX kY kl

E k=1

d

k=1

4. Cauchy's inequality. 2 (n xk yk xk2 k=1

n

yk2

(equality yk=cxk)

k=1

k=1

5. Minkowski's inequality. If p> 1, xk, yk > 0 then I

y (xk+yk)3

p

(

)( 1 n

yf)

(k 1

=

+k= 1

(equality yk = cxk)

Percent ( ) Growth factor The growth factor associated with a change of p % is (1 +p/100). The value qn of a quantity after n succesive changes of pk % (k=1, n) is

qn

q (kL TOT)A1 - 00) + TA)

Calculation of interests p=rate of interest 1. (Compounded interest) Value of capital c after t years: t ) c( + 100J 2. (Present value). The capital c falls due in t years. Present value:

c

t 2) 100J 48

2.2

3. (Annuity). Yearly instalment on a loan c, payed during t years by equal amounts: + Y• 100) 100 t 100) 4. (Inflation). An inflation of p % implies a money value decrease of 100p % 100 + p

2.2 Number Theory Number systems N Z Q R C

natural numbers integers, (Z± positive integers) rational numbers real numbers complex numbers

Natural numbers. N={0, 1, 2, 3, ... }. Sometimes 0 is omitted. Integers. Z={ 0, ±1, ±2, Rational numbers. Q= {p/q: p, qE Z, The numbers in Q can be represented by a finite or a periodic decimal expansion. Q is countable, i.e. there exists a one-to-one correspondence between Q and N. Real numbers. R= {real numbers}. Real numbers which are not rational are called irrational. Every irrational number can be represented by an infinite non-periodic decimal expansion. Algebraic numbers are solutions of an equation of the form an f + + a0= 0, ak E Z. Transcendental are those numbers in R which are not algebraic. R is not countable. (Example: 4/7 is rational, ,„3 is algebraic and irrational, e and 7t are transcendental.) Complex numbers. C= {x+iy : x, ye R}, where i is the imaginary unit, i.e. i 2=-1. 49

2.2 The supremum axiom For any nonempty bounded subset S of R there exist unique numbers G=supS and g= infS such that: (i) g G, all xe S (ii) For any E>0 there exist x1 E S and X2 E S such that x1 > G—E and x2 P2, ••• , Pn•

3. There are infinitely many primes. (Euclid) 4. For every positive integer n there exists a string of n consecutive composite integers. 5. If a and b are relatively prime then the arithmetic sequence an+b, n=1, 2, ... contains an infinite number of primes. (Lejeune-Dirichlet) The following conjectures have not been proved. 6. Every even number is the sum of two odd primes (the Goldbach conjecture). 7. There exist infinitely many prime twins. Prime twins are pairs like (3, 5), (5, 7), and (2087, 2089). Unique factorization theorem Every integer >1 is either a prime or a product of uniquely determined primes. The function gr(x) The function value ir(x) is the number of primes which are less than or equal to x. x n(x)

100 25

200 46

300 62

400 78

500 95

600 109

700 125

800 139

900 154

x tr(x)

1000 168

2000 303

3000 430

4000 550

5000 669

6000 783

7000 900

8000 1007

9000 1117

x TO) x ir(x)

10000 1229 105 9592

20000 30000 40000 50000 60000 70000 80000 90000 2262 3245 4203 5133 6057 6935 7837 8713 106 107 108 109 1010 78498

664579

x Asymptotic behavior: tr(x)-- as x --> In x 50

5761455

50847534

455052512

2,2 The first 400 prime numbers 2 3 5 7 11 13 17 19 23 29

179 181 191 193 197 199 211 223 227 229

419 421 431 433 439 443 449 457 461 463

661 673 677 683 691 701 709 719 727 733

947 953 967 971 977 983 991 997 1009 1013

1229 1231 1237 1249 1259 1277 1279 1283 1289 1291

1523 1531 1543 1549 1553 1559 1567 1571 1579 1583

1823 1831 1847 1861 1867 1871 1873 1877 1879 1889

2131 2137 2141 2143 2153 2161 2179 2203 2207 2213

2437 2441 2447 2459 2467 2473 2477 2503 2521 2531

31 37 41 43 47 53 59 61 67 71

233 239 241 251 257 263 269 271 277 281

467 479 487 491 499 503 509 521 523 541

739 743 751 757 761 769 773 787 797 809

1019 1021 1031 1033 1039 1049 1051 1061 1063 1069

1297 1301 1303 1307 1319 1321 1327 1361 1367 1373

1597 1601 1607 1609 1613 1619 1621 1627 1637 1657

1901 1907 1913 1931 1933 1949 1951 1973 1979 1987

2221 2237 2239 2243 2251 2267 2269 2273 2281 2287

2539 2543 2549 2551 2557 2579 2591 2593 2609 2617

73 79 83 89 97 101 103 107 109 113

283 293 307 311 313 317 331 337 347 349

547 557 563 569 571 577 587 593 599 601

811 821 823 827 829 839 853 857 859 863

1087 1091 1093 1097 1103 1109 1117 1123 1129 1151

1381 1399 1409 1423 1427 1429 1433 1439 1447 1451

1663 1667 1669 1693 1697 1699 1709 1721 1723 1733

1993 1997 1999 2003 2011 2017 2027 2029 2039 2053

2293 2297 2309 2311 2333 2339 2341 2347 2351 2357

2621 2633 2647 2657 2659 2663 2671 2677 2683 2687

127 131 137 139 149 151 157 163 167 173

353 359 367 373 379 383 389 397 401 409

607 613 617 619 631 641 643 647 653 659

877 881 883 887 907 911 919 929 937 941

1153 1163 1171 1181 1187 1193 1201 1213 1217 1223

1453 1459 1471 1481 1483 1487 1489 1493 1499 1511

1741 1747 1753 1759 1777 1783 1787 1789 1801 1811

2063 2069 2081 2083 2087 2089 2099 2111 2113 2129

2371 2377 2381 2383 2389 2393 2399 2411 2417 2423

2689 2693 2699 2707 2711 2713 2719 2729 2731 2741

Gaussian Primes A Gaussian Prime is a complex number of the form z = a + ib, where a, b are real integers and which contains no factor of the same form except ±1 and ±z. For example, the ordinary real prime number 5 is not a Gaussian Prime because 5 = (2 - i)(2 + i). The Gaussian Primes are of three kinds: 1. ±1±i 2. ±p and ± ip, where p is a prime number of the form 4n+3, i.e. p=3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, ... 3. ±a±ib, where a, b #0, unequal and a2+b2 is prime, i.e. a+ib=l+2i, 1+4i, 1+6i, 1+10i, 1+16i, ..., 2+i, 2+3i, 2+5i, 2+7i, 2+13i, ..., 3+2i, 3+8i, 3+10i, 3+20i, ..., 4+i, 4+5i, 4+9i, 4+11i, 4+15i, 51

2.2

Prime Number Factorizations from 1 to 999 3

4

5

6

7

8

22.3 2.11 25 2.3-7

3.11 -

22 2.7 23.3 2.17 22.11

3.5 52 5.7 32.5

2.3 24 2.13 22-32 2.23

33 -

23 2.32 22.7 2.19 24.3

3.13 72

3.17 34 7.13

22.13 2.31 23.32 2.41 22.23

32.7 3.31

2.33 26 2.37 22.3.7 2.47

5.11 5.13 3.52 5.17 5.19

23.7 2.3-11 22.19 2.43 25.3

3.19 7.11 3.29 -

2.29 22.17 2-3.13 23.11 2.72

3.23 32.11

3-37 112 3.47

2.3.17 24.7 2.61 22.3.11 2.71

3.41 7.19 11.13

23.13 2.3-19 22.31 2.67 24.32

3.5.7 5.23 53 33.5 5.29

2.53 22.29 2.32.7 28.17 2.73

32.13 3.72

22.33 2.59 27 2.3-23 22.37

7.17 3.43 -

2.3.52 25.5 7.23 2.5-17 32.19 2 2 -32-5 2.5.19 -

23.19 2.34 22.43 2.7.13 26.3

32.17 3.61 -

2.7.11 22.41 2.3.29 23.23 2.97

5.31 3-5.11 52.7 5.37 3.5.13

22.3.13 2.83 24.11 2.3.31 22.72

3.59 11.17 -

2.79 23-3.7 2.89 22.47 2.32.11

3.53 132 33.7 -

20 21 22 23 24

23-52 2.3-5-7 22.5.11 2.5.23 24.3-5

2.101 22.53 2.3.37 23.29 2.112

7.29 3.71 35

22-317 2.107 25.7 2.32.13 22.61

5.41 5.43 32.52 5.47 5.72

2.103 23.33 2.113 22.59 2.3.41

32.23 7.31 3.79 13.19

24.13 2.109 223-19 2.7-17 23.31

11.19 3.73 3.83

25 26 27 28 29

2.53 22-5.13 32.29 2.33.5 23.5.7 2.5-29 3.97

22.32.7 2-131 24.17 2.3.47 22.73

11.23 3.7-13 -

2.127 23 -3-11 2.137 22.71 2.3.72

3.5.17 5-53 52.11 3.5.19 5.59

28 2-7-19 22.3.23 2.11.13 23.37

3-89 7.41 38.11

2.3.43 2267 2.139 25.32 2.149

7.37 32.31 172 13.23

30

22.3.52

31 32 33 34

2.5.31 26.5 2-3.5-11 22.5.17

7.43 3.107 11.31

2.151 23.3.13 2.7.23 22.83 2.32.19

3.101 17.19 32.37 73

24.19 2.157 22.34 2.167 23.43

5.61 32.5.7 52.13 5.67 3.5.23

2.32.17 22.79 2.163 24-3.7 2.173

3.109 -

22.7.11 2.3.53 23.41 2.132 22.3.29

3.103 11.29 7.47 3.113 -

35 36 37 38 39

2.52-7 23.32.5 2.5.37 22.5.19 2-3.5-13

33.13 192 7.53 3.127 17.23

25.11 2.181 22.3.31 2.191 23.72

3.112 3.131

2.3.59 22.7-13 2.11-17 27.3 2.197

5.71 5.73 3.53 5.7.11 5.79

22.89 2.3.61 23.47 2.193 22.32-11

3.7-17 13.29 32.43 -

2.179 24.23

32.41

2.33.7 22.97 2.199

3-7.19

40 41 42 43 44

24.52 2.5.41 22-3.5-7 2.5-43 23.5.11

3.137 32.72

2.3.67 22.103 2.211 24.33 2.13.17

13.31 7.59 32.47 -

22.101 2.32.23 23.53 2-7.31 22.3.37

34.5 5.83 52.17 3.5-29 5.89

2.7.29 28.13 2.3-71 22.109 2.223

11.37 3.139 7.61 19.23 3.149

23.3.17 2-11.19 22.107 2.3.73 26.7

3.11-13 -

45 46 47 48 49

2.32-52 11.41 22.5.23 2.5.47 3.157 25.3.5 13.37 2.5.72 -

2.227 24.29 2.3.79 22.112 2.13.19

5-7.13 3.5.31 52.19 5.97 32.5.11

23.3.19 2.233 22.7.17 2.35 24.31

32.53 7.71

2.229 33.17 22.32.13 7.67 2.239 23.61 3.163 2.3.83 -

0

n

I

0

...

1 2 3 4

2.5 22.5 2.3.5 23.5

3.7 -

5 6 7 8 9

2.52 22 -3.5 2.5.7 24.5 2.32.5

10 11 12 13 14

22.52 2.5-11 23.3-5 2.5.13 22.5-7

15 16 17 18 19

3.67 13.17 3.7.11 -

2

22.113 3.151 2-3.7.1123.59 11.43 2.241 3.7-23 22.3.41 17.29

E.g. 432 = 2 • 33 52

-

9 32

2.2 Factorizations (continued) n

0

1

2

4

5

6

8

9

23.32.7 2.257 22.131 2.3.89 25.17

5.101 5.103 3.52.7 5.107 5.109

2.11.23 3.132 22.3-43 11.47 17.31 2.263 23.67 3.179 2-3-7.13-

22.127 2.7.37 24-3.11 2.269 22.137

3.173 232 72.11 32.61

3

7

50 51 52 53 54

22.53 2-3.5.17 23-5.13 2.5.53 22.33.5

3.167 7.73 32.59 -

2.251 29 33.19 2.32.29 22.7.19 13.41 2.271 3.181

55 56 57 58 59

2.52.11 24.5.7 2.3.5.19 22.5.29 2.5.59

19.29 3.11.17 7.83 3.197

23.3.23 7.79 2.281 22.11.133.191 2.3-97 11.53 24.37 -

2.277 22.3.47 2.7.41 23.73 2.33.11

3.5.37 5.113 52.23 32.5.13 5-7.17

22.139 2.283 26.32 2.293 22.149

34.7 3.199

2.32.31 13.43 23.71 2.172 3.193 22.3.72 19.31 2.13.23 -

60 61 62 63 64

23.3.52 2.5.61 22.5.31 2.32.5.7 27.5

13.47 33.23 -

2.7.43 22.32-17 2.311 23.79 2.3.107

22.151 2.307 24.3.13 2.317 22.7.23

5.112 3.5.41 54 5.127 3.5.43

2.3.101 23-7.11 2.313 22.3-53 2.17.19

3.11.19 72.13 -

3.7.29 25.19 2-3.103 17.37 22.157 2.11.29 32.71 11.59 23.34

65 66 67 68 69

2.52.13 22.3.5.11 2.5-67 23.5.17 2.3.5.23

3.7.31 22.163 2.331 11.61 25.3.7 3.227 2.11.31 22.173 -

2.3.109 3.13.1723.83 2.337 22.32.19 32.7.11 2.347

5.131 5.7.19 33.52 5.137 5.139

32.73 24.41 2.32-37 23.29 22.132

70 71 72 73 74

22.52.7 2.5.71 24.32.5 2.5.73 22.5.37

32.79 7.103 17.43 3.13.19

2-33.13 23.89 2.192 22.3.61 2.7.53

19.37 23.31 3.241 -

26.11 2.3.7.17 22.181 2.367 23.3.31

75 76 77 78 79

2.3.53 23.5.19 2.5.7.11 22-3.5-13 2.5.79

3.257 11.71 7.113

24.47 2.3.127 22.193 2.17.23 23.32.11

3.251 7.109 33.29 13.61

80 81 82 83 84

25.52 2-34.5 22-5.41 2.5.83 23 .3.5.7

32.89 3.277 292

2.401 22.7.29 2.3.137 26.13 2.421

85 86 87 88 89

2.52.17 23.37 3.7.41 22-5.43 2.3.5.29 13.67 24.5.11 2.5.89 34.11

90

22,32,52

91 92 93 94

17.53 2.5.7.13 3.307 23.5.23 2.3.5.31 72.19 22.5.47 -

95 96 97 98 99

2.52.19 26.3.5 2.5.97 22-5.72 2.32,5-11

3.317 312 32.109 -

32.67 7.89 3.211 -

3.229 2.73 23. 3. 29 17.41

2.7.47 22.167 2.3.113 24.43 2.349

3.223 7.97 13.53 3.233

3.5.47 5.11.13 52.29 3.5.72 5.149

2.353 22.179 2-3.112 25.23 2.373

7.101 3.239 11.67 32.83

22.3.59 2.359 23.7.13 2.32.41 22.11.17

36 7.107

2.13.29 22.191 2.32.43 24.72 2.397

5.151 32.5.17 52.31 5.157 3.5-53

22.33.7 2.383 23.97 2.3.131 22.199

13.59 3.7.37 -

2.379 3.11-23 28.3 19.41 2.389 3.263 22.197 2.3.7.19 17.47

11.73 3.271 72.17 3.281

22.3.67 2.11.37 23.103 2.3.139 22.211

5.7.23 5.163 3.52.11 5.167 5.132

2-13.31 3.269 24.3-17 19.43 2.7.59 22.11.1933.31 2.32.47 7.112

23.101 2.409 22.32.23 2.419 24, 53

22.3-71 2.431 23.109 2.32.72 22.223

32.97 19.47

2.7.61 25.33 2.19.23 22.13.17 2.3.149

32.5.19 5.173 53.7 3.5.59 5.179

23.107 2.433 22.3.73 2.443 22.7

2.3.11.1322.7.31 11.79 3.293 2.439 23.3.37 7.127 2.449 29.31

2.11.41 24.3.19 2.461 22.233 2.3-157

3.7.43 11.83 13.71 3.311 23.41

23.113 2.457 22.3.7.11 2.467 24.59

5.181 3.5.61 52.37 5.11.17 33.5.7

2.3-151 22.229 7.131 2.463 32.103 23.32.132-11.43 -

32.101 22.227 2.33.17 25.29 2.7.67 3.313 22.3.79 13.73

23.7.17 2-13.37 22.35 2.491 25.31

32.107 7.139 3.331

2.32.53 22.241 2.487 23.3.41 2.7.71

5.191 5.193 3.52.13 5.197 5.199

22.239 3.11-29 2.3.7.2324.61 2.17.29 3.7-47 22.3.83 -

2.479 23.112 2.3-163 22.13.19 2.499

53

3.172 3.13.23

32.7.13 3.283

7.137 3.17.19 11.89 23.43 33.37

2.2

Least Common Multiple (LCM) Let [a l, ..., an] denote the least common multiple of the integers a t, ..., an. One method of finding that number is: Prime number factorize a l, a,,. Then form the product of these primes raised to the greatest power in which they appear. Example. Determine A = [18, 24, 30]. 18=2 • 32, 24 =23 • 3, 30= 2 • 3 • 5. Thus, A =23 32 .5 =360.

Greatest Common Divisor (GCD) Let (a, b) denote the greatest common divisor of a and b. If (a, b) = 1 the numbers a and b are relatively prime. One method (Euclid's algorithm) of finding (a, b) is: Assuming a>b and dividing a by b yields a=qi b+ri , Ori 0

two unequal real roots

b2 —4ac 0, (ii) three real roots of which at least two are equal if D=0, (iii) three distinct real roots if D< 0. Put 64

2.4

v = 31-q - J T

-9 + , u = 11 2

2 where the roots are to be interpreted as follows: (ii) D = 0: u=v real (i) D > 0: ,P5 > 0, u and v real (iii) D < 0: ,,/i) purely imaginary, u and v conjugates of each other, i.e. u+v real.

The roots of (2.2) are x1 = u + v

U A- V 14 —v. /x23 = - — ± — 143 (Cardano's formula)

2

2

If xi, x2, x3 are roots of the equation x3 + rx2 + sx+ t= 0 then x1+x2+x3=-r xix2+xix3+x2x3=s xix2x3=-t

Binomic equations A binomic equation is of the form zn = c, c complex number 1. Special case n=2: z2 = a+ ib. Roots: r + a.4,1r - a Z=±

+ ib =

L

2

+

,b00

2 2 r = nIa + b

jr 2 al

[ 4.1r + 2a

2. General case: Solution in polar form: Set c= zn = c= rei(0+2kr)

Roots:

z = ufr ei(e+Daryn = n.ir (cos

0 +2ktr +2br + isin n n

k=0, 1, ..., n-1 Note. The roots form a regular n-gon.

a=

65

n

13= 2n n

R = n47,

3.1

3 Geometry and Trigonometry

3.1 Plane Figures Triangles a, b, c = sides, a, [3, y= angles, h= altitude, m = median, s = bisector, 2p = a + b + c = perimeter, R. circumradius, r= inradius, A = area.

xa y b Theorems 1. (a) a+/3+ y=180° (b) a0 a, b, c are right-handed) 78

3.5 Vector triple product a x (b x c) = (a • c)b - (a • b)c (a x b) x c = (a • c)b - (b • c)a

3.5 Plane Analytic Geometry P = (x; y), P1 = (x1; Yi)' ..• points, a =(ax, ay) ... vectors 1. Distance between P1 and P2= (xl -x2)2 +(y1-Y2)2 x1+x2Y1+Y2 )

- ( 2. Midpoint Pm-

2

2

3. P dividing P1P2 into ratio r/s: P=

(rx2+sx ] ry2+sy1 1 r+s ' r+s )

4. Centroid of triangle i.e. intersection of medians

P3

(xi +x2+x3yi +y2+y3) Pc3 3

P2

5. Area of triangle = +1 2

P1 -1 ax ay =+ 1 x2-x1 y2-y1 =± (x1Y2+x2Y3±x3Y1-x2Y1-x3Y2-x1Y3) 2 2 x3 —x1 y3 —y bx by 1

6. Area of polygon P1P2 Pn= j 1 , =ikxi y2+x2y3+... +xn_iyn +xnyi - x2yi - x3y2 -

7. Angle 0 between vectors: cos 0-

a•b lai ibi

Straight lines Direction vector v = (a, 13) Direction angle 0 Normal vector n= (A, B) II (-13, a) Slope k= tan 0=

p _ A Y2 - YI B x2 - x i a 79

-xnyn_i - xl yn)

axbx + ayby ai +

bx + by

3.5 Equation forms A 8. General form: AX + By+ C= 0, n= (A, B), k=— — B 9. Point-slope form: y —y1 = k(x— x1) 10. Slope y-intercept form: y= kx+ m 11. Intercept form: =T- + 1 = 1 a b +C 12. Normal form: Ax + By —0 jA2+B2 13. Parametric form: r = ro + tv 0: Ellipse case. Possible geometrical meanings: ellipse, circle, one point, nothing. (b) AC — B2= 0: Parabola case. Possible geometrical meanings: parabola, two parallel lines, one (double) line. (c) AC—B2 b) a. Center at origin, major axis along the x-axis: x2 y2 —+—= 1 a

2

b

P--(a cost, b sin t)

2

{x = a cost

Parameter form:

y=b sint

Polar form: r2 =

, 01) Asymptotes y = ± bx/a Directrices x= ± a e

82

3.6

3.6 Analytic Geometry in Space P=(x; y; z), P1= (xi; yi; z1 ) ... points, a = (ax, ay, a) ... vectors 1. Distance between P1 and P2=

z

-y2)2+(z1 z2)2

= 4/ (x1-

xi +x2 Y1+Y2.Z1+Z2) 2. Midpoint Pin - ( 2 ' 2 , 2 3. P dividing P1P2 in ratio r/s: (rx2 +sx i ry2 + sy i rz2 + sz i ) P=

r+s

r+s

S

4. Centroid of tetrahedron (X0 + X1 + x2 + X3 Pc=

4 1 —>

5. Area of triangle P 1 P2P3 = - IPI P2 x PIP31 2 a, ay az

1 6. Volume of tetrahedron = ± - [a, b,c] = ±

66

a•b = 7. Angle 0 between vectors: cos 0= —

Ial Ibl

bx by bz cx cy cz axbx + ayby + azbz

2 JO h2 4- 1,2 2 2 ja x + a y + a z c'y

8. Barycentric coordinates (a1, a2, a3) for a point P with respect to a triangle P1P2P3 are numbers such that P = a1Pi+a2P2+a3P3 and a1-Fa2+a3 = 1. Putting u = a1, v = a2, (a1, a2, a3) = (u,v, 1- u - v). Barycentric coordinates (a1, a2) for a point P with respect to a line PiP2 are numbers such that P = aiPi+a2P2 and aii-a2 = 1. Putting u = a1, (a1, a2) = = (u, 1 - u), which corresponds to usual parameterization.

Straight lines and planes Lines

Direction vector v = (a, 0, y) Line l given by —> OP= OPO 4-tv

(1) point Poe l and direction vector v.

r=r0+tv

83

3.6

{x= xo+at Equation:

y = yo +/3t

x - xo a

-

y - yo

z=zo+yt

_ z - zo

S

Y

(2) two points P1 and P2. Set v= P I P 2 =(x2 —x1, y2 —y1, z2 —zi) and use (1) Planes

n=vi xv2

Normal vector n = (A, B, C) Spanning vectors v1 =(a1, fli , yi), v2 = (a2, 132, 72) General equation form: Ax + By + Cz+ D = 0 Plane r given by (3) Poe ir and normal vector n: A(x — x0) + B(y—yo)+C(z—z0)= 0 (4) Po e ir and spanning vectors v1, v2 in the plane: (a) Calculate n= vi xv2 and use (3) or (b)

(c)

x-x0 y-yo z - Z0 al A h =0 or a2 02 72 x = xo + ai t + a2s y = yo + fli t + p2s

(parameter form)

z = zo + yi t + y2s (5) Three points P0, P1, P2E ir. Calculate n = P0P1 x P0P2 and use (3). (6) Intersections with coordinate axis: + .1 + = 1 (intercept form) a b c Angles Between two lines: cos 6=

Ivi 'v21 Ivillv21 84

3.6

Between line and plane: cos(90°— 0)= Between two planes: cos 0=

Iv • I ni v' n'

In1 •n2I I n2I

Distances From a point P1 to a line (P0 arbitrary on the line) d=

x Iv PoPi l

Ivl From a point P1 to a plane (P0 arbitrary in the plane)

d=

I n • PoP il = lAxi

+ Byi + Czi + DI

tjA2 + B2 + c2

Inl

Between two non-parallel lines (P1, P2 arbitrary on the lines respectively). d= 11311'2 • (v i xv2)1 I vl xv21 Orthogonal projection on a line and on a plane. See Sec. 4.8. Area projection

A'=A cos

Second degree surfaces General form (3.2)

Ax2+By2+ Cz2+2Dxy + 2Exz + 2Fyz + 2Gx + 2Hy +2Kz+ L=0

Analysis of (3.2): (a) D=E=F= 0: Complete the squares and compare to the standard forms below. (b) Any, D, E, F 0: Spectral methods, see sec. 4.6. 85

3.6 Second degree surfaces in standard form

tZ

z

Sphere

Ellipsoid

x2 ± y2 4. z2 = R2

x2 , yb22 - 1" a2

V = 4zR3 3

V = 4zabc 3

Elliptic cylinder z2 c2

= 1

x2 2 — + b2 = 1 a2

X

Hyperbolic Cylinder x2

y2 _

a2 b2

Elliptic Paraboloid x22 Z=

y2

+ a' 13`

Elliptic Cone 2

2 2

+———=0 a2 uL2 c2 • Z

Elliptic Hyperboloid of one sheet

Elliptic Hyperboloid of two Sheets

Hyperbolic Paraboloid

2 2 z2 x y ————= 1 a2 + b2 c2

2 2 2 z x — + — — — = —1 a2 y b 2 c2

y22 2 _b x2 Z — — 2— a2

86

3.7

3.7 Fractals Examples of self-similar fractals 1nN The fractal dimension d = — ln

N= Xd , where

N = number of new (smaller) similar geometrical figures that arise at dividing. = the linear scaling (greater than one) between the corresponding figures at dividing.

1. Serpinski's Carpet. Given a square, then do: (i) Divide the square into 9 similar squares and remove the center square. (ii) Repeat this process with the 8 remaining squares. (iii) Repeat the process with the 64 remaining squares. (iv) Continue the process indefinitely. n38 — 11n Here, N = 8, A = 3 and d = ••• gill• • MN NEM NINE! Mi• • • ••

• M • •••

• I• •

••

•111111 • .•

11•• • MEM • ••• 11.• WINN

•••

2. The Serpinski Gasket. Given an equilateral triangle, then do: (i) Remove the triangle whose vertices are the midpoints of the sides of the original triangle. (ii) Repeat this process with the 3 remaining triangles. (iii) Continue the process indefinitely. 1n3 Here, N = 3, and A, = 2 and d = — =1.585. 1n2

87

3.7 3. The Cantor Set. Given the closed interval /0 = [0,1], then do: (i) Delete the middle third open interval G,D , and let /1 = [0, flu[

.

(ii) Repeat this process succesively from the remaining closed intervals, so that a nested sequence of closed sets to D .11 D /2 D ... arises. Then the Cantor set, which is non-countable and of measure zero, is C = nr .

Ip

n=o n

Here, N = 2, A = 3 and d

1

11

ln 2 =0.631. = In 3

2

, ,3

Iterated Function System (IFS) 1. A metric space is a set X with a real-valued distance function d satisfying for all x, y and z in X: (i) d(x, y) = d(y, x)

(ii) d(x, y) 0 [= 0 x = y]

(iii) d(x, y) d(x, z) + d(z, y)

2. A metric space X is complete, if every Cauchy sequence in X has a limit in X. Below, assume that X is a complete metric space, that x, y are points in X and that A, B are compact subsets of X. 3. H(X) = all compact subsets of X1 [A set A c X is compact if every infinte sequence in A contains a subsequence having a limit in A. In IV a set is compact it is closed and bounded] 4. d(x, B) = min(d(x, y) : y E B) 5. d(A, B) = max(d(x, B) : x E A) 6. Hausdotff distance h(A, B) = max(d(A, B), d(B, A)) With this distance function H(X) will be a complete metric space. 7. A mapping f : X ---> X is a contraction mapping if there is a constant s (the contraction factor) with 0 5 s < 1 such that dVx), Ay)) 5_ sd(x,y) for all x, y E X. A contraction mapping is continuous. Thus, f maps H(X) into itself. [Notation: f(A) = {f(x) : x E A}] 8. The contraction mapping theorem: If f : X X is a contraction mapping, then (i) f possesses exactly one fixed point x E X, i.e. KO = (ii) lim f n(x) for all x E X [f n is the composition off n times] V —>00

88

3.7 9. f : X--> X is a contraction mapping with contraction factor s f : H(X) H(X) is a contraction mapping with contraction factor s. fN are contraction mappings with contraction factors sl, s2, 10. fj, f2,

sN,

N

respectively, and F(B) = U f n(B), all B E H(X) n=1 F is a contraction mapping on H(X) with contraction factor s= n= N (s 11. An Iterated Function System (IFS) is a finite set of contraction mappings fn : X —> X with contraction factors sn, n = 1, ..., N. 12. If fn, n = 1,

N is an IFS on (X, d), then

N (i) F(B) = U fn(B), all B e H(X) is a contraction mapping on (H(X), h(d)) n=1 with contraction factor s, N (ii) The unique fixed point (the attractor) A = F(A) = U f n(A), satisfies n=1 A = lim Fn(B) for all B E H(X). n—> 00 Example. Let (i) X = R (set of real numbers), d = ordinary Euclidean distance, (ii) an IFS consist offi(x) = I and f2(x) =

. Thus F= f l uf 2 is

a contraction mapping on H(R) with contraction factor s = . If Bo = [0,1], then Bn = Fn(B0), n = 1, 2, 3, ... equal the sets In in Example 3 above and lim B, equals the Cantor set C in that example. The set C is the n —>= u

attractor of F in this case, i.e,. C =

[Notation: If A cR , then xA = {xy:ye A } and A + x = fy +x:yE A 1]

89

4.1

4 Linear Algebra 4.1 Matrices Basic concepts In the following only real vectors and matrices are considered. al Column vector a=

•• •

E Rn

Row vector aT= (al, ..., an)

an Scalar product aTb =a i bi +... +anbn Norm (length) lal = aTa =

+ + an2

Matrix of order mxn: (A is square if m= n):

a ll

a1 j

A=

..., an], ai=

=(au)=

amt amn

amj

of order n x m (exchange rows and columns).

Transpose of A : AT= a ln •• • a mn

Diagonal matrix D =

ail 0 0 0 a22 ... 0 0

=diag(aii,

ann) (au = 0, i#j)

0 ...0 ann

Identity matrix I= diag(1, 1, ..., 1) of order n x n 11 0 0 ... 0 Lower triangular matrix T=

o

21 a220 nlan2

•••

(au = 0, i det(A — A/) =

1— A a12 ••• ain a21 a22 — A. •

and det A = A l A2 ... An (product of all eigenvalues). trace A = + 22 + + An (sum of eigenvalues).

=0

an,— A,

Symmetric matrices A real square matrix A is symmetric if AT A. A symmetric (a) all eigenvalues are real (b) eigenvectors corresponding to different eigenvalues are orthogonal (c) aT(Ab) = (Aa)Tb Spectral theorem for symmetric matrices Assume that (i)A is symmetric (ii) Ai, ..., An are eigenvalues (incl. multiplicities). gn such that Then there exist corresponding eigenvectors gl, lgi = 1 (pairwise orthogonal and normed) 1. gi 1 gi Set P = [g1 , g2, gn]. Then 2. P is an orthogonal matrix 3. PTAP =D= diag(A1, A2, ..., An) 4. A pDpT, Ak = pDkpT= p diag(Alk, 4k)pT 5.

xTAX 2 Ix'

(x#0) (equality •;=> x=corresponding eigenvector).

1 0 —4 Example. A= o 5 4 Characteristic equation: —4 4 3 1 —A 0 —4 0 5 —A 4 =—A3+9A2+9A-81=0 with roots A1=9, A2=3, A3=-3. —4 4 3 —A X x —8x —4z=0 —4y+4z=0 Corresponding eigenvectors: Al=9: A y = 9 —4x+4y-6z=0 - 2 1 —1 1 with general solution x=—t, y=2t, z=2t. Choose g1= 5[2 . Analogously g2= 3 2 —1 —1 2 90 1 and g3= 3 so that P= 1 2 2 —1 and D=PTAP= 03 0 3 2 —1 2 00-3 2 99

4.5

Affine diagonalization Assume that (the non-symmetric) A has n distinct real eigenvalues Al, ..., An. Then 1. the corresponding eigenvectors gl ,

gn are linearly independent.

2. L-1 AL= D = diag(Ai , ..., 4), L=[gi , •••, gn]. (Also in the case of multiple eigenvalues there may exist linearly independent eigenvectors and hence 2.)

A generalized eigenvalue problem Eigenvalue problem (A invertible): (4.5)

Bg=AAg, g#0

(This coincides with (4.4) if A =/). The number A, is an eigenvalue of (4.5) det(B-AA)=0.

Generalized spectral theorem Assume that (i) A, B symmetric, A positive definite (ii) Al , ..., An eigenvalues of (4.5). Then 1. All eigenvalues are real. 2. There exists a basis of corresponding eigenvectors gl, ai j. g

3. If L=[gi,

gn] then det L#0 and LTAL=I, LTBL=diag(A.1, ..., An)

(simultaneous diagonalization). 4. Ainin

g„ such that

= 81: gr Bg1= A j

x TBx

,x#0.

x' A x

Singular Value Decomposition Any mxn-matrix A can be factored into A= QSPT where S is an mxn "diagonal type" matrix, i.e. sii=0, i#j P is an nxn orthogonal matrix Q is an mxm orthogonal matrix

100

4.5 Determination of P, Q and S The matrix ATA is a positive semidefinite symmetric nxn-matrix with non-negative gn be an 2.,= O. Let g1 , g2, 0 and A„.“ , eigenvalues Ai. Suppose that Ai, A2, orthonormal set of corresponding eigenvectors. Construction of P: The columns of P are the vectors gl, Construction of S: sii=kti= A values of A.)

, i=1,

gn, i.e. P=[gi,

•••, gn]

r. Remaining su=0. (The pi are called the singular

1

Construction of Q: Set hi= — Agi , i=1,

r. If r0 all det Ak>0

2. positive semidefinite if Q(x)0 all Ak 3. indefinite if Q assumes positive and negative values A has positive and negative eigenvalues. A, Q are negative definite —A, —Q are positive definite.

Second degree curves (cf. sec. 3.5) Example. (ON-system) Classify the curve 6x2 +4xy + 9y2 = 1.

r

With notation as above: A = [6 21 , Ai =10, A2 =5 29 1 gl = ix = r [ , g2 = iy = 45 2 P = 1 [1 —21 . Coordinate transformation 45 2 1 Canonical form: 10V +5)72 =1 1 (Ellipse, semiaxis — and 1 ,1

.55

y=2x,

, y=—x/2.

104

P

4.6

Second degree surfaces (cf. sec. 3.6) Example. (ON-system) Classify the surface 5x2 + 5y2 + 8z2 + 8xy + 4yz — 4xz = 1. With notation as above: 54-2 A= [ 4 5 2 —2 2 8

1 2 2 ., 1 2'1 = Al= 9, A3 =0, P= [gl, g2, g31 = 5 2 1 —2 2 —2 1_

(Two eigenvalues coincide Coordinate transformation:

surface of revolution, axis along g3).

x 5, z

Canonical form: + 9-Y-2 = 1 (circular cylinder) Axis of revolution: (x, y, z) = t(2, — 2, 1)

Classification Canonical form: A I.X2 + A2? + A3i2 = c 2-1

12

A3

+ + + + + + + + + + +

+ + + + + + + —

+ + —

c + 0 +

0 0 0 0 0 0

0 + 0 + 0 + 0

0 0

Type Ellipsoid One point Hyperboloid of one sheet Hyperboloid of two sheets Elliptic double cone Elliptic cylinder One line Hyperbolic cylinder Two intersecting planes Two parallel planes Two coincident planes

105

4.7

4.7 Linear Spaces Vector spaces A set L of elements x, y, z, ..., is called a linear space or a vector space (over R) if addition and multiplication by scalars are defined so that the following laws are satisfied for all x, y, ze L and A, ye R. 3. (x+y)+z=x+(y+z) 2. x+y=y+x I. 1. x+ye L 4. there exists 0 such that x +0=x 5. there exists —x such that x + (— x) = 0 3. (A+ y)x = Ax + /ix 2. A(/./x)=(Ay)x II. 1. A,XE L 6. Ox=0 7. A0=0 5. lx=x 4. A(x+y)=Ax+Ay Test for subspace A nonempty subset M of L is a linear space itself if AXE M. x+ye M, 2. XE M, E R 1. x, ye Linear combinations. Basis 1. The vector y is a linear combination of xi, 2. The linear hull LH(xi,

xn if y-=A i xi + +Anxn.

xn) is {y : y= A ixi + + Anx„, Ai E R} .

xn are 3. xl , Ai = 0, all i (i) linearly independent if A i xi +... + Anx„ = 0 (ii) linearly dependent if there exist Ai , ..., An , not all zero, such that Aixi +...+ Anxn= 0 (4=> some xi is a linear combination of the others.) 4. el, ..., en is a basis of the linear space L and L is n-dimensional if (i) el , ..., en are linearly independent (ii) every xE L can be written (uniquely) x =xiei + +xnen Scalar product 1. Let L be a linear space. A scalar product (x, y) [other notations x • y, (xly), (x, y) etc.] is a function LxL —> R with the following properties holding for all x, y, ze L and A, y e R: 1. (x, y)=(y, x) 2. (x, Ay+ yz)=A(x, y)+ µ(x, z) 3. (x, x)_11) (Equality x=0) 2. Length of x: lx1=

x) , Icxl = Icl • Ix' (c scalar).

3. 1(x, Y)I < lx1 • IYI (Cauchy-Schwarz' inequality) 4. lx + yl < Ix' + ly1 (Triangle inequality) 106

4.7

Orthonormal basis Let L be an n-dimensional linear space with scalar product (Euclidean space). 1. A basis el, ..., en is called orthonormal (ON-basis) if 1, i=j e1)=4:5,i = {0, i

n

n

2. el, ..., en ON-basis, x=

xkek, y=

k=1

k=1

Ykek

xk2 (iii) (x, y)=

(i)xk =(x, ek) (ii)lx12 =

xkyk k=1

k=1

Orthogonal complement M subspace of L: M1= {yE L: (x, y)=0, all xe MI Orthogonal projection M subspace, e1, ..., em ON-basis of M: x' is the orthogonal projection of x on M if x = x' + X", E M, x"e

m

Moreover, x'= (x, ek)ek k=1

Gram — Schmidt orthogonalization Given: vi,

vn basis of linear space L with scalar product.

Sought: ON-basis el, ..., en of L. Construction: (Gram — Schmidt): (1) e1= — v1 1 v11

(5) f2 =V2 — (V2, ei)ei, e2=

1 f2 1'21

(k) fk —Vk —(Vk , e )e — — (vk , ek_ )ek_ ; ek — 1177,1

107

.4.

k

4.8

The space R" The set of all column vectors x = (x1, x2 ..., xn)T is called Rn. A natural choice of an ON-basis of Rn is the set of vectors e l = (1, 0, ..., 0)T, e2 = (0, 1, 0, ..., 0)T, ..., en = (0, ..., 0, 1)T, i.e. x =xiei +x2e2+ +xnen. Addition: x+y=(xi, x2 ..., xn)T+ (yi, y2 ..., yn)r= (xi +yi, X2 +y2, ..., xn +yn)T Multiplication by a scalar: cx=c(xi, x2 ..., xn)T = (cxi, cx2,

cxn)T

Scalar product: x • Y = xTY =x1Y1 ±x2Y2 + • • • +xnYn Length or norm: lxl = AixT x= Alxf +

lcxl =

... + xn2 ,

Pythagorean theorem: x • y=0 + y12 = lx12 +1y12 Cauchy-Schwarz' inequality: lx • The triangle inequality: lx + yl < lx1+ lyl Angle 0 between x and y: cos 0= x • Y xiiYi Completion to full basis Let v1, ..., vr be a set of linearly independant vectors in Rn, r M is called a linear mapping if F(.1,x+py)= AF(x)+ itF(y) for all x, y E L, pE R

Matrix representation Assumption. (i) el, ..., en basis of L,

fn, basis of M. all ••• a l j ••• aln

, A=

(ii) F(ei)= i=1

= am l

amj ... amn F(ej)

108

• • F(en)]

4.8 n

m

yi fi , the mapping x —> F(x) i=1 j=1 is represented by the matrix A, i.e. Then if x=

F(x)=

Yi

ain x1 Y=AX

• ••

Ym

anz i

amn xn

Inverse mapping If F: L —> M is invertible then F-1 :M L is represented by A-1 Composite mappings IfF:L—>M,G:M—>Nand if F and G are represented by A and B respectively, then the mapping G0 F (defined by (G o F)(x)= G(F(x)) is represented by the matrix BA. Symmetric mappings Let L be a finite-dimensional Euclidean space and F: L —> L a linear mapping. 1. F is called symmetric if (Fx, y) = (x, Fy), all x, ye L. 2. The number A. is called eigenvalue, x#0 a corresponding eigenvector, if Fx = Ax. 3. (The spectral theorem.) If F is a symmetric mapping then there is an orthonormal basis of L consisting of eigenvectors of F. The matrix representing F in this basis is diagonal with the eigenvalues along the diagonal. 4. (Projections.) A matrix P represents an orthogonal projection P2=P and PT= P P2 = P and IPx151xl, all x. All eigenvalues are 0 or 1. If a matrix A has linearly independent columns, then ATA is invertible and P=A(ATA)-1AT represents the orthogonal projection onto the column space of A. 5. (Orthogonal projection on a line.) Projection matrix P representing orthogonal projection on a line (through origin) with directional vector v= (a, b, c):

P=

a2 ab ac 1 a 2 + b2 + c2 ab b2 bc ac bc c2

x'

x =

i.e. , z

z 109

4.8 6. (Orthogonal projection on a plane.) Projection matrix P representing orthogonal projection on a plane (through origin) with normal vector n = (A, B, C):

B2+C2 —AB —AC —AB A2 + C2 —BC P= A2 + B2 + C2 — —BC A2+B2 AC 1

_ -

x' i.e.

x

y, z'

=p

Y z

Rotation around a line in space 1. Rotation of a point around a coordinate axis The rotation P —> Q, (seen counterclockwise from the positive axis) is represented by the following orthogonal matrices. Rotation of angle a around the x-axis 1

0

0

Rx(a)= 0 cos a —sin a 0 sin a cos a_ A

Rotation of angle a around the y-axis

Rotation of angle a around the z-axis

cos a 0 sin a Ry(a)= 0 1 0 —sin a 0 cos a

cos a —sin a 0 Rz(a)= sin a cosa 0 0 1 _ 0 A

z

z

H -\--- :\ /

Q

Y

..-

>

a Q

x

P

P Y ► ""

z

a2 Q Y

/I

x -.)-4---2

x

E.g. if P = (px; py; pz) and Q = (qx; qy; qz) then rotation around the x-axis is given by (qx, qy, qz)T=Rx(a)(px,py, pdT.

Composition of rotations and the Special Orthogonal group SO(3) The matrix R represents a rotation in space R is orthogonal and det R = 1 The matrix representing a composite rotation consisting of first a rotation represented by the matrix R1, and then a rotation represented by R2, is the matrix product R2R1. This new matrix represents a rotation with (in general) a new rotation axis and a new rotation angle. The set of orthogonal 3x3-matrices with determinant 1 is the special orthogonal group SO(3), the rotation group of R3. 110

4.8 2. Rotation of a point around an aribitrary line Note. Rotation is considered counterclockwise seen from the marked direction of the line L. (i) Line L characterized by angles O and T• : Rotation of angle a around L is represented by the matrix product (the result is an orthogonal matrix)

R = Rz(T)Ry(19)Rz(a)Ry(-0)Rz(-(p) (i.e. the rotation may be considered as five successive rotations around coordinate axes.)

x

(ii) Line L characterized by the equations x = at, y=bt, z=ct. v = (a, b, c)T.

Method 1. The rotation of an angle a around L is represented by the matrix R in (i) with cos e= r where

sin

cos cp= p

sin g=e r

b =P

p= Va 2 + b 2

r= Ja2 + b2 + c2 and

Method 2. Assume that the line (through origin) of rotation has the direction v = (a, b, c)T. Make a coordinate transformation from (x, y, z)-space to (x, y, space, so that the new z-axis has the same direction as v. The coordinate transformation matrix P is constructed in the following way (cf. Sec. 4.4): P 1 1 P12 p13 P=Rx , i y , iz1= P21 P22 P23 ' P31 P32 P33 P13

1 where e z = — vi v = P23 P33 P12

ey = P22 P32

1 = nia 2 + b2 + c2

1 Va 2

-b a b2 0

a b c

ex =

cos a -sin a 0 Then the rotation matrix R is R= PRz(a)PT= P sin a cos a 0 PT 0 0 1 111

4.8 Method 3. If P = (x;y;z) and x = (x,y,z)T then v•x sin a Rx = (cos a)x + (1 - cos a) —v + vxx Iv' 1111 2

Determination of rotation axis and rotation angle If R is the matrix representing a rotation with rotation angle a and with axis direction v, then cos a = (traceR - 1)/2 (R - l)v = 0 (traceR = sum of the diagonal elements of R, I = the identity matrix)

Change of basis (Cf. sec. 4.4) Let el , ..., en and e l , ..., in be two bases of L with the relationships et=Pji ej, P = (PO =[ii, i= 1 ei =

in ]

, Q=(qii), Q = P-1 = [e ,

j =1

en ]

1. (Relation between components) x1 n_

Let x= E xi ei= i=1

xi ei ,

X=

x2

, X=

. Then

i=1 xn

X= PX

x2

n _

xi = E Pij xj

xn

X - = QX xi =

j=1

j=1

gijxj

2. (Relation between matrix representations.) If F:L-->Lis represented by the matrices A and A with respect to the two bases, respectively, then A =P-1AP,

au = I gilalkPkj

A=PA P-1

k,l

A = PTAP, a y = Pli aik Pk] (P orthogonal) k,l

112

A =PA PT

4.8

Range and nullspace Given F : L M, A the corresponding matrix.

dim L =n F

dim M=m

1. (a) Nullspace N(F)= {xe L: F(x)= 0} cL

(b) N(A) = set of column vectors XE Rn such that AX = O.

N(F)

R(F)

2. (a) Range R(F)= {F(x)e M :xe L} cM

(b) R(A) = space spanned by the column vectors of A (subspace of in. Theorem

1. dim L= dim N(F)+ dim R(F)= dim N(A)+ dim R(A) 2. dim R(A) = dim R(AT)= rank(A) 3. rank (A) = rank (AT) = rank (ATA) = rank (AAT) 4. dim N(A) = n — rank (A) 5. 6. 7. 8.

N(A)=N(ATA) R(AT)=R(ATA) N(A)=R(AT)1 R(A)=N(AT)1

R(A)1=N(AT) N(A)1=R(AT)

Example Determine R(A) and N(A) of the matrix A below. Solution: By elementary row operations (see sec. 4.1), 12 1 13 12113 A= 1 2 2 3 6 — 0 0 1 2 3 =B 24238 00012 T TT T TT (i) R(A) is spanned by those columns of A which correspond to the columns (marked with arrows) with pivot elements of the echolon form matrix B. Thus the column vectors (1,1,2)T, (1,2,2)T and (1,3,3)T constitute a basis for R(A) and dim R(A)= 3.

(ii) Solving the system Ax = 0 Bx = 0

xi +2x2 +x3 +x4 +3x5 =0 x3 +2x4 +3x5 =0 x4 +2x5 =0

yields x5 = t, x4 =— 2t, x3 = t, x2 = s, x1 =— 2t— 2s or (xi, x2, x3, x4, x5) = (— 2t— 2s, s, t, —2t, t)= t(-2, 0, 1, —2, 1) + s(-2, 1, 0, 0, 0). Hence the column vectors (-2, 0, 1, —2, 1)T and (-2, 1, 0, 0, 0)T constitute a basis for N(A) and dim N(A)= 2. 113

4.9

4.9 Tensors Relations between bases: ii= E Pi j ej,

qi j ei , Q=(qi j)=p-1 ,.(pi.i)-1

Tensors of order 1 1. a = (ad covariant if ai=I pi laf (transformation as basis vectors). 2. b=(bi ) contravariant if bi

qiibj

Tensors of order 2 au = k 3. A = (au) covariant if -

PikPj lakl

4. B=(bij) contravariant if

=

(transformation as vector components).

gkigijbki k,1

5. C= (ci j) mixed if Ci f= Pi kgl Jckl k,1

Tensors of higher orders are defined similarly. The stress tensor (T1) and the strain tensor (eu) are covariant tensors of second orders. The metric tensor (4) where gu= (ei , ei) is a covariant tensor. Operations 1. Sum of tensors and multiplication by a scalar are defined component-wise. 2. Tensor product C=A O B. E.g. if A =(ail ), B= (by) then cijki=albki (mixed tensor of order 4). 3. Contraction. E.g. dfi= ckjki= E albki • 4. Trace. Tr(A)= aii.

4.10 Complex matrices In this section components of vectors and elements of matrices are complex numbers.

Vectors in el Let a, b be (column) vectors in Cn 1. Scalar product a*b=ai bi +a 2b2+...-Feinb„, b*a= a*b

alb a * b =0 (a and b orthogonal). Here the bar denotes complex conjugate. 2. Length (norm) lal2=a * a=lall2 + la212 + •• • + lan12, Ica' =1cl • lal, c= complex number. 3. Cauchy-Schwarz' inequality: la*b15_1a1 1bl. 4. Triangle inequality: la + bl 5. Pythagorean relation: la+ b12 = 1a12 +1b12 alb.

114

4.10

Matrices 6. Adjoint A* of A = (aii): A* = (a7i) . (A + B)*= A*+ B* (AB)*=B*A* (AB)-1=B-1A-1

(cA)*=A* (CE C) A**= A (A-1)*=(A*)-1

Inverse matrices Set C= A + iB, A, B real matrices. 7. If A-1 exists then C-1= (A +BA-1B)- 1 - iA-1B(A + BA-1B)- 1. •— 1 i(B+ AB-1A)-1 8. If /3-1 exists then C-1= /3-1 A(B +AB-1A) 9. A, B singular, C regular: Let r be a real number such that A+ rB is regular. Set F = A + rB and G=B- rA. Then C-1= (1 - ir)(F+ iG)-1. Continue as in 7.

Differentiation of matrices 10. If A =A(x)= (au(x)) and B = B(x), then (i) A'(x)=(aii'(x))

(ii) (A + B)' =A' + B'

(iii) (AB)' = AB' + A'B

(iv) (A2)'=AA'+ A'A

(v) (A-1)'=-A-1A'A-1

Matrix norms, see sec. 16.2. Unitary matrices 11. A square nxn-matrix U is unitary if U*U=UU*=I (/= identity matrix). 12. Let U= [u1 , u2,

an t where ui are the column vectors of U. Then

(i) U is unitary The column vectors of U are pairwise orthogonal and normed, i.e. ui*ui=Sid , j= 1, 2, ..., n. (ii) U unitary (a) U* is unitary (c) pet Ul = 1 (e) 'UM= lal

(b) U -1= U* (d) (Ua)* (Ub)= a*b (f) alb UalUb

(g) IAI= 1 for any eigenvalue X of U. (iii) U, V unitary of the same order

UV unitary.

Normal matrices 13. A (square) matrix N is normal if N*N=NN*. 14. A (square) matrix H is Hermitian if H* = H. A (square) matrix H is skew-Hermitian if H * = - H. 115

4.10

H is Hermitian = iH is skew-Hermitian H is skew-Hermitian iH is Hermitian H is Hermitian H-1 is Hermitian (if it exists) Hermitian, skew-Hermitian and unitary matrices are normal. 15. A normal matrix is (i) Hermitian all eigenvalues are real (ii) skew-Hermitian all eigenvalues are pure imaginary (iii) unitary all eigenvalues have modulus 1.

Spectral theory, unitary transformations 16. 2. is an eigenvalue of a square nxn-matrix A and g*0 is a corresponding eigenvector if Ag 17. A. is an eigenvalue of A det(A - A1)=0. 18. Gerschgorin's theorem. Let Ci denote the circular region lad11, i = 1, n. .i= 1(j#i) Then the eigenvalues of A are contained in the union of the Ci Ci=

19. Perron's theorem. A matrix whose elements are all positive has at least one positive eigenvalue. The components of the correspondent eigenvector are all non-negative. 20. Cayley-Hamilton's theorem: The characteristic equation P(A)=det(A - A/)=(- 1)n An + —1 + +Col =O. + Cn _ lAn —1 + ... +Co = 0 is satisfied by A, i.e. P(A)=(- 1)n An + Cn _ 21. Schur's lemma: For any square matrix A there exists a unitary matrix U such that T=U*AU is (upper) triangular. The elements tii of the diagonal of T are the eigenvalues (incl. multiplicities) of A. 22. Spectral theorem. (Unitary transformation) Assume that ties) of N.

(i) N is normal

(ii) 4 A2, ..., An are eigenvalues (incl. multiplici-

Then there exist corresponding eigenvectors g1 , g2 , gn such that (a) &1g./ , t *j, I gi 1=1 (pairwise orthogonal and normed) (b) U= [g1 , g2 , gn ] is a unitary matrix (c) U*NU=D= diag(A i , A2, ..., /In) 23. If N is normal then eigenvectors corresponding to different eigenvalues are orthogonal.

Hermitian forms A hermitian form is a homogeneous polynomial of second degree of the form h= h(z)= z*Hz= F hiizizi (h j =1 116

4.10 24. The values h(z) are real, all z. 25. By a suitable unitary transformation (see above) z = Uw, h= 26.

k=1

kI17 kl2•

zI2 h(z) 2, maxl z12 (Equality z = corresponding eigenvector).

27. Law of inertia. If a Hermitian form is written as a sum of squares (e.g. by completing squares or by a unitary transformation) in two different ways, then the numbers of positive coefficients as well as negative and zero coefficients are the same in both cases. 28. A Hermitian form is positive definite if h(z)>O, z#0 or all 2k>0.

Decomposition of matrices 29. For any square matrix A there exist unique Hermitian matrices H1 and H2 [H1 = (A +A*)/2 and H1 = (A — A*)I2i] such that A =H1 + iH2. 30. N is normal N=H1 -FiH2 with commuting Hermitian matrices H1 and H2 (i.e. H1ll2 =H2H1 ). 31. Let H1 and H2 be Hermitian. Then there exists a unitary matrix U simultaneously diagonalizing H1 and H2 (i.e. U*HIU and U*H2U are diagonal) s+ HITI2 = H21/1 .

Non-unitary transformations 32. Assume that the square nxn-matrix A has n linear independent eigenvectors g1 , g2, gn (e.g. this is the case if the n eigenvalues , 2, ..., .1n are distinct). Then with ..., A n ). gn], L-1AL=D=diag(A,1, L= [g1, g2, 33. Multiple eigenvalues. Generalized eigenvectors. The vector v * 0 is a generalized eigenvector corresponding to an eigenvalue A, of multiplicity k of the nxn-matrix A, if (A — )kv = O. To an eigenvalue of multiplicity k there always correspond k linearly independent generalized eigenvectors. Generalized eigenvectors corresponding to different eigenvalues are linearly independent. 34. Jordan form: For any square matrix A there exists a non-singular matrix S such that J1

0

0

J2 0

...... 0 -0

S-1AS=J= O

Jmn

-Ai 1 0 0 Ai 1 0 , Ji= ........ ....0 Xi 0 0 _O

0 0 1 ;t

where Ji are the Jordan blocks. The same eigenvalue Ai may appear in several blocks if it corresponds to several independent eigenvectors. The columns of S are generalized eigenvectors of A and constitute a basis of C.

117

5.1

5 The Elementary Functions 5.1 A Survey of the Elementary Functions Primitive function i f(x)dx

Function Y =i(x) y=xn, nE Z+ n even n odd

Domain pf

Range Rf

Inverse function x=f-1(y)

Derivative f'(x)

all x all x

y>_0 all y

x = nis5i, x 0 x = 11,1;

x nxr, - 1 n+1 n+1

y=x-n, nE Z+ n even n odd

x#0 x#0

y>0 y#0

x= 111,5, x> 0 x= 1 /1",5

y=xa, ae Z a>0 a 0 x>0

y>_0 y>0

X = y lla

y=ex y = az (a>0, a#1)

all x all x

y>0 y>0

x= In y x = °log y= —

ax ln a

6' axlin a

y= In x x>0 y=loga x (a>0, a#1) x>0

ally ally

x=eY x=aY

1/x 1/(x In a)

xlnx-x (x ln x-x)/ln a

y=sinh x

all x

all y

x=ln(y++,42 1 )

cosh x

cosh x

y = cosh x

all x

yl

sinh x

sinh x

y=tanh x

allx

iYi< 1

x=ln(y+ NTT), (x_()) 1 1 +y

y=coth x

x#0

IYI>1

x= 1 In y + 1 2 y-1

y=sinx

allx

-10) Complex case: za=e a logz Inverses y = xa ,(=> x=y1/a

Hyperbolic functions The hyperbolic functions are defined as follows. Yi

yinhx The curve y = cosh x is called catenary. y =sinhx= y'=coshx

ex - e-x e x + e-x ex - e-x y=cothx= x -x y =coshx= ex 4- e-x y=t anhx= 2 2 ex + e' e -e y'=sinhx

y'=1/cosh2 x 122

y'=-1/sinh2 x

5.3 Transformation table

sinh x =

cosh x

_

+ jcosh 2x- 1

cosh x = Nil + sinh2x

-

tanhx

//coth 2x - 1

1

I cothxl

V1 - tanh 2x

Vcoth 2x - 1

sinhx Jcosh2x - 1 + 1- cosh x 41 + sinh2x

coth x=

Ail + sinh2x sinhx

i

coshx

Aicosh2x

-1

1

+

j1 - tanh 2x

tanh x=

+

coth x

tanh x

sinh x

1 cothx

1 tanhx

Geometrical interpretation

Hyperbola A2 —y2=1 t/ 2=area \ \ of the region

2. cosh2x - sinhx = 1 sinhx tanh x = cosh x 1 o cshx coth x = . si nhx tanhx

1. sinh(-x) =- sinh x cosh(-x) = cosh x tanh(-x) =- tanhx coth(-x) =- coth x

3. sinh(x ±y)= sinhx cosh y ± cosh x sinhy cosh(x ±y) = cosh x cosh y ± sinhx sinhy tanhx ± tanhy tanh(x ± y)= 1 ± tanhx tanhy 1 ± cothxcothy coth(x ± y) = cothx ± cothy 123

-

5.3

sinh = 2

4. sinh 2x = 2 sinhx cosh x cosh 2x= sinh2x+ cosh2x

cosh

x jcosh x + 1 2 2

tanh 2x= 2 tanhx 1+ tanh 2x

tanh

x coshx- 1 -± 2 Alcoshx + 1

sinhx coshx + 1

coth 2x + 1 2 cothx

coth

x jcoshx + 1 -± 2 coshx- 1

sinhx coshx - 1

coth 2x-

5. sinhx+ sinhy =2 sinh x

2Y

cosh x y

x+ y x y sinhx- sinhy = 2 cosh — sinh 2 2 xx+y cosh x + cosh y = 2 cosh — cosh 2 2y x+y x cosh x- cosh y = 2 sinh — sinh 2 2y sinh(x y) sinh(x ± y) cothx± cothy = tanhx±tanhy= sinhx sinhy coshxcoshy 1 6. sinhx sinhy= [cosh(x+y)- cosh(x -y)] 1 sinhx cosh y = - [sinh(x+y)+ sinh(x -y)] 2 1 cosh x cosh y = - [cosh(x+y)+ cosh(x -y)] 2 Complex case: 7. sinh iy = isin y, cosh iy = cos y, tanh iy = itany, coth iy=- i cot y 8. sinh(x+iy), cosh(x+iy), tanh(x+iy), coth(x+ iy): Use 3 and 7. Inverses y= sinhx x=arsinhy=ln(y+ Vy2 + 1) y = cosh x, x0 x= arcosh y= ln(y + ./y2 - 1), y?_ 1 y = tanh x x= artanh y-- In ", IYIi y = cothx x= arcothy _ 124

5.4

5.4 Trigonometric and Inverse Trigonometric Functions Trigonometric functions The trigonometric functions are defined with the aid of the unit circle. The angle a is measured in degrees or radians. One rotation or 360° is 27r radians. 180° 1° = Ir- radians...0.017453 rad. 1 radian= jr 57.295780° 180 Degrees: 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 ir 7T ir Jr 27r 37r 57r 77r 57r 47r 37r 57r 77r 117r , LIt T. w TC -T T. i -T, -3- T. Radians: g 4 j -j: Definitions sin a=y

cos a=x

tan a=x

cot a=

Y

Derivatives. D sin x = cos x

D cos x=- sinx

D tan x= 1 + tan2x =

Related functions:

D cot x =-1- cot2 x=

1 Cos 2 x

1 sin 2x

1 1 secx =, csc x = cosx sinx

Trigonometric functions for some angles of the first quadrant. Angle 0° = 0

15°=,r/12 30°=n/6

sin 0 1 /7 /-,7, - 42. ) 4(4° 1/2

cos 1 1 1— /7.-. -(46 + 4z) 4 ,s/j/2

tan 0

cot -

2-.f

2 + .,/j

1/,/j

,sn

45°=n/4

1/„i2..

1/,12-,

1

1

60° =7r/3

„fi/2

1/2

,fi

1/ ,,/j

1 (V; - 4 ) 4 0

2 + jj

2-

75° = 57r/12 90° =7r/2

4 1

( A + ,i2.)

125

0

5.4 Y yosx --_, -1 ,

Graphs

.

y=sinx , _

O

\ ik \

1,

/

/

\

2n

1,

"ir

\

• -- — (periods=2n)

—1

,

•

Â

YI y=tahx

I I I I I

n

O

I

Geometrical interpretations

I I I I I Dm-

t3 o o o

I

I I I I I

. sinx hm =1 x

x—>0

(x in radians)

sinx x=sec y,

csc-1 x= sin-11X

x

arccot 1 arccot x

1

Principal values of sec-1 and csc-1

sec-1x = cos-11X

Ail -x2 x

1

x2 _

6.1

6 Differential Calculus (one variable) 6.1 Some Basic Concepts f(x) even

Intervals

[a, b] = {x :

closed interval

(a, b)=]a, b[= {x : a 0)

(B # 0)

5. lim h(f(x))= h(A) (h(t) continuous) 7. A = B, f(x) 00) Then lim

f(X) = lim

x —> a g(x) (x —> 00)

if the latter limit exists.

f, (x)

x.- a g f(X) (x —) ...)

Indeterminate forms u (x), u(x)v(x), [u(x)]*) and u(x)- v(x) are not well v (x) defined at a (or at 00) if f(a) has formally any of the forms

Functions f(x) of the forms

0 00 (1) - (2) (3) 0 • 00 (4) [01° (5) 00° (6) r''' (7) 00 - 00 0 00 133

6.2 However the limit lim f(x) may exist and may be found by the use of 1'Hospital's x-*a

rules, rewriting the functions as follows: (3) uv=

1

. (4), (5), (6) /iv = evin ". Here the exponent will be of the form (3).

v

Some standard limits lim Il + x -> +.0

=e

=et lim zilx =1

lim l + x x -> +00C

(lnx)P xP -0 (q>0) lim — = lim xP e-qx =0 (a>1, q>0) lim s->0. x q x->«, ax x-*oo /i nt lim xP Iln xlq =0 (p>0) lim = =0 m ->,>0 m! x->0+ s inax

lim x->13 x

=a

lim

ax -1

1 (1+x) = 1 x->o x l im n

= lna

x->o x 0 if m b,xn + + bo ±. if m>n

Examples I. (1'Hospital's rule) (a) hill 1 - cosx x -> o x2 (b)

lim lnx

x

[0 = sinx 0 x->o 2x 1

.H=

lim

x -*=

x

1

co;x 1 0= = 20 x -> o

2 = O. = lim -> co vx

2,fx

2 1-I- + 0( x4 )) 2 _ lim II. (Taylor expansion) lim 11 cosx - lim x -> 0 x2 x -> o x x2

+ 0(x2)) = ) 2

Continuity Definitions A function y=f(x) is said to be 1. continuous at x0 if x0 E Df and lim f(x)=f(x0),

x-* x0 2. continuous in an interval I if f(x) is continuous at every point of I, 3. uniformly continuous in an interval I if for any E>0 there exists a 45>0 such that 1f(xi)-f (x2)I 0, fog Lipschitz continuous 2. f'(x) bounded in an interval I

f(x) Lipschitz continuous in I.

3. f is Lipschitz continuous f is uniformly continuous (the converse is not true).

Example. la. f(x) = x2 is Lipschitz continuous on [—a,a] with L = 2a, because lx12 —x22.I = I( x i — x2)(xi + x2)I < 2alxi —x21 lb. f(x) = x2 is not Lipschitz continuous on (0, 2. f(x) = 4 is uniformly continuous in [0, 1], but not Lipschitz continuous.

135

6.3

6.3 Derivatives The derivative c(x) of a function y=f(x) is defined by (x) = lim f (x + Ax)- f (x) Ax Ax —>0

= lim

Ax-3

AV

Ax

Alternative notation: dy d y'=f'(x)= — dx =— dx f(x)=Df(x), dy y=

dt

(y a function of time)

Higher derivatives d2 v y"=f"(x) = -=,=D2f(x), dx` y(nc f (n)(x) d ny

dxn

Dn

(x )

[clef: DO-if (x)1].

Differential Difference Af=f(x+ Ax) - f (x) Differential df=f'(x)dx f (x) differentiable

df=f(x)Ax+E(x)Ax, where e(x)

0 as dx —> 0

Equation of tangent of curve y = f (x) at (a, f (a)): y-f(a)=f'(a)(x -a) Equation of normal of curve y =f (x) at (a, f (a)): y-f(a)=

1

(x a)

Lagrange's mean value theorem Assume that f(x) is (i) continuous in [a, b], (ii) differentiable in (a, b). Then there exists 1, E (a, b) such that

f(b)- f(a)= (b - a)f'(4)

Cauchy's mean value theorem Assume that f (x) and g(x) are (i) continuous in [a, b], (ii) differentiable in (a, b). Then there exists E (a, b) such that

[f(b)-f(a)]e(4)=[g(b)- g(a)lr()

136

6.3

Differential formulas f=f(x), g = g(x), D=d/dx Sum, product, quotient fig 2fg' g

D(f+ g)=f'+ g' D(fg)= f' g +fg' D ck)f (n-k)g(k)

Dn(fg)

D(f ugm)=f n-1gm-1(ngr + mfg.)

k=0

D gm

fn 1 (ngr-mfg') g m+1

Logarithmic differentiation. Powers f(x) = u(x)a v(x)b w(x)c (i) lnlf I= a lnlul + b lnivi+c lnlwl + (ii) -c11 -=a-4-1-4-+bc-Lv, +c 412 +... fu v w (x) + w'(x) + u'(x) +b (iii) f'(x) =a w(x) u(x) v(x) f(x) D(fg)= Deg ln f =fg

lffg +

g'ln

Composite functions. (Chain rule) , Df(g(x))=f'(g(x))g (x)

d _dy d

dz _ dz dy dx ;1y dx

dx dx

d2z )2 + dz d 2(dY) z D2f(g(x))=f (g(x))[g'(x)]2 +ft(g(x))g"(x) d x 2 _ dy2ksdx dy Integrals -d - 1 f(t)dt =f(x) dx J f(t)dt = -f(x) dx a x iL ix c yr) f(t)dt=f(v(x))v'(x)-f(u(x))u'(x) u(x) dv d v(x) - F(x, J F(x, t)dt= F(x, v) dx u(x)

du F(x, t)dt + vr) d u(x) ax

137

d2y dx2

6.3 Inverse function d3x _F-3(d23,)2 dy d3 y1 I(dy ) dy3 L dx2) dx dx3i wx y

_ (dy)-1 d 2x =_ d 2y(dyy dy (1.)c) dy2 dx 2l Vlx) Implicit function y = y(x) given implicitly by F(x, y) = 0:

dy = _Fx d2 Y _ 1 [FxxFy 2 -2FxyFxFy+ FyyFx21 dx dx2

Basic derivatives f (x) xa 1 xa

f'(x) axa-I a

- Xa+1

f (x) sinhx coshx

f (x) sinx

f'(x) cosh x sinh x

cosx

1 1 — tanh x ,, = 1 - tanh2x cosh x 2 ,,,rx II .4.2 x =1- cool — _1 coth x x sinh2x x2 2 1 1 -— arsinhx x 3 x2 ,,/x2 + 1 1 eX arcoshx ex n/x2—I ax lit a1 ax artanhx 1 - x2 I 1 In ' xi arcothx x 1 -x2 1 toga Ix' x In a

,rx

tan x cot x

1 =1+ tan2x cos2x - I = -1- cot2x sin 2x

sec x

sin x sec2x

csc x

-cos x csc2x

arcsin x

1

arccos x

,j1—x2 1

arctan x arccot x

138

f'(x) cos x -sin x

,11-x2 1 1 +x2 1 1 +x2

6.4 Derivatives of composite functions (u = u(x), u'= u'(x), D = dl dx)

Df (u)= u'f '(u) Dua = aua-1 u' D Dlnlul = — u

ua

au' ua+l

Deis= eu

Dsin u = u'cos u Dsinku = ku' sink-1u cos u

Dcos u =- u'sin u Dcosku =- ku'sin u cos" u Dtan u= Dcot u=

u' sin2u

Darcsin u =

u'

Darctan u=

u' 2 COS U

u' 1+u2

Higher derivatives

f(k)(x)

f(x)

n(n - 1) ... (n- k +1)(x- a)", k n 0

(x- a)n, n positive integer 1 (x _ ar , n positive integer

E 1)k . n(n + 1) . ..(n + k - 1)

eax inixi

(x - ark a(a -1) ... (a - k +1)x" akeax (_. ok-i(k _ 0! x-k

sin ax

aksin (ax + 1 1

cos ax

akcos (ax + 1(2) 2

xa

6.4 Monotonicity. Extremes of Functions Let f(x) be differentiable in an interval I. Then (in I): f'(x) > 0 f(x) strictly increasing f'1 (Holder's inequality)

1

b

IT +[fi giP1P P , pl

(Minkowski's inequality)

a

_a

Improper integrals The following integrals are said to be convergent if the limit exists, otherwise divergent.

Infinite interval: f(x)

(a) ff(x)dx= lim 7f(x)dx R-->00 a

a

I f(x)dx= lim

(b) Cauchy principal value: (CPV)

R f(x)dx

R-oo —R

Unbounded function b

lim 1 (a) if(x)dx= a

f(x)dx

e-->0+ a+E

c—E

b

b

(n) Cauchy principal value: (CPV).1 f(x)dx= lim ( J f(x)dx + f f(x)dx) aa

E—>0+

... convergent if p> 1 dx are T d—x and I divergent if /31 xP 2 x(lnx)P 1 convergent if p 0 as R —> oo

xe/ R

Test 00

(i)lf(x, t)Ig(t), XE I, (ii) J g(t)dt convergent a 00

J f (x, t)dt uniformly convergent for xE I

7.3 Applications of Differential and Integral Calculus Plane curves (Curves in space, see sec. 11.1) 1

A = area, l = arc length, ic= curvature, p= — = radius of curvature, TO = centre of curvature Curves in parametric form

y . dx

ds= •

(Dot denotes differentiation with respect to t, e.g. x = —

dt

Curve

C: fx=x(t)

2+5,2 dt

t=b

, at (ii) vertical x = xo if lim x(t)= xo, lim y(t)= ±00 t-3t0 t—>t 0

- xy (i2+5,2)3/2

u

V 3/2

=X— 5 -2 xv n =Y+ u

K= da

ds

The evolute is the curve consisting of all centres of curvature of the given curve. 148

7.3

Curves in function form y =y(x) b

[f (x) — g(x)]dx (f(x),g(x))

A= a

All 4-f

1=

(x)2 dx

(x) Asymptote y = kx + m: k= lim Z— , m= lim (y(x)—kx) x—>+00 x

g(x)

{4=x y/(1 +y'2)

y" (x) [1+y' (x)2 ]

y"

1+1/2 11=y+ ; Y

3/2

Tx= -1A- il x[f(x)— g(x)]dx a

Centroid T = (Tx, Ty)

2 l 2 T =— j [f (x)— g (x)]dx Y 2A a about the x-axis

Moments of inertia about the y-axis (i) of curve y=f (x) with density p(x)

Ix = f (x)2 p(x)J1 + 1(x)2 dx

x2 p(x) I 1+ f'(x)2 dx

Iy =

a

(ii) of plane region with constant density Po 1 x2 [f(x)— g(x)]dx (f(x),g(x))

y =p 0 1

Ix=p0131 [fix)3 — g(x)3[dx

a

Curves in implicit form Fx dy =— — , C: F(x, y)= 0, — Fy dx

d2y =—F;F xx + 2Fx Fy Fxy — F.Fyy dx2

F3

—Fy2Fxx +2Fx Fy Fxy —Fx2Fyy K=

U

v3/2

(F+ F'))312 Fx v 1;=x+ — u

F vv ' u 149

7.3 Curves in polar coordinates, x = r cos 0, y =r sin 9 C: r= r(0), ocf3 Remark. The curve r = r(0) may be transformed to rectangular parametric form by x= r(0) cos 0 { y = r(0) sin 0

, a 0 13

dy _ AO) _ I'M sin 0 + r(0) cos 0 dx x'(6) - r'(0) cos 0 - r(6) sin 0 P 1 fi 2 1 A= - f r-(9)de 1= i Al r(0 )2 + r'(0)2 d9 2 a a r(9) tang_ r'(9)

4=x

71

—y

lc =

r2 + 2r'2 - rr" u [r2 + r,213/2 = v3/2

v (rcos 0 + r'sin 0) u

I

x= r cos e y=rsin 0 r2=x2+y2

ffl tangent ke _ ds= ,I r2i-r'2 de r=r(8) 9=a

v (r sin 9-r'cos 0) u

dA=1 r2 de 2

Family of curves Family given by F(x, y, A) = 0, A, = parameter:

Envelope The envelope is the solution (eliminating A) Y of the system F(x, y, .1)=0 I F;,(x, y, X) = 0

x Envelope: Tangent to every curve of the family. 150

7.3

Orthogonal trajectories Differential equation of Family of curves Rectangular coordinates F(x, y, y')=0 Polar coordinates F(6, r, r')=0

Family of orthogonal trajectories F(x, y, —1/y')=0 Orthogonal trajectories: Intersect the curves of the family perpendicularly.

F(6, r, — r210 =0

Solids and surfaces of revolution General volume formula A(x) = area of section V=.br A(x)cbc a

Volume of a solid of revolution Rotation of region D about x-axis: x=b V= f Ity2C1X x=a

Rotation of D about y-axis: x=b V=

f 27rxlyidx (0 0) c ax c

93. fx 1a2x2 ± c2dx = 1 (a2x2 ± c2)3/2 3a2 94. .fx2 ja2x2 ± c2dx = x 2x2 ± ,_ c) 2 0/2 = —(a 4a2 s ja2x2 .4_ e2 95.

x

96. .1. 97. f

c2 x ja 2x2 + c2 _ 8a c43 lnlax + ja2x2 ± c21 8a

---

2 2 2 dx= ja2x2+c2 - cln ,la x + c + c x

2 2 2 j a2 x2 — c2 dx = Ala2x2 c2 carctan,ja x - c x c

1 dxx2 ja2x2 + c2

ja2x2 ± c2 c2 x

98. f xn = xn-1 ja2x2 ± c2 (n-1)c2 r xn - 2 dx dx (n>0) T J ja2x2 ± c2 na2 i ja2x2 ± c2 na2

99. f xn ja2x2 ± c2dx = - 1) c2 f xn-2 n I a2x2 ± c2dx - xn- I (a2x2 + c2)3/2 (n T (n+2)a2 (n + 2)a2 f 2x2 ja ± c2 100.

xn

(n > 0)

(a2x2 ± c2)312 (n-4) a2 r Ja2x2± c2 dx = T (n_ i)c.2xn-1 T dx (n-1)c2 J x n-2

1 101. f dx= J xn ja2x2± c2

=T

ja2x2 + c2 (n - 2) a2 r 1 dx (n > 1) T (n-l)c2x" (n-1)c2 J xn-2 ja2 x2 ± c2 159

(n > 1)

7.4 fx+ b b x-b

1

102.

dx (x-b)A1x2-b2 1

103. 5

(b has arbitrary sign)

1

dt

Al(pb2 +q)t 2+2pbt+ p

(x-b),Ipx2 +q (-ifx>b,+ if xb,+ ifx 0) dxna2 ,I1c2 _ a 2x2 nag "42_ a 2x2 162

7.4

147. ixn jc2 - a2x2dx = -

x n-i (c2 _ a 2 x2 \312

' + (n-1)c2 ixn-2 Jc2 - a2x2dx (n + 2)a2

(n + 2)a2

(n>0)

(c2 _ a2x2)3t2 (n4)a2 r jc2 _ a 2x2 dx (n> 1) 148. fjc2 - a2X2dx = ' ' + xn (n-1) c2 xn-1 (n-1)c2 i xn-2

1

149. 5

1 (n-21a2 r dx (n > 1) _ a 2x2 jc2 (n- 1) c2xn-1 (n -1) c2 J xn-2

,I c2 - a2x2

dx =

xn jc2 - a2x2 150.

+

1

1

dt dx = [x - b = 11- T f 1 t (x- b)," px2 + q ,,l(pb2 + q)t2 + 2 pbt + p (- if x >b, + if x b, + if x 0)

b

2.

f(x-a)m -1(b- x)n

dx= (b _ a )m+

n _ t T((n)r(n) F(m+n)

(a 0)

a

xn

3.

f1+x

0

4. f

P 5.

1

dx =(-1)n[1n2 - 1 + 2

dx

0- x2

(n= 1, 2, 3,

(n> 1)

(1- xn) un n

x-„

+

dx =

sinE n

InF((a + 1)/2) 2F ((a + 2)/2)

(a>-1)

178

...)

7.5 1 6.f

a-1

dx -

sinair

dx

^ firF Q

7.

+

a

AI 1 - xa

8.

(0-1)

(n -1)!! n=1, 3, 5, ... n!! (n-1)!! x2 n=2, 4, 6, ... 2 ' n!! n+1 n 3/2 2 n>-1 2 21, r,1n+ 2

n/2

20. f sin 2a+l x cos 211+ 1 x dx =

.1-(a + 1) F(I3 + 1) 2F(a + + 2)

0 (m # n integers) sinmxsinnx dx = z - (m = n integers) 2

21.

0 (m # n integers) 22. f cos mx cos nx dx 0

= {7r

- (m= n# 0 integers), ir (m = n = 0) 2

(m, n integers, m + n even)

0 23. f sinmxcosnx dx = 2m 0 m 2 _ n2 24.

x/2 r

x/2

dx dx Jo 1 + a cosxJo 1 + a sinx

25. f

dx - 2arccos a 1 + a sinx

26. f

dx

x/2

arccosa (Ial-1, a>0) (n = 2k, k integer, a> 0) (n = 2k +1, k integer, a > 0)

(a > 0)

0

181

7.5

(a>0)

44. e'cosbx dx = a z a+ b2 o

45. J xe sinbx dx = (a22ab 46. xe" cosbx dx = 0 47.

f e-ax sinbx 0

a 2 b2 (a2 b2)2

b dx = arctan -

48. f(lnx)ndx (-1)nn! 0

(n=1, 2, 3, ...)

49. flnllnx! dx= Je'lnx dx =

50.

0

0

f

dx = x1 6

o 51.

—y

2

f 0

(a>0)

(a> 0)

a

x

(a>0)

x +1

2 , a x = -12

52. r lnx gji_x2

53. fxiln 0

1

dx = --ln 2 2 + 1) - r(n (m+ 1) " 1

(m> -1, n> -1)

54.f — sinfinx dx = -11y x 2 0

ir/2

ir/2

55. 5 ln(sinx)dx = ln(cosx)dx = -fin 2 2 0 0 n/4

56. 5 1n (1 + tanx)dx = 711n 2 8

182

8. I

8 Sequences and Series 8.1 Sequences of Numbers Notation: lan ri or a i , a2 , a3, ..., an, ... Limit: The sequence fan17 has the limit A, lim an =A or an —> A as n 00, if for n—>«, any number E> 0 there exists an integer N such that lan —Al N. If the limit exists the sequence is convergent, otherwise divergent. (Laws of limits and rules for determining them, cf. the corresponding laws and rules for functions in section 6.2.) lim an = lim sup an = lim (sup ak) exists for all sequences (possibly ± 00).

n—*oe k

n-*oo

lim an= lim inf an =s lim ((kinf ak) exists for all sequences (possibly ±..). n—>0. n n—>o. Cauchy sequence: A sequence fan I7 is called a Cauchy sequence if for any E> 0 there exists N such that la. — anl N.

Theorems 1. fan I7 monotone and bounded

lim an exists (finite).

n—>«, 2. tan ri is a Cauchy sequence lim an exists (finite). n—>00 Examples 1. lim an = n —>"

0 if lal1

an

2. lim , =0 (a constant)

n

3. lim nja = lim 11,Fla =1 (a = positive constant)

n 00

n

4. an= (1 +

57.an+i=

11 n

nJl

is increasing and an —> e = 2.71828... as n —>

411 +an , a1=0 recursively given sequence. y 2

{an}7 is increasing and bounded lim an=A exists and

n —>

A= 11 + A

nt 2

A=1 183

8.2

8.2 Sequences of Functions Pointwise convergence The sequence { fn(x)} is said to converge pointwise to f (x) on the interval I if for every fixed xE I: lim fn(x)= f (x) [f (x) = limit function]. n —>

1. Arzela's theorem

Assume (i) fn(x)

(x) pointwise on [a, b] 1 0 there exists N such that 1 fn(x) — f (X)I < E for n> N and x E I, i.e. if sup XE s

1 fn(x) — f (x)I —> 0 as n —>

Example. fn(x)—

nx + 1

—> 0 uniformly for x e [0, 1] because

1 sup 1.fn(x) I = — —> 0as n —> oo. n+1 [0,1]

XE

2. Dini's theorem

Assume (i) {fn(x)17 increasing, i.e. fn(x)fn +1(x), all n, x (or decreasing) (ii) fn(x) f (x) pointwise on [a, b] (iii) fn(x), f (x) continuous on [a, b]. Then the convergence is uniform. Further results

Assume that fn(x) is continuous for each n and fn(x) —> f (x) uniformly on [a, b]. Then 3. f (x) is continuous on [a, b]. b

b

4. lim J fn(x)dx = J f(x)dx n—>0.

5.

a

a

fn'(x)}7 converges uniformly = f(x) exists and is equal to lim fn'(x). n--400

184

8.3

8.3 Series of Constant Terms ak= al + a2+a3+ ... is convergent with sum s if the sequence

An infinite series k=1

ak is convergent with limit s. (Otherwise the series is

of partial sums sn= k=1

divergent.) CO

1.

an —> 0 as n

an convergent n=1

Summation by parts (Abel) akbk=Anbn+i - F Ak(bk+1-bk), where An=

2.

k=1

k=1

ak k=1

Integral estimates n+1

n

3. f(x) increasing:

j f(x)c/x. i f(k) < f f(x)dx k=m m-1 m n

n+1

f f(x)dx < i f(k) f f(x)dx

4. f(x) decreasing:

m

k=mm-1

Standard series 1 — is convergent p > 1 (p constant) n=1 nP 1 x-n = — is convergent Ix' < 1 1 -x n=0

Convergence tests Series with non-negative terms 5. (Comparison test). Assume 05_an bn. Then (a) 1bn convergent Ian convergent 1 1 (b) Ian divergent 1

Ebn divergent 1 a, = c 0, 0. (or an- bn). Then 6. Assume an , bn>0, lim n—>oe on

Ian convergent

IN convergent 1 1 7. (Integral test). Assume that f (x) is positive and decreasing for xl‘r. Then f(n) convergent f f(x)dx convergent n=N

185

8.3 Series with arbitrary (complex) terms

8. Elan

an convergent

convergent 1

1

an+i

9. (Ratio test). Assume lim n->co an

(a) c1

1

Ian divergent 1

c. Then 10. (Root test). Assume lim n->c- ' (a) c 0, n —> o. Then 00

E (-1)nan convergent 12. (Dirichlet's test). Assume (i) An =

n

E

ak is bounded (complex) sequence,

k=1

bn y 0, n —> 09. Then anbn convergent

13. (Abel's test). Assume (i) Zan convergent (ii) On) monotone and convergent.

Then

00

¥ anbn convergent 1

Examples 1. 2.

I cony. for p>1 div. for p 1

1

jl

n1 n(ln n)P 1cos -

n=1 (\

n-1 )

(integral test)

cony.

1 1 1) 1 (Comparison test: 1 -cos - = — +O (— - — and E n2 cony. ) 2 n2 n 2n 2 n4 3.

cony. (Leibniz' test) 1 , 7 ikr conv. for x# 2mir (Dirichlet's test, cf. 8.6.18)

4. k=1

4k

Infinite products 00

An infinite product IT (1 + ad= (1+ a1 )(1 +a2) ... with 1 +ak #0 converges to p#0 if k=1

fi (l+ak)=p. n—>co k=1 lim

1. no -1-lak l) convergent

II + ak) convergent

2. ft (1 + lak l) convergent lad convergent 186

8.4

8.4 Series of Functions Uniform convergence CO

The series I fk(x) [=s(x)] is uniformly convergent for xE I if the sequence k=1 .... n Sn(X)= I fk(x) converges uniformly to s(x), xE I, i.e. sup I 1 fk(x)I —>0 as n —> °°. xe I k=n k =1

Tests 1. (Weierstrass' majorant test). Assume (i) I fn(x)lt 1 /1n, xE I (ii) IMn convergent.

i

Then 00

y, fn(x) is uniformly convergent in I.

i

2. (Dirichlet's test). Assume (i) Fn(x)= i fk(x), IFn(x)I_M, xe / k =1 (ii) gn+ i(x)ga(x), xe I (iii) gn(x)-->0 uniformly, xE L Then I, fn(x)gn(x) is uniformly convergent, xe I. 1

Theorems Assume s(x) = 1 fn(x) (fn(x) continuous), is uniformly convergent in [a, b]. Then n=1 3. s(x) is continuous.

b

b

4. i s(x)dx = I f fn(x)dx Q

n=1 a

00

00

5. I f,:(x) uniformly convergent = 1

s'(x)=/,fn'(x).

1

Power series A power series of a real (or complex) variable x is of the form (an may be complex)

c.

f (x)= 1 an(x—x0)n, an = n=0

PO (x0 ) n!

In particular if x0 =0: (8.1)

f(x)=

... I anxn= ao+aix+a2x2+..., a

n

n= o

=

f(n)(0) n!

Divergence

Radius of convergence R 1 ini = [ lim njF ini = lim ,„ = lim sup nfr n—>c,0 ' n—>oo ' " n—>m

I
0) 1. The power series (8.1) any e>0. (i) converges for lxlR (Convergence investigations for lxl =R: Look for tests in sec. 8.3). 2. The series (8.1) may be differentiated or integrated term by term arbitrary many times and the received new series also have radius of convergence= R, i.e. f'(x)=

nanxn-1,1xl 2

!Inn In n In n n — —> u as n —> 001- 1+ — + an= en - [note: In n 2n2 8. Differentiation and integration of a given series, e.g. d[ 1 = cd(1+x+x2 +...)=1+2x+3x2 +... dx1 - x

1

(1 x)2

Operations with series Let s=aix+a2x2+a3x3 + t =b0 +b1x +b2x2 +... /22

b3

b4

al

a2 +a1b1

a3+a2b1 +aib2

a4+a3b1 +a2b2 +aib3

! al

1_ a — —1 a2 2 2 8 l

r=

bo bi

1 1— s a +s)1/2

1 1

1 a1a2+ — 1 3 --4 1 3 -a 2 16a1 5 3 1 (1 +s)-1/2 1 - 2 2 34 aia2- a3- — 2l a1 38 a21 - 1a 16a1 1 3 es a2 + 1-a 2l a3+ - 1 1 at 2 ala2± 6a 2

ln(1 + s)

0 al

cos s

1

sins

0 al

0

1 2 + Tsa 3 21(22 - — 5 4 -1 a4 -,. aia3- §a2 128a1 3 32 4a 03 +§a2

152 1 35 4 - .a4- 16a1a2 + 128a1

1 azt+aia3+-ia22 + 1j,a ia2

1 4

+Tel

1 21 a2 - -a 2

1 2 a3-aia2+-a1 3

-t a i

-a 1 a2

2 1 4 1 2 -aia3+aia2- 1 a4--a2 2 4a 1 2 - a2-aia3+ 1 ai4 2 24

a2

1 3 a3-6-a1

2 a4- 21a1a2

191

8.6

8.6 Special Sums and Series Euler-Maclaurin summation formula, see sec. 16.6.

Miscellaneous sums and series Arithmetic series: an=an-i±d=ai +(n-1)d (d= difference) 1.

[ai +(k-1)6]-

ak= k=1

k =1

n(ai+a ) n Ra i + (n-1)d] n = 2 2

Geometric series: an =an_ i • x = aoxn (x = quotient) n-1,,

2. 1 ax's=a+ax+...+axn 1=a k=0

3. 1 axk =a+ax+ax 2+...= k=0

4.

kmxk = (1 -x)-m-1 k=1

j =1

a 1 —x

a(M)x-i

xn —1 =a x-1

1—xn , (x# 1) 1-x

(-10

of

af

ax

OX) y(= constant)

Alternative notations: fx'..fx =Dxf= ,—= (— ., Higher derivatives:

, ar

a2 f

, a2 f

fx,=fx"x=— axi2 =Dxfx, fyx= fy'x ' =ax Ly =Dxfy, fyy= f;Y ='2 =DY•f; etc. ay Remark. fry=fyx if these functions are continuous. Notation: f E Ck f has continuous partial derivatives of order Example. f(x, y)=x3- 2xy2+ 3y4 f); = -

k.

fx,= 3x2_ 2y2, j,y,= _ 4xy+12y3, f.,7x= 6x, Ky=f),7„= -4y,

4x+ 36y2.

Differentiation of f(x, y) with respect to g(x, y), keeping h(x, y) constant. 01) (p)= lim g h,

f(Q)- f(P)

Y

Q-*P g(Q)-g(P)

1-1= (Set u=g(x, y), v= h(x, y). Express f in terms of u and v (ag and differentiate f with respect to u, keeping v constant} = of . E.g. (a (ax(x÷Yy) )1y =fu=x+y,v=x-yx= f = xY -

u+v

h(x, y)=C

u-v ,y= 2 ;

""X

u2 - v2 ( of ) u x+y 4 2 L. i )„ 2

Alternatively, using the chain rule, (L =(aLxf

)y

(Mv+(g1 (t)v

Differentiability (Linear approximation) f (x, y) is differentiable at (x, y) if

/h2 + k2 s(h, k)

Af=f(x+h,y+k)-f(x, y)= hfx'(x, y)+kfy'(x, y )+,s

where E(h, k) —>0 as (h, k)--4(0, 0). (Analogously for f: Rn -4 R)

af

af

Differential: df= — dx+ — dy ax ay f (x, y) is differentiable at (x, y) if the partial derivatives off (x, y) exist in a neighborhood of (x, y) and are continuous at (x, y).

224

10.4 n=grad f

Gradient R). The vector Given f: R3 —> R (analogously for f: V f= grad f = (fx f y' , 4') is orthogonal to the corresponding level surface f y, z) = C. Directional derivative Given the directional vector e = (ex , ey, ez) of length 1. 1 fe'(a, b, c)= l im - [f (a + tex, h + te c + te )-f (a b,c)]= 1 -4() t d dt f(a+ tex , b +teY' c + tez)i=o= e • grad f (a, b, c) =—

(if grad f is continuous). f 'grad f I

1. fe' is maximal [minimal] in the direction e= grad

2. max fe = I grad f I, min fe

re =

L

grad f Igrad flj

I grad fl.

The chain rule 1. z= z(x, y), x= x(t), y = y(t): dz _ az dx+az dy Tr — ax dt ay dt

d = dx a dy a dt it ax + dt ay

d2 z az d2 x dx 1a2 z dx a2 z dy 2+T it , x . 1 + axTy dtJ + dt = ax •• dt az d2 y dy a2 z dx a2 z dy + _= • — + — — — + — • — dy dt2 dt axay dt ay2 dt

4. x, y, z depending on each other:

3. z = z(x, y), x=x(u, v), Y =y(u, v):

{ aZ _ aZ ax aZ

m

4 3it - Fx • i -F T)-7 • iTt-e az _ az ax az ay .J-1; Fi • av

(11) um), k = 1,

4. f = f (xi , ..., xn), xk=xk(ui , of = of axk , auk k =1 axk au,

_ '•••9

(ix-) = 1/(I etc. ax oY z

(Mz(Mx(M, n

m. •

The mean value theorem If f: Rn —> R has continuous partial derivatives, then f(x+h)-f(x)= h • grad f (x + Oh), 0 < 0 0, — 00 0 (IC 1) u(r, 0)=f (r), 0 rR (IC 2) u,f(r, 0) = 0, 05 rR

Separation of variables (an zeros of J0(x), see sec. 12.4) u(r, t)= E (An cosc —c4 t + Bn sin c—a J (aR r), RL't) 0 R n=1

where J0 is a Bessel function and An-

2

R

R2 J1(an)2 la

are determined as Fourier-Bessel coefficients by (IC).

242

a nr

, i rf(r)Jobil dr, (Bn= 0)

10.9

Transform representation of solutions Example 6. (12(w) Fourier transform of u(x)) { (PDE) Au=u;x +ucy =0, —.0 X3 ) —4.7r I XI

Green's functions (Green's functions for ordinary differential equations, see Sec. 9.4.) Consider the problem (10.8)

Lu(x)=f (x), xe D, Bu(x)= 0, XE aQ

If G(x, y) (Green's function) solves the problem LxG(x, y)= 8(x —y), x, ye 0, BxG(x, y) = 0, 'LE an, yE sz then u(x) = f G(x, y) f (y)dy is a solution of (10.8).

Dirichlet's problem for Laplace's operator If G(x, y) is the Green's function for the problem —du =f

u = 0 on an,

i.e. —dxG (x, y) = 45(x — y), x, yE S2, BxG(x, y)= 0, XE af2, yE D then 1 ac u(x)= f G(x, y) f (y)dy — .1 — (x, y) g(y)dS

ao any

solves the problem —du=f in D, u=g on a(2.

244

10.9

Example. Green's functions for Laplace's operator. Problems below: —AxG(x, y)=8 (x — y) x, ye 12, G(x, y) = 0, xe aa, ye 12 (i) Half plane 12= lx =(xi, x2): x2 > 0} c R2:

X2

G =(x, y)=G(xi, x2, yi, y2)= — 27r (ln Ix —y I —ln Ix— ir I),

Y= (Yi, Y2), i = (Y1, —Y2) (ii) Half space 12= {x= (xi, x2, x3): x3 >0} c R3: l( 1 1 G=(x, y)-=G(xi, x2, x3, Y1, Y2, Y3) = ZRIx — yl Ix — il)' Y= (Y1, Y2, Y3), Y =(Y1, Y2, — Y3) (iii) Disc 12= Ix =(xi, x2): I x I = ,ixi +x2

0

Length of curve=s= $ds= flr(t)Idt = i 2 + C

a

gdt

a

Arc length element ds=ldri=li Idt=vdt Motion of particle Below, r = r(t) = position vector, t = unit tangent vector, n = unit principal normal vector, at= tangential component, an = normal component: Velocity v= r = vt Speed v = s = Irk Acceleration a= v =r = art+ann v•a at = v = — =

an= icv2 = ivxai =

fxil 11'1

Rotation round an axis: v=r =o xr 246

Differential geometry

principal normal normal plane

osculating plane's

rectifying plane binormal Think of C, t, n in the plane of paper

Arc length t s= fj.i2 + )i2 + i2dt

Arbitrary parameter t

Concept

c as

parameter

Unit tangent4 i r 1 = -vector lel - v

t=

Unit principal normal vector

f— vt n= if-141

n

Unit binormal vector

b=txn

b=txn

lc= li x il

K=Iel

Curvature

dr r i= — ds

r" = lel

11'13 Radius of Curvature

PK=

Torsion

2=

1 p,= K

1 K

f• (ix f)

1-=

li X i:12 Radius of torsion

11'12

11 P r= r Pr=

Frenet's formulas =xvn,

ri = —

r' • (r"x r"')

+2 vb, b

= —r vn

(v=1 for s = arc length as parameter) Remark. C straight line x = O.

C plane curve

O.

247

11.2

Example. (Helix) r(t) = (a cos t, a sin t, bt), c=,1a2 + b2 r (t)= (—a sin t, a cos t, b), v=c r (t) = (— a cos t, — a sin t, 0) 'f(t)= (a sin t, — a cos t, 0) r x (t) = (ab sin t, — ab cos t, a2) a tc= , 2= Cz

Cz

to

S= $l r(t)Idt=cto 0

1

t = 1 (— a sin t, a cost, b), n=— (cost, sin t, 0), b= — (b sin t, —b cost, a)

11.2 Vector Fields r = (x,

y, z), F = (P, Q, R), ex, ey, ez ONR-basis.

Vector field. F(r) = (P(x, y, z), Q(x, y, z), R(x, y, z)): R3 —> R3 Scalar field. 0(r)= 0(x, y, z): R3 —> R z F(r)

oz') aaa Del operator V = (— — , ax' ay az a a a =ex.Fx +ey ay +ezz

1. grad 0 = V0 =

e2 r=(x, y, z)

=

) e

y

Every point is assigned a vector

2. F is a gradient field there exists 0 such that F = grad 0 [0 is potential of F] 3. div F = V- F = P.; + Q; + R; 4. curl F =rot F = VxF = (R; —Q;, P;— R; , Q;— P ;) 5. Laplacian AO =V • V 0= div grad 0 = 0x; + Oy;+ Oz'z 6. Biharmonic operator a4

A 20 .v4 = u

ax4 7. F = grad 0 curl F =0

aLi

ay4

a4.

,D4

a1/4,

+ 2 ax2 y2 ay2 z2 ax2 z2

(e.g. in a convex domain) [0 is scalar potential of F]

8. F =curl G div F=0 (e.g. in a convex domain) [G is vector potential of F] 248

11.2 Laws for operations with the operator V Linearity 1. V (a0 0'10= aV + V V' 2. V• (aF +fiG)=aV• F+,6 V• G 3. Vx(aF-F/3G)=aVxF+/3 VxG

grad(a0 +MP) = a grad (I,+/3 grad VI div(aF +PG) =a div F+f3 div G curl(aF +/3G)=a curl F+/3 curl G

Operations on products 4. V(07)=0 V -HP V 5. V(F • G)=(F • V)G + (G • V)F + +Fx(VxG)+Gx(VxF) 6. V• (OF)=OV• F+(VP) • F 7. V• (FxG)=G • VxF—F • VxG 8. Vx(OF)=0 VxF+(VO)xF 9. Vx(FxG)= (G • V)F — (F • V)G + + F(V• G)— G(V• F)

grad(V11) =0 grad '11 + i' grad ch grad(F • G) = (F • grad)G + +(G • grad)F + F xcurl G + G xcurl F div F+F • gradch div(OF) div(F xG)=G • curl F— F • curl G curl(OF)=0 curl F+ (grad 0.)x F curl(F xG)=(G • grad)F (F • grad)G +F div G — G div F

Double application of V 10. V• (VxF)= 0 11. Vx (V0)= 0 12. Vx (VxF) = V(V• F)—V2F

div curl F= 0 curl grad 0 = 0 curl curl F = grad div F — d F

F(r)

Orthogonal curvilinear coordinates Assume that, in a domain of a Cartesian (x, y, z)-coordinate system with unit basis vectors ex, ey, ez, a one-to-one correspondence between (x, y, z) and (u1, u2, u3) is given by (11.1)

Ix=x(ui , u2, u3) y=y(ui , U2, U3)

F(P)=F,ex-FFyey+Fzez= =Ful eu1+Fu,eu 2+Fu,eu3

Z =Z(Ui, U2, U3)

Given P = (x; y; z), assume that the surfaces ui = constant, i =1, 2, 3, intersect at P under right angles. The intersection curves are called coordinate lines. The local unit basis vectors eul, eu2, eu3 form a local ON-system. Condition for (11.1) to define an orthogonal (u1, u2, u3)-system: ar ar _ . . ,,j —0, /*/, r= (x, y, z) au; The system is right-handed

y, z) , >O.

C ( Ui, U2, U3)

249

11.2

Differential formulas in orthogonal coordinate systems General coordinates (u1, u2, u3) eui

_ Vuiar/ ar F u ji aui 5t7;

Let F(P)=Fxex+Fyey+F,e,= Fu eui +Fu2eu2+Fu3eu3 be vector valued. Let u(P) be scalar valued. )2 + ( az )2 , 1 _ i(ax y + i = 1, 2, 3. Set hi= ar atsi Pui aui) A au; Then (i)

Z

Fui =h 11Fxau + Fy

au (vector component relationship)

i =1, 2, 3.

(displacement vector)

(iia) dr =hidui eui +h2du2eu2 + h3du3e„3

(arc length element) (b) ds 2 = I dr I2 =h i2dui 2 + h22du22 + h32du32 (c) h1h2 duidu2, h2h3 du2du3, h3hi du3du1 (surface elements) (volume element) (d) dV = hih2h3 duidu2du3 (iii) grad u=Vu=

aU — —eu j=1 hi au;

(iv) div F= V• F=

a hih2h3 i=l aui 3

r i h2h3 F hi

u)

h i eui h2eu2 h3e u3 1 a a a (v) curl F= VxF= h 1 h2h3 au1 au2 au3 hi Fu, h2F,,2 h3Fu3 (vi) Au =V 2u =

(hih2h3 au

3 a h i h2h3 i=1 aui

h?. aui)

Cartesian (rectangular) coordinates (x, y, z) hi =h2=h3 = 1 (ii) ds2 = dx2 dy2 dz2 all aU (iii) grad u= aU —ex + —ey+ —ez (ex =1, ey = j, ez =k) ax ay az aFx aFz (iv) F +— ax ay az aF) aF, aF) (aF a x (v) curl F ( --- — e +—(aFe ———— + aX ay Z ay az x az ax a 2 t, a 2u a2 u (vi) Au=+ ax2 aye+az 2 250

11.2 Translated and rotated coordinates (4, ri , C) x=40+a11 -"lg.) -1- ai3c Y = 710 + an +a2217 +a23 (ay) orthogonal matrix z=c01-a31+a32r7+a334. { h = h2= h3 = 1 (i)

F4=a11Fx+ a2iFy+ a3iFz Fri =a12Fx+ a22Fy + a32Fz Fc = ai3Fx +a23Fy +a33Fz

(iia) dr= rge4+ en + 4ec (b) ds 2 =4 2 + dri 2 + dc 2 (c) 4117, dridc, dcd4 (d) dV=41774" au au au (iii) grad u= e4+Fli en+Fc ec aF aFr, aFc (iv) div F= — + — + arj ac (aF, aF aF„) (aF4 aFc) (v) curl F. (— - — e+ — - — e + - —) er an ac ac a4 a4 u u u

(vi) Au=a42 —+— —2 n2 + a4

Cylindrical coordinates (p, rp, z) (Polar coordinates in the plane: Neglect terms with z.) Coordinate transformations x=pcosT, y= psimp, z=z p = jx2 + y2 , (p= tan-1 (suitable branch), z=z h1 =1, h2= p, h3=1 Basis vector relationship ex= ep cos ci-e,c, sin q) ey= ep sin 9 + eq,cos (p { ez= z e {ep= ex cos cp + ey sin q) e(p= -ex sin y) + ey cos (p ez = ez 251

11.2 (i) Vector component relationship: {Fx=Fp cos (p- T.,sin (p Fy=Fp sin cp + F1, cos p Fz= Fz

Fp= Fx cos (p+ sin p (p Fy,=-Fx sin (p+ Fz = Fz Fy

Fy cos

(displacement vector) dr= epdp+eq,pd(p+ezdz (arc length element) ds 2=dp2 + p2d(p 2 + dz 2 (surface elements) pdpd(p, pdTdz, dzdp (volume element) dV= pdpd(pdz au 1 au du grad u = Vu = — e + - — e + — e °P dz z ap P P F(P+aFz ( PFP ) + 1 a_ aq. az P ap aF„\ (aFp aFz curl F= rot F= VxF = ( 1 Fz 13) e9+ aZ eP+ 0 U3' -a71 (a( pFT ) aF1 ap aø ez 1 1 au) 1 a2u a2u Au =v2u= - up + p72 u(p,p+ uzz (vi) 1-313' a 492 ± z2 Pa Spherical coordinates (r, 0, yo) div F=V

= Upp+ p

Coordinate transformations x=r sin 0 cos (p,y=r sin0sin cp,z=r cos6 r = x2 + y 2 + z2 , 0= arccos (z/Vx2 + y2 4_ z2

= tan -1(ylx) (suitable branch) F

h1 =1, h2= r, h3 =rsin e Basis vector relationship {ex = er sin 0 cos p + ee cos °cos cp-e(p sin p ey =er sin 0 sin p + ee cos 0 sin p+ ey, cos p ez = er cos 0- ee sin 0 er = ex sin 0 cos (p+ey sin 0 sin (p+ez cos 0 ee = ex cos 0 cos q)+ey cos 0 sin p-ez sin 0 { ev = -ex sin (p +ey cos v Vector component relationship: {Fix = Fr sin 0 cos q)+ Fe cos 0 cos q)- Fc, sin q) Fy= Fr sin 0 sin (p+ Fe cos 0 sin p+ FT cos q) Fz = Fr cos 0- Fe sin 0 Fr = Fx sin Cocos p + sin 0 sin (p+ Fz cos 0 Fe= Fx cos ()cos (p+ Fy cos 0 sin (p- Fz sin 0 { FT= -Fx sin (p+ cos cp Fy

Fy

252

11.3 (displacement vector) (iia) dr = erdr + eerd0 +e prsin0dcp (arc length element) (iib) ds 2= dr2+ r2dO 2+ r2sin2echp 2 (surface elements) (iic) rdrd0, r2sin0d04, rsinOdcpdr (volume element) (iid) dll = r2sinOdrdachp 1 du 1 au u (iii) grad u = V u= a— e + - — e 0 + r sin 0 dcp e(1) ar r r ae 1 aF, 1 a(Fe sine) 1 a(r2Fr ) + . + r sm 0 dcp ae r2 ar r sine A aF e) sin (a(Fc, 1 (v) curl F =rot F = VxF = r sin 0 -)) e r+ a0

(iv) div F= V. F=

+

(vi)

r sm e

(aFr sin ea(rF9)) e + aq) ar

1(a(rFe ) aFr) ) e9 ar

a2u 1 1 a (. au + + sin 0 =V 2u = I qr2 r2 ay au aeJ r2sin20 acp2 at' r2 sinOae 1 [a (

a2u 2 au

=— a r2 +

N-

1R) ± 1

a2u1 if =cos 61

2 42

11.3 Line Integrals Differential forms f, g:

R, h: R —> R. Differential form: ll ae =fdg =f (=dx + + axi 1 - ax0 xn

1. d(af+ bg)= a df+ b dg 3. d ( n _ g df - f dg g

2

2. d(fg)=f dg + g df 4. d(h(f))=h'(f)df

Exact differential forms In R2: w = P dx + Q dy is exact if there exists 0(x, y) called primitive function of w such that Vx=P, Oy =Q, (i.e. w=d0). (in a simply connected domain). Test: P dx + Q dy exact P =

In R3: co= P dx+ Q dy+R dz is exact if there exists 0(x, y, z) [primitive function] such that aix =P, Oy Q, O'z = R, (i.e. =d0). (in Test: P thc + Q dy + R dz exact curl(P, Q, R)=0 13; = Q;, P 'z = R ;, Q ;= R a simply connected domain). 253

11.3

Line integrals Given: Curve C: r= r(t)= (x(t), y(t), z(t)), a t dr = (dx, dy, dz), ds =Idri= "112 + Y 2 + i2 dt

Vector field F = F(r) = = (P(x, y, z),

y, z), R(x, y, z))

Scalar field OW =0(x, y, z) There are four kinds of line integrals: 1. f F•dr

2. f Oldri

C

C

3. F xdr

4. f Odr

C

C

1. Tangent line integral f co= f F • dr= f Pdx+ Qdy+Rdz= c c = fI2 a

dt +Q i d c+RMdt

Properties: (i) 5 co =- f co

(ii)

C

-C

(iii) f d0 = f V0 • dr = 0(B) -1,(A) C C

f co= f co+ f co c,÷ C2 CI C2 (iv) co exact

f co= o(B) -0(A), 0 primitive function of co C

Example. Calculate /= f y dx + zdy-x2dz, C: (x, y, z) = (cos t, sin t, t),(21

22t.

2,r

C

Solution: dx=- sin t dt, dy =cos t dt, dz= dt. Thus, 1= f (- sin2t + t cos t-cos2t)dt= - 27r

Green's formula in the plane Assume (i) C closed curve with positive orientation, (ii) P, Q continuously differentiable on C and in D. Then fPc/x+Qdy=ff(-2—LP)dxdy

Area of D= = xdy=- J ydx= 1 - Jxdy-ydx -

254

C

11.3

Theorem Assume that P, Q are continuously differentiable in a simply connected plane domain D. Then the following conditions are equivalent. (i) P cbc + Q dy is exact

— = aQ (ii) OP ay ax P dx + Q dy = 0 (any closed C in D)

(iii)

c (iv) P dx+Q dy depends only on the initial point and the end point of the curve C1

ci (lying in D)

Stokes' theorem (see sec. 11.4) b

2.f 0 I dr I =1 0 ds = T'(r(t)) dr dt= dt

a b

( -yr (dzy = f o(x(t), At), z(0) ,j(dxy c12' WO + WO

dt

a

3. SF x dr = f F(r(t))x d dt a

i

4.f 0dr =f di(r(t))5/1t dt

(vector valued)

(vector valued)

a

Applications (i) Arc length /(C) of curve C: /(C)= f ds = f i2 ))2 i2 dt C a (ii) Centroid T= (XT, yT, zT) of homogeneous curve C: xT= l( yc, zc similarly.

255

1 C)

xds,

11.4

11.4 Surface Integrals

zI

Surface Surfaces are given in one of the following three ways: A. Graph of a function S: z= z(x, y), (x, y)E D

n

=+

(ex, zy' , - 1)

Y

, (1111=1), dS = fidS =

D

Al + (4)2 + (4)2

_DI

= ± (4, zy, -1)dxdy, dS = ,i1 + (zD2 + (zy')2 dxdy B. Parametrization S: r = r(u, v) = (x(u, v), y(u, v), z(u, v)), (u, v)E D r,,' x r„' n=+ , 1 ), dS = 'VS = ±rL x r'„ dudv lru x r ,v l 41111= dS = I x r',,Idudv C. Level surface

rad "fl,( 1111= 1) -Igrad grad f l

Y

S: f(x, y, z) = C, 11 = + -

Surface integrals Given: Surface S as above.

Vector field F = F(r) = (P, Q, R) Scalar field 0=0(r). There are four kinds of surface integrals:

i. ff F • dS= If F • il dS s

s

2.

If 0 dS s

3.

LI F xdS s

1. Normal surface integral A. ff F•dS= SIF- fidS=±5.1(P -Q +R)dxdy ax ay S S D (+ refers to the "upwards" direction of the normal vector) B. ff F • dS = fi. F • fi dS = Si F(r(u, v)) • (ri'd0 in (a, b) (ii) p(x)>O, q(x) > (iii) A, B, C, D real non-negative constants, A2+ B2> 0, C2+D2> 0

260

12.1 The Sturm-Liouville eigenvalue problem Find eigenvalues A and corresponding eigenfunctions cp(x) 0 of the boundary value problem dx {- ± c {p(x)yo'(x)} + q(x)p(x)=Aw(x)(p(x) Aci'(a)-13(p(a). 0 Cp'(b)+ Dyci(b)= 0 5. The eigenvalues Ai, satisfy 0 00) B2n _ (_ 1)n+1

2(20 (2702n

( 1r+1 . 4 41 )2n V) re

n

Bn

n

Bn

n

Bn

0 1 2 4

1 - 1/2 1/6 -1/30

6 8 10 12

1/42 -1/30 5/66 - 691/2730

14 16 18 20

7/6 -3617/510 43867/798 -174611/330

Euler polynomials En(x) Generating function

tn 2ext = 1 E (x) — n! +1 n 0 n Relations. En'(x)=nEn_ 1(x); E„(x +1)+ En(x)=2.xn

E0(x)= 1

Ei(x)=x-

E4(x)=x4-2x 3+ x

1

E2(x) = x2- X

2+1 3x E3(X)= X 3-4 2

E5(x)= x 5-5 - x4+5 - X2-1 2 2 2

269

12.4 Euler numbers En (2n)!22n+ 2 7r 2n + 1

En= 2n En (51) = integer. E2n, = 0, E2n =

0

(-1)k(2k+1)-2n-I , n>_ 0

Genrating function tn 2ett2 2-n En et +1 n=0 Asymptotic behaviour (n —> .0) 8 ,T (21ityn

(022n+2

E212- (-1)n

22 n+1

It4e)

n

En

n

En

n

0 2 4 6

1 -1 5 - 61

8 10 12 14

1385 - 50521 2702765 -1993 60981

16 18 20 22

En I 93915 12145 - 240 48796 75441 37037 11882 37525 -69 34887 43931 37901

12.4 Bessel Functions Bessel functions Jp(x) x +2k (-1)k (p# integer) (0 0.: J„(x)=

-zx 2- [cos(x - 1-

Yn(x)=

; 1 rx [sin(x - ni _

igx)=

ex

2 gx

1+ 0(5

11+ oG)]

[1 + ()GI)]

Kn(x)= ± 2rx-e-x[l + 0(.01

2. x —> 0+: •I n(X) — (i)n

ll

4(9 — t Y

2 Yo(x)- Tr In x Ko(x)--lnx

Yli(x)

(n-1)! (2)n

Kn(x) (n2

ir

1)!

X

, n> 0

py ,n>0

Lx)

Zeros The functions Jn(x) and Jn'(x), n= 0, 1, 2, ... have inifinitely many simple positive zeros. Some of these are tabulated below for 0 5n 57. Orthogonal series of Bessel functions Assume p0 and weight function w(x)=x in the interval [0, a], a >O. (i)

Let a1, a2, a3, ... be the positive roots of the equation Jp(x)= O. a ) r is a complete orthogonal system on [0, a] with Then { J pW respect to w(x) =x and a 2 jp+ i(ak)2 jp(akx) 2 = . Thus for 0 5x 5 a: a ) w =x 2 ...

f(x)= I, ck.f p k=1

rkx)

2 f ftx\ rp(akx) x A., , ck = 2 a .1 p+ 1(ak) 2 oj j‘jj a u''''

275

12,4 (ii) Let pi, 02, 03, ... be the positive roots of the equation ap(x) + x J (x) = 0 (c > -p). Then f(x) = 1 ck f p k= 1

(PkX)

pki

2,62 k , Ck = 2 2 2 af f (x) -IP(1 2 a (fik - P +c2)4690 0

xdx

(iii) Let /31, $2, /33, ... be the positive roots of the equation -pJp(x)+xJ'(x)=0. (Note the special case JWx)=0 for p =0) Then

fp (13+x)

I ck , k=1 a + r 2 (x)xP±i dr, Ck= Co= a 2 2P- 2 f ,2 IA 0

f(x) =coxp+

I ( 14 pkil k)

a 2

1x dx,Icl f(x) J 4131 a

An application of Bessel functions on oscillation of a circular membrane, see Example 5, sec. 10.9. Examples of eigenvalue problems 1. Dirichlet condition. (Singular) eigenvalue problem:

(xy'(x))' + (2x- n21x)y(x)=0,

0 < x 20 [sin(x - 71 4 4 8x

- , error 20 - + 3-cos (x -3711 Y1(x) --- ii2 [sin(x -31 4 8x 4 7CX 2(n - 1) 4-1(x)-4-2(x) Jo'(x)=-Ji(x) x 2(n - 1) Yn- 1 (X) - Yn-2(x) Yd(X) =- Yi (x) Y n(x) = x

Jn(x)=

For x > 20, /0(x),-. ex (1 + -F1x )Alrx

I i(x) ...-- ex (1 - M/411Frx

Ko(x) ...-- e-x j

K1(x)=eCx x ni ;11 + A)

2x(1 - 8x) 2(n - 1) 4_1(x) In(x) = In_2(x)-

ex) = ii(x)

x

Kn(x) = Kn_2(x) +

2(n); 1

4(x) .- K i(x)

Kn-1(x)

283

12.4 Zeros ak = ank of Jn(x) and associated values Jn'(ank ) = k

aOk

Maok)

alk

am

Ji(avd

1 2 3 4 5

2.4048 5.5201 8.6537 11.7915 14.9309

- 0.5191 0.3403 - 0.2715 0.2325 - 0.2065

3.8317 7.0156 10.1735 13.3237 16.4706

- 0.4028 0.3001 - 0.2497 0.2184 - 0.1965

5.1356 8.4172 11.6198 14.7960 17.9598

- 0.3397 0.2714 - 0.2324 0.2065 - 0.1877

6.3802 - 0.2983 0.2494 9.7610 13.0152 - 0.2183 16.2235 0.1964 19.4094 - 0.1800

6 7 8 9 10

18.0711 21.2116 24.3525 27.4935 30.6346

0.1877 - 0.1733 0.1617 - 0.1522 0.1442

19.6159 22.7601 25.9037 29.0468 32.1897

0.1801 - 0.1672 0.1567 - 0.1480 0.1406

21.1170 24.2701 27.4206 30.5692 33.7165

0.1733 - 0.1617 0.1522 - 0.1442 0.1373

22.5827 0.1672 25.7482 - 0.1567 28.9084 0.1480 32.0649 - 0.1406 35.2187 0.1342

k

ii(

au)

n+1(ank)

am

J3(a3k)

li(lYara

ask

a6k

Jg( a 61J

1 2 3 4 5

7.5883 11.0647 14.3725 17.6160 20.8269

- 0.2684 0.2319 - 0.2064 0.1877 - 0.1732

8.7715 12.3386 15.7002 18.9801 22.2178

- 0.2454 0.2174 - 0.1961 0.1799 - 0.1671

9.9361 13.5893 17.0038 20.3208 23.5861

- 0.2271 0.2052 - 0.1873 0.1731 - 0.1616

11.0864 - 0.2121 14.8213 0.1948 18.2876 - 0.1794 21.6415 0.1669 24.9349 - 0.1566

6 7 8 9 10

24.0190 27.1991 30.3710 33.5371 36.6990

0.1617 - 0.1522 0.1442 - 0.1373 0.1313

25.4303 28.6266 31.8117 34.9888 38.1599

0.1567 - 0.1480 0.1406 - 0.1342 0.1286

26.8202 30.0337 33.2330 36.4220 39.6032

0.1521 - 0.1441 0.1373 - 0.1313 0.1261

28.1912 0.1479 31.4228 - 0.1405 0.1342 34.6371 37.8387 - 0.1286 41.0308 0.1237

as

A(a 5k)

a7k

Ji (a 7 k)

Zeros Pk= fink of Jn'(x) and associated values Jn( fink) k

fink

JO(P0k)

Pik

Ji(Pu)

Pv,

1 2 3 4 5

0.0000 3.8317 7.0156 10.1735 13.3237

1.0000 - 0.4028 0.3001 - 0.2497 0.2184

1.8412 5.3314 8.5363 11.7060 14.8636

0.5819 - 0.3461 0.2733 - 0.2333 0.2070

3.0542 6.7061 9.9695 13.1704 16.3475

0.4865 - 0.3135 0.2547 - 0.2209 0.1979

4.2012 0.4344 8.0152 - 0.2912 0.2407 11.3459 14.5858 - 0.2110 17.7887 0.1904

6 7 8 9 10

16.4706 19.6159 22.7601 25.9037 29.0468

- 0.1965 0.1801 - 0.1672 0.1567 - 0.1480

18.0155 21.1644 24.3113 27.4571 30.6019

- 0.1880 0.1735 - 0.1618 0.1523 - 0.1442

19.5129 22.6716 25.8260 28.9777 32.1273

- 0.1810 0.1678 - 0.1572 0.1484 - 0.1409

20.9725 - 0.1750 24.1449 0.1630 27.3101 - 0.1531 30.4703 0.1449 33.6269 - 0.1378

k

P4k

.12(P2k)

J 6(136k)

P3k

J7(P7k)

14(P4k)

P5k

J 5(P5k)

P6k

1 2 3 4 5

5.3176 9.2824 12.6819 15.9641 19.1960

0.3997 - 0.2744 0.2296 - 0.2028 0.1840

6.4156 10.5199 13.9872 17.3128 20.5755

0.3741 - 0.2611 0.2204 - 0.1958 0.1785

7.5013 11.7349 15.2682 18.6374 21.9317

0.3541 - 0.2502 0.2126 - 0.1898 0.1736

8.5778 0.3379 12.9324 - 0.2410 16.5294 0.2059 19.9419 - 0.1845 23.2681 0.1693

6 7 8 9 10

22.4010 25.5898 28.7678 31.9385 35.1039

- 0.1699 0.1587 - 0.1495 0.1417 - 0.1351

23.8036 27.0103 30.2028 33.3854 36.5608

- 0.1653 0.1548 - 0.1462 0.1388 - 0.1326

25.1839 28.4098 31.6179 34.8134 37.9996

- 0.1613 0.1514 - 0.1432 0.1362 - 0.1302

26.5450 - 0.1576 29.7907 0.1482 33.0152 - 0.1404 36.2244 0.1338 39.4223 - 0.1281

284

P7k

J3(P3k)

12.4 Zeros ak = ank ofjjx) and associated values jn'( ank)= n +1,( den) k

iAtrok)

alk

ji(crik)

a2k

1 2 3 4 5

3.1416 6.2832 94248 12.5664 15.7080

- 0.3183 0.1592 - 0.1061 0.0796 - 0.0637

4.4934 7.7253 10.9041 14.0662 17.2208

- 0.2172 0.1284 - 0.0913 0.0709 - 0.0580

5.7635 9.0950 12.3229 15.5146 18.6890

- 0.1655 0.1079 - 0.0803 0.0641 - 0.0533

6.9879 - 0.1338 10.4171 0.0933 13.6980 - 0.0718 0.0585 16.9236 20.1218 - 0.0493

6 7 8 9 10

18.8496 21.9911 25.1327 28.2743 31.4159

0.0531 - 0.0455 0.0398 - 0.0354 0.0318

20.3713 23.5195 26.6661 29.8116 32.9564

0.0490 - 0.0425 0.0375 - 0.0335 0.0303

21.8539 25.0128 28.1678 31.3201 34.4705

0.0456 - 0.0399 0.0354 - 0.0319 0.0290

0.0427 23.3042 26.4768 - 0.0376 29.6426 0.0336 32.8037 - 0.0304 0.0277 35.9614

izi(ae)

a5k

j;(a5k)

a6k

k

aok

a4k

igavd

.N(a6k)

a3k

a7k

ß(a3k)

.Ma7k)

1 2 3 4 5

8.1820 11.7049 15.0397 18.3013 21.5254

- 0.1123 0.0822 - 0.0650 0.0538 - 0.0459

9.3558 12.9665 16.3547 19.6532 22.9046

- 0.0966 0.0735 - 0.0594 0.0499 - 0.0430

10.5128 14.2074 17.6480 20.9835 24.2628

- 0.0848 0.0664 - 0.0547 0.0465 - 0.0405

11.6570 - 0.0754 15.4313 0.0606 18.9230 - 0.0507 22.2953 0.0435 25.6029 - 0.0382

6 7 8 9 10

24.7276 27.9156 31.0939 34.2654 37.4317

0.0401 - 0.0356 0.0320 - 0.0291 0.0266

26.1278 29.3326 32.5247 35.7076 38.8836

0.0378 - 0.0338 0.0305 - 0.0278 0.0256

27.5079 30.7304 33.9371 37.1323 40.3189

0.0358 - 0.0322 0.0292 - 0.0267 0.0246

0.0340 28.8704 32.1112 - 0.0307 0.0280 35.3332 38.5414 - 0.0257 41.7391 0.0238

Zeros ßk = ßnk ofjn'(x) and associated valuesin(ßnk) k

.12( ß2k)

ß3k

j3(ß3k)

.100k)

filk

J1(ßlk)

ß2k

2 3 4 5

0.0000 4.4934 7.7253 10.9041 14.0662

1.0000 - 0.2172 0.1284 - 0.0913 0.0709

2.0816 5.9404 9.2058 12.4044 15.5792

0.4362 - 0.1681 0.1086 - 0.0806 0.0642

3.3421 7.2899 10.6139 13.8461 17.0429

0.3068 - 0.1396 0.0950 - 0.0726 0.0589

0.2417 4.5141 8.5838 - 0.1205 11.9727 0.0850 15.2445 - 0.0663 0.0545 18.4681

6 7 8 9 10

17.2208 20.3713 23.5195 26.6661 29.8116

- 0.0580 0.0490 - 0.0425 0.0375 - 0.0335

18.7426 21.8997 25.0528 28.2034 31.3521

- 0.0534 0.0457 - 0.0399 0.0355 - 0.0319

20.2219 23.3905 26.5526 29.7103 32.8649

- 0.0496 0.0428 - 0.0377 0.0337 - 0.0305

21.6666 - 0.0464 24.8501 0.0404 28.0239 - 0.0358 0.0321 31.1910 34.3534 - 0.0292

ß5k

1

k

ßOk

134k

ß6k

5.6467 9.8404 13.2956 16.6093 19.8624

0.2016 - 0.1067 0.0772 - 0.0612 0.0509

6.7565 11.0702 14.5906 17.9472 21.2311

0.1740 - 0.0961 0.0709 - 0.0570 0.0479

7.8511 12.2793 15.8631 19.2627 22.5781

0.1537 - 0.0877 0.0658 - 0.0534 0.0452

8.9348 0.1380 13.4720 - 0.0808 17.1175 0.0614 20.5594 - 0.0503 0.0429 23.9064

6 7 8 9 10

23.0828 26.2833 29.4706 32.6489 35.8205

- 0.0437 0.0383 - 0.0341 0.0308 - 0.0280

24.4748 27.6937 30.8960 34.0866 37.2686

- 0.0413 0.0364 - 0.0326 0.0295 - 0.0270

25.8461 29.0843 32.3025 35.5063 38.6996

- 0.0393 0.0348 - 0.0313 0.0284 - 0.0260

27.1992 - 0.0375 0.0333 30.4575 33.6922 - 0.0300 36.9099 0.0274 40.1151 - 0.0251

i4( 34k)

285

.16( ß6k)

ß7k

:17( ß7k)

j5(ß5k)

I 2 3 4 5

12.4 Numerical tables of Ber, Bei, Ker and Kei Functions x

Ber(x)

0.0

1.000000 0.000000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.999998 0.999975 0.999873 0.999600 0.999023 0.997975 0.996249 0.993601 0.989751

0.002500 0.010000 0.022500 0.039998 0.062493 0.089980 0.122449 0.159886 0.202269

2.420474 1.733143 1.337219 1.062624 0.855906 0.693121 0.561378 0.452882 0.362515

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.984382 0.977138 0.967629 0.955429 0.940075 0.921072 0.897891 0.869971 0.836722 0.797524

0.249566 0.301731 0.358704 0.420406 0.486734 0.557560 0.632726 0.712037 0.795262 0.882122

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

0.751734 0.698685 0.637690 0.568049 0.489048 0.399968 0.300092 0.188706 0.065112 - 0.071368

3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

Bei(x)

Ker(x)

Kei(x)

x

Ber(x)

- 0.776851 - 0.758125 - 0.733102 - 0.703800 - 0.671582 - 0.637449 - 0.602175 - 0.566368 - 0.530511

5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

- 6.230082 - 6.610653 - 6.980346 - 7.334363 - 7.667394 - 7.973596 - 8.246576 - 8.479372 - 8.664445 - 8.793667

0.116034 - 0.346663 - 0.865840 - 1.444260 - 2.084517 - 2.788980 - 3.559747 - 4.398579 - 5.306845 - 6.285446

- 0.011512 - 0.009865 - 0.008359 - 0.006989 - 0.005749 - 0.004632 - 0.003632 - 0.002740 - 0.001952 - 0.001258

0.011188 0.011052 0.010821 0.010512 0.010139 0.009716 0.009255 0.008766 0.008258 0.007739

0.286706 0.222845 0.168946 0.123455 0.085126 0.052935 0.026030 0.003691 - 0.014696 - 0.029661

- 0.494995 - 0.460130 - 0.426164 - 0.393292 - 0.361665 - 0.331396 - 0.302565 - 0.275229 - 0.249417 - 0.225142

6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9

- 8.858316 - 8.849080 - 8.756062 - 8.568793 - 8.276250 - 7.866891 - 7.328688 - 6.649176 - 5.815515 - 4.814556

- 7.334747 - 8.454495 - 9.643739 - 10.900737 - 12.222863 - 13.606512 - 15.046993 - 16.538425 - 18.073624 - 19.643992

- 0.000653 - 0.000130 0.000319 0.000699 0.001017 0.001278 0.001488 0.001653 0.001777 0.001866

0.007216 0.006696 0.006183 0.005681 0.005194 0.004724 0.004274 0.003846 0.003440 0.003058

0.972292 1.065388 1.160970 1.258529 1.357485 1.457182 1.556878 1.655742 1.752851 1.847176

- 0.041665 - 0.051107 - 0.058339 - 0.063670 - 0.067373 - 0.069688 - 0.070826 - 0.070974 - 0.070296 - 0.068939

- 0.202400 - 0.181173 - 0.161431 - 0.143136 - 0.126241 - 0.110696 - 0.096443 - 0.083422 - 0.071571 - 0.060825

7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

- 3.632930 - 2.257144 - 0.673695 1.130800 3.169457 5.454962 7.999382 10.813965 13.908912 17.293128

- 21.239403 - 22.848079 - 24.456480 - 26.049184 - 27.608771 - 29.115712 - 30.548263 - 31.882362 - 33.091540 - 34.146834

0.001922 0.001951 0.001956 0.001940 0.001907 0.001860 0.001800 0.001731 0.001655 0.001572

0.002700 0.002366 0.002057 0.001770 0.001507 0.001267 0.001048 0.000850 0.000671 0.000512

- 0.221380 - 0.385531 - 0.564376 - 0.758407 - 0.968039 - 1.193598 - 1.435305 - 1.693260 - 1.967423 - 2.257599

1.937587 2.022839 2.101573 2.172310 2.233446 2.283250 2.319864 2.341298 2.345433 2.330022

- 0.067029 - 0.064679 - 0.061985 - 0.059033 - 0.055897 - 0.052639 - 0.049316 - 0.045972 - 0.042647 - 0.039374

- 0.051122 - 0.042395 - 0.034582 - 0.027620 - 0.021446 - 0.016003 - 0.011231 - 0.007077 - 0.003487 - 0.000411

8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9

20.973956 24.956881 29.245215 33.839755 38.738423 43.935873 49.423085 55.186932 61.209725 67.468741

- 35.016725 - 35.667081 - 36.061120 - 36.159401 - 35.919830 - 35.297700 - 34.245761 - 32.714319 - 30.651388 - 28.002868

0.001486 0.001397 0.001306 0.001216 0.001126 0.001037 0.000951 0.000868 0.000787 0.000710

0.000370 0.000244 0.000134 0.000038 - 0.000044 - 0.000115 - 0.000174 - 0.000223 - 0.000263 - 0.000295

- 2.563417 - 2.884306 - 3.219480 - 3.567911 - 3.928307 - 4.299087 - 4.678357 - 5.063886 - 5.453076 - 5.842942

2.292690 2.230943 2.142168 2.023647 1.872564 1.686017 1.461037 1.194601 0.883657 0.525147

- 0.036179 - 0.033084 - 0.030108 - 0.027262 - 0.024557 - 0.022000 - 0.019595 - 0.017344 - 0.015248 - 0.013305

0.002198 0.004386 0.006194 0.007661 0.008826 0.009721 0.010379 0.010829 0.011097 0.011210

9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

73.935730 80.576411 87.349953 94.208443 101.096360 107.950032 114.697114 121.256066 127.535652 133.434460

- 24.712783 - 20.723570 - 15.976414 - 10.411662 - 3.969285 3.410573 11.786984 21.217532 31.757531 43.459153

0.000637 0.000568 0.000503 0.000442 0.000386 0.000333 0.000285 0.000240 0.000200 0.000163

- 0.000319 - 0.000337 - 0.000349 - 0.000355 - 0.000357 - 0.000356 - 0.000351 - 0.000343 - 0.000333 - 0.000321

0. - 0.785398

For x> 10, see next page.

286

Bei(x)

Ker(x)

Kei(x)

12.5 For x 10: Ber(x),-= ex,1.4 .._ [cos(,, 2- 8 Lr) 8x + I cos x_ -L °oT 2 rrx 42 Bei(x),-- exL 5 [sin (:;__ - 71") + --1— sin (_ - LT 2 8 8x 28 2/rx rc cos x r) - cos (,±_. + ')] Ker(x),-- e-xi'4 — 2x[ 2 i r, ir [ x Kei(x),-- e-x/44 2— sin — i') + x sin ( + Y)1 x 2

12.5 Functions Defined by Transcendental Integrals F x)

The Gamma function F(z) Az) is an analytic function in the whole plane except for simple poles at 0, -1, -2, ... nzn! urn n-->o. z(z+1)(z + 2) ...(z+n)

F (z) =

r(z)= i tz-le-f dt, Re z> 0

0 co (—on F(z)= f tz — le—tdt+ I n=0 n!(Z+0' 1

z# 0, -1, -2, ...

F'(z) ,.._ 1. -. _ y+ i (1 _ 1 ), y ---= 0.5772, Euler's constant. r(z) z n=in z+n) F (z + 1) = zF(z) .1(n)=(n-1)!, n=1, 2, 3, ... F (D = ,s5r 1(-1)n2nfir qn+2 1) _(2n- 1)!!,fir , n=1,2,3,... TI—n+ (2n - 1 )!! 21 2n n n F (z)F (1 -z) = r Gi +z)r G - z) -

sin ICZ

F (2z) = — 22z-1r (z) Jr

COS AZ

q

z+) r(1)=-7 2 287

12.5

Asymptotic behavior (Stirling's formula)

(z) = 127r CZZZ-112 [1+

()] 139 n1 + I arg z 1 12z 288z2 51840z3÷

,

The Beta function B(p, q) r/ 2

1

B(p, q)= r(p)1-(q) = (p + )

tP-1(1-0-idt=2 J Sin2P-i t Cos2q-i t dt (Re p, Re q> 0) 0 0

Numerical table of 1(x) Values of r(x) outside the interval vely.

5_ 2 may be obtained using F(x+1)=xF(x) recursi-

x

r(x)

x

F(x)

x

1-(x)

x

F(x)

1.00 .01 .02 .03 .04

1.00000 0.99433 0.98884 0.98355 0.97844

1.25 .26 .27 .28 .29

0.90640 0.90440 0.90250 0.90072 0.89904

1.50 .51 .52 .53 .54

0.88623 0.88659 0.88704 0.88757 0.88818

1.75 .76 .77 .78 .79

0.91906 0.92137 0.92376 0.92623 0.92877

1.05 .06 .07 .08 .09

0.97350 0.96874 0.96415 0.95973 0.95546

1.30 .31 .32 .33 .34

0.89747 0.89600 0.89464 0.89338 0.89222

1.55 .56 .57 .58 .59

0.88887 0.88964 0.89049 0.89142 0.89243

1.80 .81 .82 .83 .84

0.93138 0.93408 0.93685 0.93969 0.94261

1.10 .11 .12 .13 .14

0.95135 0.94740 0.94359 0.93993 0.93642

1.35 .36 .37 .38 .39

0.89115 0.89018 0.88931 0.88854 0.88785

1.60 .61 .62 .63 .64

0.89352 0.89468 0.89592 0.89724 0.89864

1.85 .86 .87 .88 .89

0.94561 0.94869 0.95184 0.95507 0.95838

1.15 .16 .17 .18 .19

0.93304 0.92980 0.92670 0.92373 0.92089

1.40 .41 .42 .43 .44

0.88726 0.88676 0.88636 0.88604 0.88581

1.65 .66 .67 .68 .69

0.90012 0.90167 0.90330 0.90500 0.90678

1.90 .91 .92 .93 .94

0.96177 0.96523 0.96877 0.97240 0.97610

1.20 .21 .22 .23 .24

0.91817 0.91558 0.91311 0.91075 0.90852

1.45 .46 .47 .48 .49

0.88566 0.88560 0.88563 0.88575 0.88595

1.70 .71 .72 .73 .74

0.90864 0.91057 0.91258 0.91467 0.91683

1.95 .96 .97 .98 .99

0.97988 0.98374 0.98768 0.99171 0.99581

2.00

1.00000

288

12.5

Elliptic integrals Elliptic integrals of the first kind r

F(k, (p) =

de

o ,,11 - k2 sin 20

x

=

dt

(k2 < 1, x= sin (p)

0 ,16-1-2)(1_k2t2)

Elliptic integrals of the second kind (13' E(k, co) =I , 11 - k2 sin2 0 de= f

0

0

1 - k2t2 dt (k2 0)

p-xt 1 2. En(4= 5=dt (x> 0, n=0, 1,2, ...) En+i(x)-= - [e-x-xEn(x)], n= 1, 2, ... n 1 tn x - xn 3. Ei(x)= J et dt= 7+ In x + E — (x > 0, Cauchy P.V.), 7= Euler's constant t n =1 nn! x

4. li(x). s dt =Ei(lnx), (x> 1, Cauchy P.V.) In t

.Y.

-4 Asymptotic behaviour (x --> .0): e-x En(x)- — x

Ei(x) --.e'v x

nx ii(x)-- ix

Error function 5. erf(x)= 2, Je x -t2dt = 2 ei (-1)n X2n+ 1 Cf. table, sec. 17.8. jirn =0 n!(2n + 1) ,Pr 0 6. erfc(x)= 1 -erf(x) 291

12.5

Sine and cosine integrals smt . t

0 00

8. Ci(x)=—

c, 0)

Fresnel integrals

9.C(x)= cos 0

2

771

2

dt.

(-1)n(g/2)2n 4n , X (2n)!(4n+1) n=0

.„.t 2 Hyl(g/2)2n +1 4+3 10.S(x)= sinr=dt = E n =0(2n+1)! (4n +3) x 2 0 YÄ o.

292

12.5 Numerical tables of Exponential, Sine, Cosine and Fresnel Integrals x

E1(x)

.

Ei(x)

Si(x)

Ci(x)

C(x)

S(x)

--

0.000000

--,

0.00000

0.02 0.04 0.06 0.08

3.354708 2.681264 2.295307 2.026941

- 3.314707 - 2.601257 - 2.175283 - 1.866884

0.020000 0.039996 0.059988 0.079972

- 3.334907 - 2.642060 - 2.237095 - 1.950113

0.02000 0.04000 0.06000 0.08000

0.00000 0.00000 0.00003 0.00011 0.00027

0.10 0.12 0.14 0.16 0.18

1.822924 1.659542 1.524146 1.409187 1.309796

- 1.622813 - 1.419350 - 1.243841 - 1.088731 - 0.949148

0.099944 0.119904 0.139848 0.159773 0.179676

- 1.727868 - 1.546646 - 1.393793 - 1.261759 - 1.145672

0.10000 0.11999 0.13999 0.15997 0.17995

0.00052 0.00090 0.00144 0.00214 0.00305

0.20 0.22 0.24 0.26 0.28

1.222651 1.145380 1.076235 1.013889 0.957308

- 0.821761 - 0.704195 - 0.594697 - 0.491932 - 0.394863

0.199556 0.219409 0.239233 0.259026 0.278783

- 1.042206 - 0.948988 - 0.864266 - 0.786710 - 0.715286

0.19992 0.21987 0.23980 0.25971 0.27958

0.00419 0.00557 0.00723 0.00920 0.01148

0.30 0.32 0.34 0.36 0.38

0.905677 0.858335 0.814746 0.774462 0.737112

- 0.302669 - 0.214683 - 0.130363 - 0.049258 + 0.029011

0.298504 0.318185 0.337824 0.357418 0.376965

- 0.649173 - 0.587710 - 0.530355 - 0.476661 - 0.426252

0.29940 0.31917 0.33888 0.35851 0.37805

0.01412 0.01713 0.02053 0.02436 0.02863

0.40 0.42 0.44 0.46 0.48

0.702380 0.669997 0.639733 0.611387 0.584784

0.104765 0.178278 0.249787 0.319497 0.387589

0.396461 0.415906 0.435295 0.454627 0.473898

- 0.378809 - 0.334062 - 0.291776 - 0.251749 - 0.213803

0.39748 0.41679 0.43595 0.45494 0.47375

0.03336 0.03858 0.04431 0.05056 0.05737

0.50 0.52 0.54 0.56 0.58

0.559774 0.536220 0.514004 0.493020 0.473173

0.454220 0.519531 0.583646 0.646677 0.708726

0.493107 0.512252 0.531328 0.550335 0.569269

- 0.177784 - 0.143554 - 0.110990 - 0.079986 - 0.050441

0.49234 0.51070 0.52878 0.54656 0.56401

0.06473 0.07268 0.08122 0.09037 0.10014

0.60 0.62 0.64 0.66 0.68

0.454380 0.436562 0.419652 0.403586 0.388309

0.769881 0.830226 0.889836 0.948778 1.007116

0.588129 0.606911 0.625614 0.644235 0.662772

- 0.022271 + 0.004606 + 0.030260 + 0.054758 + 0.078158

0.58110 0.59777 0.61401 0.62976 0.64499

0.11054 0.12158 0.13325 0.14557 0.15854

0.70 0.72 0.74 0.76 0.78

0.373769 0.359918 0.346713 0.334115 0.322088

1.064907 1.122205 1.179058 1.235513 1.291613

0.681222 0.699584 0.717854 0.736031 0.754112

0.100515 0.121879 0.142296 0.161810 0.180458

0.65965 0.67370 0.68709 0.69978 0.71171

0.17214 0.18637 0.20122 0.21668 0.23273

0.80 0.82 0.84 0.86 0.88 0.90

0.310597 0.299611 0.289103 0.279045 0.269413 0.260184

1.347397 1.402902 1.458164 1.513216 1.568089 1.622812

0.772096 0.789979 0.807761 0.825438 0.843009 0.860471

0.198279 0.215305 0.231568 0.247098 0.261923 0.276068

0.72284 0.73313 0.74252 0.75096 0.75841 0.76482

0.24934 0.26649 0.28415 0.30228 0.32084 0.33978

0.00

293

12.5

Si(x)

Ci(x)

x

E1(x)

Ei(x)

1.00 1.05 1.10 1.15 1.20

0.219384 0.201873 0.185991 0.171555 0.158408

1.895118 2.031087 2.167378 2.304288 2.442092

0.946083 0.987775 1.028685 1.068785 1.108047

0.337404 0.362737 0.384873 0.404045 0.420459

0.77989 0.77591 0.76381 0.74356 0.71544

0.43826 0.48805 0.53650 0.58214 0.62340

1.25 1.30 1.35 1.40 1.45

0.146413 0.135451 0.125417 0.116219 0.107777

2.581048 2.721399 2.863377 3.007207 3.153106

1.146446 1.183958 1.220559 1.256227 1.290941

0.434301 0.445739 0.454927 0.462007 0.467109

0.68009 0.63855 0.59227 0.54310 0.49326

0.65866 0.68633 0.70501 0.71353 0.71111

1.50 1.55 1.60 1.65 1.70

0.100020 0.092882 0.086308 0.080248 0.074655

3.301285 3.451955 3.605320 3.761588 3.920963

1.324684 1.357435 1.389180 1.419904 1.449592

0.470356 0.471862 0.471733 0.470070 0.466968

0.44526 0.40177 0.36546 0.33880 0.32383

0.69750 0.67308 0.63889 0.59675 0.54920

1.75 1.80 1.85 1.90 1.95

0.069489 0.064713 0.060295 0.056204 0.052414

4.083654 4.249868 4.419816 4.593714 4.771779

1.478233 1.505817 1.532333 1.557775 1.582137

0.462520 0.456811 0.449925 0.441940 0.432934

0.32193 0.33363 0.35838 0.39447 0.43906

0.49938 0.45094 0.40769 0.37335 0.35114

Si(x)

Ci(x)

C(x)

S(x)

X

XexEi(X)

xe-xEi(x)

C(x)

S(x)

2.0 2.1 2.2 2.3 2.4

0.722657 0.730792 0.738431 0.745622 0.752405

1.340965 1.371487 1.397422 1.419172 1.437118

1.605413 1.648699 1.687625 1.722207 1.752486

0.422981 0.400512 0.375075 0.347176 0.317292

0.48825 0.58156 0.63629 0.62656 0.55496

0.34342 0.37427 0.45570 0.55315 0.61969

2.5 2.6 2.7 2.8 2.9

0.758815 0.764883 0.770637 0.776102 0.781300

1.451625 1.463033 1.471662 1.477808 1.481746

1.778520 1.800394 1.818212 1.832097 1.842190

0.285871 0.253337 0.220085 0.186488 0.152895

0.45741 0.38894 0.39249 0.46749 0.56238

0.61918 0.54999 0.45292 0.39153 0.41014

3.0 3.1 3.2 3.3 3.4

0.786251 0.790973 0.795481 0.799791 0.803916

1.483729 1.483990 1.482740 1.480174 1.476469

1.848653 1.851659 1.851401 1.848081 1.841914

+ 0.119630 + 0.086992 + 0.055257 + 0.024678 - 0.004518

0.60572 0.56159 0.46632 0.40569 0.43849

0.49631 0.58182 0.59335 0.51929 0.42965

3.5 3.6 3.7 3.8 3.9

0.807868 0.811657 0.815294 0.818789 0.822149

1.471782 1.466260 1.460030 1.453211 1.445906

1.833125 1.821948 1.808622 1.793390 1.776501

- 0.032129 - 0.057974 - 0.081901 - 0.103778 - 0.123499

0.53257 0.58795 0.54195 0.44809 0.42233

0.41525 0.49231 0.57498 0.56562 0.47520

4.0 4.1 4.2 4.3 4.4

0.825383 0.828497 0.831499 0.834394 0.837188

1.438208 1.430201 1.421957 1.413542 1.405012

1.758203 1.738744 1.718369 1.697320 1.675834

- 0.140982 - 0.156165 - 0.169013 - 0.179510 - 0.187660

0.49843 0.57370 0.54172 0.44944 0.43833

0.42052 0.47580 0.56320 0.55400 0.46227

4.5 4.6 4.7 4.8 4.9

0.839887 0.842496 0.845018 0.847459 0.849822

1.396419 1.387805 1.379209 1.370663 1.362196

1.654140 1.632460 1.611005 1.589975 1.569559

- 0.193491 - 0.197047 - 0.198391 - 0.197604 - 0.194780

0.52603 0.56724 0.49143 0.43380 0.50016

0.43427 0.51619 0.56715 0.49675 0.43507

294

12.5

1.353831 1.345589 1.337487 1.329538 1.321754

1.549931 1.531253 1.513671 1.497315 1.482300

- 0.190030 - 0.183476 - 0.175254 - 0.165506 - 0.154386

5.5 5.6 5.7 5.8 5.9

0.862562 0.864473 0.866331 0.868138 0.869895

1.314144 1.306714 1.299471 1.292416 1.285554

1.468724 1.456668 1.446198 1.437359 1.430184

- 0.142053 - 0.128672 - 0.114411 - 0.099441 - 0.083933

6.0 6.1 6.2 6.3 6.4

0.871606 0.873271 0.874892 0.876472 0.878012

1.278884 1.272406 1.266120 1.260024 1.254115

1.424688 1.420867 1.418707 1.418174 1.419223

- 0.068057 - 0.051983 - 0.035873 - 0.019888 - 0.004181

6.5 6.6 6.7 6.8 6.9

0.879513 0.880977 0.882405 0.883799 0.885159

1.248391 1.242848 1.237482 1.232290 1.227267

1.421794 1.425816 1.431205 1.437868 1.445702

+ 0.011102 + 0.025823 + 0.039855 + 0.053081 + 0.065392

7.0 7.1 7.2 7.3 7.4

0.886488 0.887785 0.889053 0.890292 0.891503

1.222408 1.217709 1.213166 1.208774 1.204527

1.454597 1.464433 1.475089 1.486436 1.498345

0.076695 0.086907 0.095957 0.103789 0.110358

7.5 7.6 7.7 7.8 7.9

0.892688 0.893846 0.894980 0.896089 0.897174

1.200421 1.196452 1.192615 1.188905 1.185317

1.510682 1.523314 1.536109 1.548937 1.561671

0.115633 0.119598 0.122246 0.123586 0.123638

8.0 8.1 8.2 8.3 8.4

0.898237 0.899278 0.900297 0.901296 0.902275

1.181848 1.178493 1.175247 1.172106 1.169068

1.574187 1.586367 1.598099 1.609278 1.619807

0.122434 0.120017 0.116440 0.111767 0.106071

8.5 8.6 8.7 8.8 8.9

0.903234 0.904174 0.905096 0.906000 0.906887

1.166127 1.163279 1.160522 1.157852 1.155266

1.629597 1.638570 1.646655 1.653792 1.659934

0.099431 0.091936 0.083679 0.074760 0.065280

9.0 9.1 9.2 9.3 9.4

0.907758 0.908611 0.909450 0.910272 0.911080

1.152759 1.150330 1.147974 1.145690 1.143474

1.665040 1.669084 1.672049 1.673930 1.674729

0.055348 0.045069 0.034555 0.023913 0.013252

9.5 9.6 9.7 9.8 9.9

0.911873 0.912652 0.913417 0.914169 0.914907

1.141323 1.139236 1.137210 1.135241 1.133329

1.674463 1.673157 1.670845 1.667570 1.663384

+ 0.002678 - 0.007707 - 0.017804 - 0.027519 - 0.036764

295

S(x)

C(x)

2 .0 ' .0 7,

0.852111 0.854330 0.856481 0.858568 0.860594

Ci(x)

LA

5.0 5.1 5.2 5.3 5.4

Si(x)

ti o z 0, ›.t -,•,', v ,-.>

xe-xEi(x)

2 „.„.2 9 (1-• -lxs>in = 2 - Cr,.1, - 5 7)cos irx +E(x), where l e(x)I 0 5. 0' (t)= 8(t),

dt

5(t)=" h-0 sgn(t) = 250

1 h2

f(t) continuous at t = a: 6. f(t)3(t a) = f(a)(5(t - a) 7.

f f(t)6(t - a)dt = f(a)

h

j

(t - a)dt = (t - a) + C

1(t) continuous at t= a: 8. f(t)(5' (t - a) = f(a)8' (t a) - f'(a)8(t a) 9.

1 h2

j f(05' (t - a)dt =- f f'(t)8(t - a)dt = - f '(a)

S(x)=" lim Sh'(x)" h—>0

-60

f (n)(t) continuous at t = a: 10. f f(t)(P)(t - a)dt = (- l)nf(n)(a) fw gnkt _ a) =

n

k= 0

(_ 1)k i (k)(a)gn-k)(t _ a) (k ).

11. f * b"('')(t)= f (n)(t), n =0, 1, 2, ... 12. elf(t)= 0

"

(t a)n g'')(t a). (-1)nn!8(t - a)

f(t)= CO(t) + Cr:VW + +Cn_ 6(n-1), 297

n=1, 2, 3, ...

't

12.7

Example

Let f(t) =

I. Calculate the (distributional) third derivative f'"(t). Solution: f (t) - sgnt e-It I f"(t)= -28 (t)e- fr I + (sgnt)2e-It I = -23(t)+e-It I f (t) = - '(t)-sgnt e-It I

12.7 Functional Analysis Spaces 1. 2. 3. 4. 5.

Linear space or vector space. Metric space: Set with a distance function. Normed space: Linear space with a norm (distance). Banach space: Complete normed space. Hilbert space: Banach space with an inner product.

1. Linear spaces. See sec. 4.7. 2. Metric spaces A set M of points u, v, ... is a metric space if there is a distance function d(u v) on M satisfying for all u, v, we M: (i) d(u, v)>_0 [=0 u = v] (ii) d(v, u)= d(u, v) (iii) d(u, v) d(u, w)+ d(w, v) Convergence 1. un -> u as n -> 00 if d(u„, u) --> 0 as n ---> .0 (u, u„E M) 2. A sequence {u„}7 in M is a Cauchy sequence if lim d(uni , un)= 0, i.e. for any e> 0 there exists an N that d(u n, un) N. m' n-->co 3. M is complete if every Cauchy sequence in M has a limit in M. Topology

Let S be a subset of M. 1. ape S is an interior point of S if there exists a 3> 0 such that 2. 3.

4. 5. 6. 7. 8. 9.

uE S for all uE M satisfying d(u, up) < b. uoe M is an accumulation point of S if there is a sequence {un}, tin e S, 14, uo with lim un = u0. S is open if every point of S is an interior point. The closure S (of S) is formed by adding to S all its accumulation points. S is closed if S = S, or equivalently, if the complement M\ S is open. S is compact if any sequence in S contains a subsequence that converges in S. S is relatively compact if any sequence in S contains a subsequence that converges in M. S is dense in M if every point of M is the limit of some sequence in S. Or, S is dense in M, if for any uE M and any E> 0 there is a VE S such that d(u, v)< e. M is separable if it contains a countable dense subset. t 1. S is convex if, for any u, ve S, to + (1- t)ve S for alla t such that

298

12.7 3. Normed spaces A linear space L of elements u, v, . . . is a normed space if there is a norm 11 • 11 on L satisfying for all u, vE L: [=( .4=> u= 0] Ilull (a scalar) 1°211=lal (iii) With the distance d(u, v) = 11u— v11 the space is a metric space. 4. Banach spaces A complete normed space is called a Banach space. Examples of Banach spaces 1.Every finite-dimensional normed space (e.g. R"). 2. C([a, b]) = { continuous functions on [a, b] with norm 11f11 a m b If(x)1 00, with ordinary Lp-norm. 3. The Lp(Q)-spaces, 1 Completion B is a completion of a metric space M, if B is the smallest complete space of which M is a dense subset. (i) Every metric space M has a completion. 4. The set of polynomials on [a, b] is dense in C[a, b] (sup-norm). 5. C(0), Co(0), C°°(S2) and CI'1 (0) are dense in Lp(S2), 1513< 00. 6. The set of trigonometric polynomials with period T is dense in Lp[O, T], 1

uo (ii) bounded if there exists a constant C such that all UE B The smallest bound C is called the norm of T, denoted 11 T11, i.e. Tull IITII = supll u.B111411 o 1174 s IITII Ilu II

=

sup 11Tull Ilull= 111'1 7'2 11

111'1 II 111'211 IlaTII=Ial IITII

11 Ti+ T2115 IlTI 11+117'211 (With this norm the space consisting of all bounded linear operators on B is itself a Banach space.) T is continuous at u0 T is continuous at 0 .=> T is bounded

A linear operator T is compact (or completely continous) if for every bounded sequence {un} the sequence Tun } contains a convergent subsequence. 300

12.7

1. If T, Tn compact and S bounded linear operators, then (i) T is bounded, (ii) T1+ T2 , TS and ST are compact, (iii) Tn -> A, i.e. 11T„-A 11 -> 0 A is compact. 2. If T is linear and the range of T is finite dimensional, then T is compact. Examples 6. If B is infinite dimensional, then the identity operator /u = u is not compact. 7. Let K(x, y) be an integral operator on [a, b], i.e. b

Kf (x) = f K(x, y)f (y)dy. a

Then (i) K(x, y) is continous in the square a yb K is compact on C [a, b] and 11K11.maxIK(x, y)1 (ii) K(x, y)E L2 in the square a x, yb K is compact on L2 [a, b] and 11K11 q1K(x, y)11 L2 Inverse operator T-1: v = Tu u= T-1v; T7-1 =T-1T=I. 7-1 exists if (i) T is surjective and 11 Tull i clI u ll, c> O. Then 11T-111 1/c. (ii) T= I - K, IIKIl< 1. Then T-1=(/-K)-1=/+K+K2+... and 11T-111

1

1 - 11K11

Symmetric operator Let T: H --> H be a linear operator. The adjoint operator T* is defined by (Tu l v) = (u l T*v) T**= T (TS)* = S*T* (T*)-1= (7-1)* liT*11=11711 T is self-adjoint or symmetric (hermitian in the complex case) if PK= T, i.e. (Tu l v) = (ulTv). Example 8. The integral operator in example 7 above is symmetric if K(x, y) = K(y, x). Projection Let H1 be a subspace of H and Pu the orthogonal projection of u onto H1, i.e. (u - Pu Iv) = 0, all ve P is an orthogonal projection P2= P and P*= P

H2

u

>' Pu • PUEH1, IIPII=1

Spectrum Let T be a linear operator on H. The number A, is an eigenvalue of T and u the corresponding eigenvector if Tu = Au, u# 0. 1.Eigenspace HA.= {ue H: Tu= Au} 2. T symmetric = all eigenvalues are real and eigenvectors corresponding to different eigenvalues are orthogonal. (14 1H4 Al *A.2). 3. T bounded IAl ~llTll. 4. T symmetric and compact (i) His finite dimensional if /1.#0 (ii) 11 TI1 or -11 TI1 is an eigenvalue. 301

12.7

A spectral theorem. (Hilbert) Assume that T# 0 is a linear, symmetric and compact operator on H. Let > 0 be the non-zero eigenvalues of T and let { ed be a corresponding 1 At I > 1 A2 I >141 orthonormal family of eigenvectors. Then (ulei )ei +up, where tioE Ho, i.e. Tuo= 0 (i) u

Tu=F 2,i(ulei )ei

The Fredholm alternative in Banach spaces Let K be a compact (integral) operator on B. Then the equation

f— Kf= g has a unique solution fe B for any ge B if the equation f— Kf= 0 admits only the trivial solution f= O.

The Fredholm alternative in Hilbert spaces Let K be a compact (integral) operator on H. Then the equation

f—Kf= g has a solution fE H if and only if g is orthogonal to every solution of the equation

K*f = 0 The solution is unique if this equation admits only the trivial solution f= O.

Fixed point theorems of continuous mappings Let U be a subset of a Banach space. An operator T: U —> U is a contraction mapping if there exists a positive number a< 1 such that 11 Tu —Tv115_allu—vIlfor all U, VE U

Fixed point theorems 1. If U is a subset of a Banach space and if T: U —> U is a contraction mapping, then (i) there exists exactly one UE U such that Tu = u (i.e. u is a fixed point of T) (ii) For any u0E U, the sequence { un } defined by un+i = Tun converges to u. 2. Every contionuous mapping of a closed and bounded (i.e. compact) convex set in Rn into itself has a fixed point. (Brouwer) 3. (i) If S is a convex compact subset of a normed space, then every continuous mapping of S into itself has a fixed point. (ii) If S is a convex closed subset of a normed space, and if Rc S is relatively compact, then every continuous mapping of S into R has a fixed point. (Schauder)

302

12.8

Linear functionals A linear operator F: B (or H) —> R (or C) is called a linear functional (or linear form).

Theorems 1.(Hahn-Banach): Let F1 be a bounded linear functional on a linear subspace B1 of B. Then F1 can be extended to a linear functional F on B such that 11F11 = 11 F111. 2. (Riesz): Every bounded linear functional F on H has the form Fu=(u1v) for a unique vE H, and VII=

Bilinear forms An operator A(u, v): HxH —> C (or R) is a bilinear form if

A(alui + cr.,2 u2, v)= aiA(ui, v)+ a.2A(u2, v),

1A(u, fl1v1 + B2v2) = )3 1A(u, v1) +ff 2A(u, v2). The form is bounded if IA(u, v) 1 < C11 u 11 • 11 v II. The norm IIA 11 =

ll i Ilull = Ilvsip

IA(u, v)1

A(u, v) is elliptic if IA(u, u)I cll u 112, c >O. Theorem (I nr-Milgram) (i) Every bounded bilinear form A on H has the form A(u, v) = (Tulv) for a unique linear operator T, and IIA 11 = 11 T11(11) Let A be a bounded and elliptic bilinear form on a linear space V and let F be a bounded linear form on V Then the variational problem A(u, v) = F(v), all VE V has a unique solution UE V.

12.8 Lebesgue Integrals Lebesgue measure Set measure Let S be a subset of an interval I=[a, b] and let S'=1\S. Set m(I) = b —a. Exterior Lebesgue measure me(S)=inf E m(In), Sc U In , In interval. n=1

1

Interior Lebesgue measure mi(S)= (b — a) — me(S'). If me(S)=mi(S), then the set S is Lebesgue measurable and the Lebesgue measure is

m(S) = me(S) = mi(S) 303

12.8 An unbounded set S is measurable if Sfl (- c, c) is measurable for any c> 0 and ni(s). lim in[sn(-c,c)] C->00

Sets of measure zero

A countable set S. {p,,}7 e.g. Q, has measure zero, because, for any to an interval of length e • 2—n it follows

E> 0,

letting pn belong

00

me(s)< L. E • 2-n= E n= 1 On the other hand, a set of measure zero does not have to be countable. Measurable functions

A function f(x) is measurable if the set {x: f (x).>_ c} is measurable for every c. A property which holds for every point (of a set) with the exception of a set of measure zero is said to hold almost everywhere (a.e.).

The Lebesgue integral 1. Assume that (i) f (x) 0 and measurable (ii) S measurable. Set S(y)= { x: f (x)y}. Then m(S(y)) is decreasing and hence Riemann integrable: 00

f (x)dx= f m(S(y))dy

Definition:

s

o

If this value is finite, then f (x) is summable over S. 2. f (x) arbitrary sign. Set fl,(x)= max[f (x), 0] 0 and f_(x)= max[—f(x), 0] 0. Then f (x) = f+(x)—f_(x) and f (x)dx = f f+(x)dx — f _(x)dx

s

s

s

Fubini's theorem Iff (x, y) is summable over R2, then f ff(x, y)drdy = f dx f f (x, y)dy. R R

R2

(The formulation implies that the right hand side makes sense.) Limit theorems Fatou's lemma (i) fn(x) > 0 summable over Q (ii) fn(x) —> f (x) a.e.

If

(iii)

ffn(x)dx

then f (x) is summable and ff(x)dx C Sz 304

12.8 Lebesgue's theorem

If (i) f,i(x), g(x) are summable over K2 (ii) g(x) (iii) f,i(x)—> f(x) a.e.,

then f(x) is summable and f f (x)dx = li m f„(x)clx n -> Approximation

If f(x) is summable, then for any

E> 0

there exists a piece-wise constant function (with a

finite number of steps) (p(x) such that f If (x)- 0(x)Idx < E.

Some function spaces in R" Notation. a. Below 12c R" denotes a domain (open and connected). All functions are complex valued and defined on a subset of le. b. Multi-index a= (ai, ..., an ), I al = + c. Da = D°1e1

+ a„,

a

all i.

, Di = alaxi .

Leibniz' formula: Da (uv) =

(73) DfluDa-i3v, (73) = (s ail)

ip ann)

)3 a d. S21 c c Q means that C21 c SI and Qi is bounded. e. Let u be a function. The support of u is the set supp u = {xE

u(x)# 0} .

f. u has compact support in 52 if supp u c c Q.

Spaces of continuous functions C°(S2)= C(Q)= {0: 0 is continuous in š2} Cm(S2) = {0: Dad is continuous in 0, I al 5 fli} C.'"*(5.2)= {0: Dart) is continuous in LI, all a} Co (0) = {Functions in Cm(S2) with compact support in K2}, 0

00.

Cm (K) = {OE Cm(S2) : Da0 is bounded and uniformly continuous in S2, l alm} CB (S2)= {OE Cm(0): Da0 is bounded in K2, I al m} 1. With the norm 11011 = max sup 1020(x)I, xEi2 Cm (S2) is (i) a Banach space (ii) separable if Q is bounded. 2. Let 02 c c Q1 c cll. Then there exists a OE Co (L2) such that 0(x)= 1 in 1'2 2 and 0(x)= 0 outside 521. 305

12.8

LP-spaces A function f (x) belongs to LP(S2) if

f if (x)IP dx < 00. LP (52) is a Banach space with the norm

iviip=ivilL,(0)=(fv(x)1Pdx)11P A function f(x)EL —(0) if f(x) is measurable over 52 and if there is a constant C such that a.e. The smallest possible value of C, essup If (x)I, is the L°°-norm off (x), i.e. I f (x)I XE

Ilf 1100 = essuj If (i) I Also L—(0) is a Banach space. Furthermore, L2(Q) is a Hilbert space with the inner product (u, v) = f u(x)v(x)dx. Q

Approximation 0, m, and p there exists a constant C such that for 0 5.j 5m —1,

1u Imp — I u lmp,n =

Clu10,p, all uE linn'P(0). 8. Poincare-Friedrich's inequality: If Q is bounded then there exists a constant C= C(Q) such that lu 10,25CI u 11,2, all uEI-101(Q). 9. Sobolev imbedding theorem. Notation. A normed space X with norm 11' Ilx is imbedded in another normed space Y, written X Y, if (i) X is a vector subspace of Y (ii) There exists a constant C such that Clix Ilx, all xEX. Ilx Let Q be "sufficiently regular" (e.g. convex). Then A. If mp WM(51), p q 500. In particular, Cll u Il[n/2]+1,29 [6] = integer part.) C. If mp > n, then Wi +mP(Q) --> Cgi(Q). 307

12.9

12.9 Generalized functions (Distributions) * Test functions. (The class S) 1. A test function is a complex-valued function (p(t) on R=

.0) satisfying:

(i) 9(0 is infinitely differentiable, i.e. TE C°°(R)

(ii) lim tPip(q)(t)= 0 for all integers p, O. Id -4 — The class of all test functions is denoted S. (Another class of test functions is D, consisting of all infinitely differentiable functions, which are zero outside a bounded subset of R. Note that Dc S.) [Example: (p(t)= e-t2 S] 2. A sequence DoE S is a zero sequence if lim max ltPip(7b)l =0 for all p, n.-->0. [ER

[Example: of S

pn(t) =

0

1 + -) - (p(t) is a zero sequence.]

Generalized functions. (The set S') 3. A functional on S is a function f which maps (pc S to a complex number, denoted (f lip) or .f (W)•

4. A generalized function (g.f.) (or temperate distribution) is a continuous linear functional f on S, i.e. = c(.fl(p)+ P(f10, a, e C; (1); vie s. lim (fk n) = 0 for any zero sequence roe S.

(fia(P+

The set of all temperate distributions is denoted S'. (The corresponding functionals on D are called distributions.) 5. f = g (fl(p)=(gl(p) for all TES.

6. The support of (p e S is the smallest closed set outside of which (p(t)= O. 7. Let Ac R be an open set. Then f= g in A if (f IT) = ( gl(p), all (p e S with support in A. 8. The support offE S' is the smallest closed set outside of which f= O. 9. Let f (t) be a piece-wise continuous function such that

J(1+t2)-m If (t)Idt < oo for some integer m. Then (fi(p)=

rf (t)T(t)dt

defines a regular g.f. A non-regular g.f. is called singular. Also for singular g.fs. the notation f (t) (instead off) is used as well as (fl(p)= f (t)(p(t)dt

* Essentially listed from Jan Petersson: Fourieranalys, Chalmers University of Technology, Goteborg, 1994. 308

12.9 10. Dirac's delta function 6(t) is a singular g.f., defined by (SIT) = (P(0) 11.The g.f. f (at + b), a * 0, is defined by 1 -b f (at + b)cp(t)dt= — f(t)(p -7a-)dt lal C"' and for each integer 12. The g.fs.f+ g, cf, /and wf(where ger p such that CI' 00(t) --> 0 as Itl —> are defined by (f+ gfrp.)=(fl(p)+ (Op)

(cfl(P)= c(f19))

(f

(Ivfl(P)=(fiviv)

= (fIFP)

0 there exists an inte-

13. v(t)6(t-a)= yi(a)5(t - a) 14. (t)= 0 f (0= c8(t)

Derivatives 15. The derivative f '(t) is defined by (f'l cp) = - ( f

16. (te)'=wf'+iv'f 17. 8'(t)= S(t) 18.The singular g.fs.

, n=1, 2, 3, ... are defined by

t

(-1 )n 1

_„ = (n - 1)!

Dn ln It'

19. tt-1=1

A Fourier transforms 20. The Fourier transform (P (w) of a test function cp(t) is defined by 43(co)= f cu(t)e-` indt

21. cpe.5.4=> 4G ES Wn is a zero sequence.

22. (pn is a zero sequence

fr( of a g.f. f(t) is defined by 23. The Fourier transform co)

(110 = (flti9) g. 24.fe S' E S', f(t)= 271f (- t), = 25. The laws F3-F11, sec. 13.2, hold for g.fs.

26.h (co =1, (co)=7115(0Convolutions

00

27.f€ S', (pE S f * (p(t)=

- (r)drE Cm(R) CO

28. The convolution h=f * g, f, gE S' is defined if fr:iE S', by n

nn

29. If the convolutions exist: f*g= g*f, f*(g+h)=f*g+f*h, (f*g)'=f' *g=f*g' 30. f * .5(n) = f (n), all fe S'

309

13.1

13 Transforms 13.1 Trigonometric Fourier Series Fourier series of periodic functions T = period of function f (t) S2= — -= basic angular frequency T Cosine - sine - form Orthogonality T

{0, k S2tcosnS2tdt= T,n=k=0 Jcosk TI2, n= k>0 o rT {T12, k=n>0 J sin k.S2t sin n.f2t dt = 0, kin 0

sin IcS2t cos mat dt =0 0 ao f (0 = — +nE 1 (an cos nS2t + bn sin ni2t)* 2 a+T

2 a = - $ f (t) cos nS2t dt (n 0) n T a

2 b=T

a+T

f (t) sin nS2t dt (n 1) a

Special case. Period T=2.7c r-

f(t)=

a0 Z.

1)11) u (a cos nt +13, sin nt) + n1 n

U(kt V) Yntt

'1`u

AAAO 0:

1 an = -

f (t)cosittAlt

bn = f f (t) sin nt dt

11.-7` T12

4

ti ,0

f(t) even

4r = - f (t) cos nOt dt, T0

bn = 0

oa

an = 0,

Sy

174,4

( I Cr1

1 ' erf)U0 If I il ?CO CA.

TI2

f (t) odd

I 10 ,, t l: a n

r b n=1 j f (t) sin nS2t dt T 0

......1, hat orko,N,ir ir,T,.„AS

11, 3 C , l -)

* For bounded and piece-wise differentiable functions, the equality holds at points t where f tinuous. At jumps the Fourier series equals (f(t +)+ fit —))12. '\0\ ,0

,

iht. _221.10) Ch

I

310 r

CV/ 1

is con-

13.1 Approximation in mean (cf. sec. 12.1) a0 n Sn(t)= — n + I (ak cos kf2t+ bk sin WO k= 1 Mn= 7J LT (t) - sn(t)12 dt (T=2,7r/f2) 0 Qn minimal ak, bk are Fourier coefficients. T

2 n min Qn = 2- J f 2(t)dt- -k=1(ak2 bk) 2 0 Amplitude - phase form (f (t) real, S2= 27r/ T) 00

/

A, cos(nS2t+ an), An 0 for n=1 Calculation of An, an, see below. f(t)=Ao+

Complex form (0= 27c/T) Orthogonality T

0

etkate-inprdt= { 7; k= n 0, 1c# n l at.T f( _.n _

00

cneinQt, .ft0= 11=-0.

Cn =

t)e

melt

a

Relations between Fourier coefficients sin, cos form - amplitude, phase form (f(t) real) A0 = ap, An = + 2 ao =2A0, an = An cos an sin, cos form - complex form 1 cn = - (an- ibn), co= 21- ao, 2 an =cn +c_ n, ao = 2c0, f(t) real: a0 = 2co,

an= arg(an -ibn)= bn = - An sin an

{-arctan(bn /a„), an> 0 , n>1 aretan(bn /an), a„1

1 c, = (an+ ibn) , rz• 1 bn =i(cn-c„) , a„.2Re cn, bn =-2Im cn

c_ n= c,„ ,n >_ 1

complex form - amplitude, phase form (f(t) real) an =arg cn ,n>1 An=21cni, Ao = co, Aneian= 2cn, 1 = - Ane_ • co = Ao, cn = 1- Aneia^ = - An(cos an + i sin an), c_n 2 2 2 311

,

1

13.1 Parseval's identities [Below, a„, bn, An, an, cn refer to f (t),

an , cn , refer to g(t)]

a+T

1 ao 1 v7 I J f 2(t)dt =(a 2 + b2 ). A2 o 2 n1 A2 4 +- n= n a a+T

T a

1 1f(t)g(t)dt = apaiS (ana;,+ bab;)= 4 +n t

a+T

T a 1

=

if(0 I2dt= E I cn I2

1

1 + - 1, A A;, cos(ota - a;,) ° 2 n =1 a

a+T

r

f(t) g(t) dt =

CnC'

a

a+T - cn c' einot J f (t - r)g(r)dz = E

i a

"

27c T

(Periodic convolution)

Periodic solutions of Differential Equations Problem. Find a periodic solution of the differential equation y"(t)+ ay'(t) + by(t) = f(t), where a and b are constants, and f(t) is periodic with period T. 2r Solution. Let f(t) have the Fourier series expansion f(t)= CneinQt, = — . T n=— =, Substitute y(t) =

I

ynein°' into the differential equation. Identifying coefficients will give

a particular T-periodic solution with yn =

-n 2S22 +ian12+b cn

Sine and cosine series Orthogonality

• knx ntrx sin — dx f sin — L L

0

{ LI2, k= n> 0 0, k#n

L0, n#k kn-x nrx 5 cos— cos—dx = L, n = k =0 0 LI2, n = k> 0 f(x) given in the interval (0, L):

f(x)= E b sin n.1 n

L

x

2 r nnx bn = - f (x) sin — dx L0

ao ntrx f(x)= —+ a, cos — , L 2 n=1

2 nnx an = - f f (x) cos— L0 L

312

13.1

Special Fourier series Fourier series: f(t) =

Function f(t) (1)

T=2L al

2h - sinnga

ah+ — I tt n=1

ih

i

1, -L -aL

al L

(3) T=2L 1 -L -aL

(4)

tint 2h Z 1-cosnira sin— n L g n=1

— t

ah —+ 2

A

al

1h aL L

2h,-, - Orsinnga 1 - cosnira)cos ngt n ant ) L

+ 7, 7 r ,,L1

T=2 L a< i -L-aL 0_,

,

ngt cos—, L

't

(2) T=2L

cie'l h -L -aL i 1aL L h

n

th aL L

2h \-, ( Kcosnna+ sinnira)sinnirt 1-' 1\ n 762 n=1 ant ) L t

Fourier series for further rectangular and triangular periodic functions can be obtained by combining (1)-(4). (5)

T-2L

•.

_

(6)

h (L-t) L L

h (L-t) h -

T=2 L

, _ (7)

7r n= i n

L

1 (2n -Dirt -Dirt t -h 4h `.* cos + y, L 2 ir2 n = 1 (2n -1)2

T-2 L t(L-t) '. AlkL -W

1 (2n -1)nt 8L2 sin n3 n=1 (2n -1)3

T=2 L ..

- 1 L2 - L2 I, —

L

(8)

2h - 1 flirt — I - sin

t(L-t)

6

_ 313

7r2 n=1 n2

cos

2n la L

13.1 (9)

T-2L

L

1 h h zt 2h cos2n7rt - + - sin— -- I L L ir n= 1 4n2- 1 z 2

Tu`

2h 4h c's - — y,

la

h sin —

_

(10)

T=2 L

h sin —

L

7

1

,

n=1

cos 2nzt L

-

(11)

8h -,

2ngt n sin — L g n= 1 4n2 - 1 L

/st hcos—

T=2L

L -,

-L

(12) f(t)=t, -L0, n=1,2,3,...)

co2 +a2

2a

F41b.

(a > 0)

( a —icor

1 „alt!

(a>0)

(a > 0)

t2 + a2 2

Cal' lsgn t

(02+ a2

(a > 0)

irre-colsgaa,

t2+a 2 2a

kco+ b (a>0, c real) (co-c)2 + a2

cal tl+ict(iak sgnt+b+ kc)

1 caltl+ielia2kt +(b+ kc)altl+b+ kc] 4a3

F45. eint

kco + b (a> 0, c real) [(o) - c)2 + a2]2 27r5(co- (2)

F46. cos S2t

746(co+ s2)+ 5(0)—(2)]

F47. sin

in[8(co+ 0) - 8(w-

1 F48. - — sinat sgnt 2a

1 (.02_ a2

F49.

CO

2 cosat sgnt

(02_ a 2 2 sina co

F50. 0(t+a)- 9(t -a)= 1 1, a

319

13.2 F(co)

f(t)

1, 0 a jl0,

F60

1 cosh t

F61.

1 sinht

F62.

sinh at sinhbt

F63. F64

4

DnAca

rJo(aco) - ia irJi(a co) tt

trot cosh 2 7CCO

- intanh — 2

resin (a 7r1b) b cosh (con b)+ b cos (a irl b)

(0 < a 0)

13.2

Cosine and sine transforms 00

f(x)= n TFO)cos fix dP

Fjp)= f f(x)cos fix ch

o

o . F(P)= f f(x) sin fix&

f(x) = rFs(p)sinfix 0

o Plancherel's formulas If(x) 12dx =3' r Fc(pi2ds 0

Ir 0

,

riFs(16)120 0

Relations between Fourier transforms

If F(/3) is the Fourier transform of f (x), -.0 (a >0)

sin a

Fc2.

(a >0)

a a 2 + #2

Fd.

(a >0)

1 ,f_r e-/32/4a

_ air (00 {1, xa -ax

F.52.

e

F53.

xe

(a >0)

-ax2

cos axe

Ffi.

sin axe

a2 + )32 1-r — e -/32/4a

(a>0)

F54. xa-1 F.55.

1 - cos a 13

(a >0)

0 Below 8(n)=

0, n < 0

x(n), n>_0

X(z) x(n)z-n n=0 aX(z) + bY(z)

zl. x(n) z2. ax(n) + by(n) k-1 z3. x(n - k)0(n - k)= x f °(n -k), n> k z4. x(n + k) (k > 0)

z

-k X(z)

zkX(z)-z kx(0)-zx(k- 1)

z5. anx(n)

Xlal

z6. (- 1)k (n -1)(n- 2) ... (n - k)x(n - k)0(n - k -1)

k _ d k X(z) dz — Zr(Z)

z7. nx(n) z8.

k =0

x(n - k)y(k) =

k =0

x(k)y(n - k)

X(z)Y(z)

00

* Alternatively for sequences (x(n))*s the z-transform is defined by X(z) = E x(n)z'. In n =.0 —00

that case, some of the above laws are replaced by, z3*, z4*.

ZkX(Z)

x(n + k) (k arbitrary integer) 00

z8*.

x(n) * y(n)= E x(n - k)y(k) k = —=

X(z)Y(z)

327

13.4

x(n), n 0

X(z)

r 1, n=k z9.

1

8k(n)= 0, n k

zk

z10.

an

zu 1.

0, On 0, r = b b (p= arctan - if a >0, (p= ir-F arctan - if a 0) rt-T), t> T 0

, t 0) f(- ax) (a > 0) f '(x)

-f(y) aF(y)+ bG(y) F(y + a) F(ay) - F(- ay) F'(y)

Hi7 .

xf(x)

f(x)dx YAY)+ I CO

Hi8.

Hi9 . Hi10.

{

1, a0) sin ax (a> 0) x

Hil7 .

= bb y 1 - In a-y

Ir

(5(x- a) 1

Hill.

- (F * G)(y)

(f * g)(x) = J f(t)g(x -t)dt

y+a

(Re a> 0)

oe(y 2

oc2)

a

(Re a> 0)

y2+ a2 cos ay - sin ay cos ay - 1 Y

343

13.8

13.8 Wavelets* Notation 1. H(z) =

I hkz-k (two-sided z-transform)

kEZ

2. A Hilbert space V1 = V0 W0 if every element in V1 can be written uniquely as the sum of an element in V0 and an element in Wo, where V0 and W0 are closed subspaces of V1. 3. A set as f =

k e Z1 is a Riesz basis of L2(R) if every f E L2(R) can be written uniquely aktpk and there are positive constants A and B such that

(pk : kEZ

Allf 112

E kez

112-

If A = B = 1 the basis is orthonormal.

Multiresolution Analysis A multiresolution analysis (MRA) of L2(R) is a sequence of closed subspaces V1 of L2(R), jE Z, with the following properties: 1. Vj c Vi+i , 2. f (x)EVi f (2x) E Vi + , 3. f(x)EV0 f (x + 1)E Vo, 4. U Vj is dense in L2(R) and fl Vi = { 1

5. There exists a scaling function tpe V0 such that the set {9)(x — k) : k EZ} is a Riesz basis of V0.

1. Dilation and translation: tpi,k(x)= 2112T(2- x — k). 2. The set { tpj k : k€ Z} is a Riesz basis of 3. Since ve V0c V1 the scaling function satisfies the dilation equation (1)

his)(2x — k) or

T(x) = 2 k

ep( o. H(co12)Co(co12),

for some sequence (hk)E f2(z), where H(w) = k

4. The scaling function is uniquely defined bythe dilation equation (1) and the normalization J tp(x)dx = 1. It follows that T(0) = 1 and H(0) = 1. Example: The so called Haar MRA is defined by the subspace 1/0 = IfE L2(R) : f(x) constant on [k,k + 1), kE Z}, and the scaling function {1 1:1.x a (m -1 I )!c/z)

z=a

{(z-a)mf(z)}.

Calculation of definite integrals 27t

1.

R z- z1 z + z-iyz

R(sin (9, cos 61)(119=[z= e"9]=

2i ' 2 iz o lzii Im z>-0 except for a finite number of points al, 2. If f(z) is analytic in the upper half-plane ..., an above the real axis, and if I zf(z)I -> 0 as z -> 00, then

J

Ax)dx = 27ri E Res f(z)

ic=lz = ak

3. If CR: z=Reie, 07.c and if if(z)I M • /2-k, (M, k> 0 constants), then J .flz)e imzdz -› 0 as R -> 00. CR

c••= cos x Example. I= 5 dx=Re J

Set f(z) =

eix

_ e., x2+a2 dx'

__ x2 + a2

a>0.

eizz-ia : Res f(z) = e-a lim 2 2 = [l'Hospital's rule] z ->iaz +a z2 + a2 z = ia

IZIe-Y < 1 e- a z1 -> 0 as z -› 0.0. = Ca lim - = — . Furthermore, lzf(z)I= z -> ia 2z 2ia lz2 + al 1z2 + al . Ca ne-a Thus, /= Re (2itt • =

2ia)

a

353

14.3

Calculation of sum of infinite series Assume that If(z)I constant lz I ', a >1 as z -> co. 00

1.

f(n)=- [sum of residues of itf(z) cot nz at all poles of f(z)].

2.

(-1)nf(n)=- [sum of residues of Irfiz) at all poles off(z)]. sin gz •

Example.

ircotrcz ascot gz) g (z _,,a z2 a2 + Res z2 a2 = - coth ira a 2 - Res a z=-ia

1 n2

14.3 Power Series Expansions singular point

Taylor series If f(z) is analytic in a neighborhood of z= a, then an(Z-

f(Z) =

n=0

an

.ff (n) (‘ n!

Radius of convergence R= distance to the nearest singular point, or 1

.

an+I

R = 11M -n -> sup 1,11a n1 = [ n11M

n11M ->

if they exist] .

an

Example. Sought: Taylor series of Log(2z-i) about z = 0; Log(2z- i)= Log[- i(1+ 2iz)[= = Log(- i)+Log(1+ 2iz)=-iit/2+2iz-1/2(2iz)2+

Table of series expansions, see sec. 8.6.

Laurent series If f(z) is analytic in an annulus about z = a, then .i(z)=

n

1 ▪ ca(Z — a)n , ca= 2gi

flz) dz

c (z—a)n+1

R1 and R2 radii of convergence: 1 = lim sup nAcn1;

R 1 = lim sup VIC _ n -> 00 f(z) has singular points on the circles lz—al=Ri,i= 1, 2. R2

n ->

Example.

2 the in annulus l< lz - 2I< 3. z2 - l 1 1 1 1 11 = Solution. f(z) = = [z-2= w]= w+1 z -1-z+ 1 w+3 41+-1) 3(1+1 w 3 w w2 1 1( 1 1 1 ( = 1 (-1P(Z-2)-'1 3) (z-2r Sought: Laurent series expansion off(z)=

W

W

j- 5(1 -

-

=

n=0

354

1 I- (- n n=0 .)

14.4

14.4 Zeros and Singularities Zeros Assume that f(z) is analytic (and 0) in a neighbourhood of z= a. The point a is a zero of order n if f(z) = (z — a)"g(z), where g(z) is analytic and g(a) # 0.

Remark. a is a zero of order n f(a) = f (a) = • . . =f0-1)(a)= 0, f (n)(a)# 0 Singularities z = a is a singular point of f(z) if f(z) fails to be analytic at a. It is isolated if there is a neighbourhood of a in which there are no more singular points.

Classification of isolated singularities. The point z = a is (i) a removable singularity if lim f(z) exists. z —> a (ii) a pole of order n if f(z)=(z—a) ig(z), where g(z) is analytic, g(a)# O. [The Laurent series expansion about a contains finitely many negative power terms.] (iii) an essential singularity otherwise, in which case there are infinitely many negative power terms in the Laurent series expansion about a. Furthermore, branch points of multiple-valued function are examples of non-isolated singular points.

1. (Picard's theorem) The point z = a is an essential singularity of f(z) = Every neighborhood of a contains an infinite set of points z such that f(z) = w for every complex number w (with the possible exception of a single value of w). [Example. f(z)= ell'. Essential singularity at z = 0, exceptional value w = 0]. 2. An isolated singular point z = a is a pole lim I f(z)I = z —> a

The argument principle Assume that f(z) is analytic inside and on a simple curve C except for a finite number of poles inside C, f(z) # 0 on C. Let N= number of zeros, P = number of poles inside C (including multiplicity). Then p

IV

(z) dz = 1 Ac argfiz) 27rq f'(z) f

A arg f(z)=4r N— P=2 Rouche's theorem Assume (i)f(z), g(z) analytic on and inside a simple closed curve C (ii)l g(z)I< f(z) on C. Then f(z) and f(z)+ g(z) have the same number of zeros inside C. 355

14.5

14.5 Conformal Mappings Assume that f(z) is analytic. The mapping w =f(z) is conformal (i.e. preserves angles both in magnitude and sense) at zo if /(zo) # 0. Remark. The Jacobian

, a(u, v) If = y) a(x,

Riemann's mapping theorem Assume that SI is a simply connected region with boundary C. Then there exists a mapping w =f(z), analytic in SZ, which maps CI one-to-one and conformally onto the unit disc and C onto the unit circle.

The bilinear (Miibius) transformation The mapping w = az+b(ad— be 0) maps cz+d (i) circle —> circle or straight line (ii) straight line —> circle or straight line Invariance of cross ratio (w — wl)(w2 — w3) = (z — zi)(z2 — z3)

(w — w 3) (w2 — w

(z — z3)(z2 — zi)

[Wk = w(zk)]

Inverse points z and z* are inverse points (i) with respect to a circle if (z* — a)( — a) = R2 d1 d2 =R 2

(ii) with respect to a line if (b — a )(z* — a) = (b — a)(Z —a)

a \Z• Invariance of inverse points respect to cor(with points inverse of Pairs of inverse points are mapped to pairs responding circles or lines).

Preservation of harmonicity by conformal mappings Assume that (i) h(u, v) is harmonic in w-plane. (ii) f(z) = u(x, y) + iv(x, y) is an analytic function mapping S2 conformally into Sr.

356

14.5 Then H(x, y) = h(u(x, y), v(x, y)) is harmonic in Q. ah Remark. = 0 on 0 on

air = an=

au.

Cf. Poisson's integral formulas, sec. 10.9. Example. (Solving a Dirichlet's problem by conformal mapping.) Problem: Determine the electric potential U(x, y) in the unbounded shadowed region .0 if the potential is given on the boundary as indicated in the figure, i.e. solve the following Dirichlet problem: (*)

z-plane

AU= 0 in Q (i.e. U is harmonic in Q) U=0 on Fi and T2 U = Uo on 1-3

.Q .AU= 0

r1

r3

u =0 Solution. Set z =x + iy and w= u+ iv. By the bilinear transformation x+iy-1 x2 +y2 -1+2iy _ , - u+iv, + iv, w = z-1 z+1 x+iy+1 (x+1)2 +yL

u= u0 r2 X 1 U= 0

12 is conformally mapped onto D'= the first quadrant of the w-plane. Furthermore, F1 l->Fi':0— 0' j =

n

Problem transformation (SLP) transforms into a (CLP) problem by the use of surplus variables si, m, defined by s =Ax — b: si 0, i = 1,

* Compiled by Michael Patriksson, Chalmers University of Technology, Gothenburg.

371

sni,

15.2 min cixi +c2,x2 + + cnxn

•(=>

a12x2 + +ain xn —si =bi a21xi+a22x2 + +a2n xn —s2=b2 s.t.

min cTx

s.t.

{Ax - s = b xon, soni

+an,„xn-sm= bm amlxl+am2x2 1,...,n, i =1, ...,m

The system {Ax b; x On} is equivalent to {Ax + s = b; x On, s Om ); for a constraint, si is called a slack variable. Variables xj 0 transform into non-negative variables by substitutions - xj for xj, where x7> O. A free variable x. is substituted by x . = xt - x: where xt x7 > 0 J— J J J' J, J A maximization problem with objective cTx is transformed to a minimization problem with objective - cTx. Note: LP solvers take care of all necessary substitutions automatically. Linear programming duality For each LP problem (P) there is a related dual problem (D) with as many variables (constraints) as there are constraints (variables) in the primal problem (P). To (CLP) and (SLP), the dual problems are: (DCLP) max bTy s.t.

{AT y c y free

(DSLP) max bTy s.t.

{A Ty c y O'n

The dual of a dual problem is the primal problem. Optimality Here, a problem (P) [(D)] is a minimization [maximization] problem. A feasible solution to an LP problem (P) [(D)] is a vector x [y] that satisfies the constraints. An optimal solution x* [y*] is a feasible solution with a minimal [maximal] value of the objective function among the feasible solutions.

372

15.2 1. (Weak duality) If x [y] is feasible in (P) [(D)] , then (i) bTy < cTx; (ii) bTy = cTx

x [y] is optimal in (P) [(D)] .

2. (Strong duality) If both (P) and (D) have feasible solutions, then there exist finite optimal solutions x* [y*] to (P) [(D)], and bTy* = cTx* . 3. (Optimality characterization) A vector x is optimal in (SLP) if and only if the following is true. (i) x is feasible in (SLP). (ii) There exists a vector y which is (a) feasible in (DSLP), and (b) such that (x, y) together satisfy the complementarily conditions: aiix f -bi)= 0, i =1, ...,m j=i . Iaif yi -c4= 0, j=1, ...,n xii =1

The simplex method Let X = {xe Rn : Ax = b; x On ). X is a polyhedral set. An extreme point of X is an xe X that cannot be written as x = 1,x1 + (1 - A)x2 with x1 x2, both in X, and 0 < A < 1. Extreme points and basic feasible solutions Suppose A is of type mxn, n>m, with rank(A,b) = rank(A) = m (can always be ensured by deleting redundant equations if necessary). Partition A's columns to (B,N), where B of type mxm is invertible, and partition x likewise to (xi, xiDT. The solution to the system Ax = b given by xB = B-1 b and xN = On-m is a basic solution. The matrix B(N) is the basic (non-basic) matrix, and xB (xN) the vector of basic (non-basic) variables. If xB 0m, then x is feasible; (xi, x,I)T is a basic feasible solution (BFS). If xB > Om, then it is non-degenerate, otherwise degenerate.

373

15.2

1. If X is nonempty, it has a nonempty and finite set of extreme points, at n) most I I. 2. For every extreme point of X there corresponds at least one BFS. Conversely, for every BFS, there corresponds a unique extreme point. 3. If (CLP) has a finite optimal solution, it has at least one optimal extreme point. Notes: If an LP problem has an unbounded solution, it is probably not a correct model. If it has no feasible solution, the model may be incorrect but not necessarily. An optimal solution need no more than m positive variables.

The simplex method moves between adjacent (neighbouring) extreme points — described as BFSs, cf. 2 above — so that the objective value improves monotonically, until an optimal one is identified, which will happen after a finite number of steps (cf. 1. and 3. above). Cases of degenerate bases are handled through anti-cycling rules.

A summary of the revised simplex method Consider (CLP), and assume that x =

xiDT is a BFS.

1. Solve Bb = b (gives the value of the BFS, xB =

).

2. Solve BTy = cB (gives a temporary dual vector yT = 3. Let cl = c —yT N (the reduced cost vector; note: c13' = ci; — yT B =(0m)T ). 4. If

, then stop. The current BFS is optimal.

5. Let j* E arg min {e .} (gives the incoming variable x1*). jEN 6. Solve Ba - j* = al* (ai* is column j* of A; gives the direction of change in the basis space). 7. Let i*E arg min bi/ ar: ic B with aii*> 0 } (gives the outgoing variable xi*). 8. If

5. 0, then stop. The LP problem has an unbounded solution.

9. Replace column i* of A in the current basic matrix B with column j* of A; replace xi* with xj* in xB. A new BFS x = (xT3, xk) T with the corresponding basis matrix B is identified. Go to 1. Notes: LP solvers use special linear algebra software for the systems in 1., 2. and 6., and may use other criteria for choosing j* than the one in 5. Solvers: CPLEX, MPL, XPRESS-MP, MATLAB, GAMS, LINDO, LOQO, OSL, EXCEL. 374

15.2

Example. Solve the maximization problem of the form (CLP), with A

2 1 1 01 =[2 2 0 1

= [61 ; c = [1500, 1000, 0, Of'. 8

(This is a two-variable SLP with added slacks.) A starting BFS is xB = [x3, x4]T. The first iteration is very simple when the slacks are basic: 6= 6 1. Bb = b [106 = 01 8 8 2. BTy _ B [01 Oil y = [001 y 4001 T [211 = [1500, 1000] 3. CN = [1500, 1000] - [0, 0]

22

4. CA, is not 5 O. Continue. 5. j* = 1. x1 is an incoming basic variable. 6. Bw* = aj*

7. min

{6 8

22

[1 01 01 =3

* = [21 =

* = [21

2

2

i* = 3. x3 is an outgoing basic variable.

8. al* is not < O. 9. x_B =

1. B6

R - 2 0 rx xi x_4,T,:x_N = ,__3, __2, T, _ 21

b 1[_22 01 6 = [61 6 = [2,1 1i 1_8j

2. ey = cB [2 Y= [15001 y = [7501 0 01 0 3. c-ATT = [0, 1000] - [750, 0] [201 = [-1500, 1000] 21 4. cA, is not O. Continue. 5. j* = 2. x2 is an incoming basic variable.

375

15.2

6. Bii* = ai*

0 Ol

16 8 7. min - - = 3 2' 2 8.

= [21 ii* = [21 2 2

i* = 3. x3 is an outgoing basic variable.

is not < O. rx3, x41 T, _ = 2 1 [2 2]

= l, 9. xB[x

1. Bb =b [2 22

= [61 8

= [1 2

y [ 01 =[ 15005 2. BTy = cB [2 250 1000] 2] 12 3. cNT = [0, 0] - [500, 250] [1 01 = [-500, -250] 01 4. cN O. The current BFS is optimal. x* = [2, 2, 0, Cl] T ; optimal value: yTi) = 1500. Dual. In the above example, y* = [500, 2501T is the vector of shadow prices for the two constraints. Finding a starting basic feasible solution Finding a solution to the system Ax = b; x On is equivalent to solving an LP, for which the simplex method can be applied.We assume that b Om (otherwise, we multiply equations with bi < 0 by — 1). In each row, we introduce artificial variabm. We then solve the artificial LP problem les wi 0, i = 1, (AP)

min a = 17'w

min a = wi + w2 + + ai ixi + a12x2 + ..• + aix, + wi = bi a21x1 + a22x2 + + a2nxn + W2 = b2

s.t. amixi + am2x2 + + amnxn +wm = bm m n,0, i =1, xj ?_ 0, j =1,

{Ax + w = b x 0, w 0

(AP) always has a finite optimal solution (x*, w*). If a* = 0, then w* = 0 is an optimal vector of artificial variables, and the vector xB in the optimal BFS includes only x variables (barring degeneracy). This is then a BFS for the problem (CLP). If a* > 0, then X is empty (the problem (CLP) has no feasible solution). 376

15.2

Sensitivity analysis Consider (CLP) and let x = (x13., xilOT be an optimal BFS. Shadow prices and right-hand side pertubations The optimal value z equals c Tx = c B7.3‘,6 +4xN = 4B-1 b = y Tb , cf. step 2. of the simplex method. If the component yi of the corresponding solution to (DCLP) is unique, then the rate of change in the optimal value z as a function of bi is the partial derivative az=a(yTb)

=

If y is unique, therefore, with z = z(b), we

have that Vz (b) = y. For maximation problems, the value dz(b) is known as the abi shadow price of constraint i. If yi is not unique, then the rate of change in z with an increase (decrease) in bi is the one-sided right (and left, respectively) directional derivative a+ z(b) = max { : yk is an optimal dual vertex}( ab

i

z(b)= min { ylt}, respectively). a bi

Suppose b changes to b' . As long as xi; =B-lb' 0, the same basis is optimal. Let bi change to bi + obi. Then, xB = B-1b' = B-lb + SE• column i of B-1. So,

, where (B-1)i is

0 .=> Sbi (13-1)- Irl b. This defines an interval of fea-

sible 81,i , which includes zero. Cost vector perturbations Suppose ci changes to c j =ci +Sci . As long as cN 0 (min problem), the same basis is optimal. Suppose xi is a non-basic variable. Then, only Z../ changes. xB is then still optimal if C. = ci + Sc./ - yTaj = ej +sci 0, that is, if oci - e:. Suppose xi is a basic variable. Then, the vector cN changes. xB is then still optimal if (cN

.a• - scp3-1iv);?..0T, where (TIN)i is row

j of B-1N. This defines an

interval of feasible Ai, which includes zero. Notes: These sensitivities are normally calculated automatically by commercial LP software, which provides the shadow prices and feasible ranges for individual changes of any right-hand side or cost coefficient. Sensitivities for changes in the matrix A and to the addition of a new variable or a new constraint are normally not provided, but the latter two are easily performed by reoptimizing the problem from the current optimal basis. 377

15.2

Network flows Minimum cost network flow problem Notation. G = (N,A): directed network of nodes N = {1, 2, ..., m{ and n directed arcs A = { (i, j), (k, l), (s, t) } joining nodes in N. Arc (i, j) is incident to nodes i and j and is directed from i to j. Each node iE N has a number bi, the node's supply when bi > 0 (a source node) or the node's demand when bi < 0 (a sink node). It is assum

med that

bi = O. i=1

xij:

flow of an item on arc (i, j).

cif :

unit cost for shipping an item along arc (i, j) .

visi:

capacity of flow on arc (i, j) .

(MCNFP) min /ciixii (i, De A

s.t.

1I xij — Ix ji = bi, i = 1,..., m mi, j)E A j:(j, 0 E A Vii- Xif-.— 0, (i, j) € A

Instances: (1) find the shortest path between node s and node t.

Let bs = 1; bt = —1; bi = 0, iE N\ {s, t}; v = oo, (i, De A Optimality: x* is the flow corresponding to a shortest path if there exist node prices m, such that c : = cif + yj — yi 0, (i, j)E A, while c = 0 for all ij ij (i,j)E A with .0 > O.

yi, i = 1,

(2) the transportation problem. Let the network G be bipartite: the set N is a union of disjoint sets N1 and N2, all arcs (i, j) have i E N1 and je N2; vij = 09; (i, j) E A. For i E N1, bi is called the capacity of node i, for jE N2, is called the demand of node j. The maximal flow problem Find the maximum flow f between node s and node t; (MFP)

max f

s.t.

1 xii — I x ji = j:(i, D€ A j:(j, °EA V u _. Xij

0, (i, j) E A

378

f, i=s 0, i iz N\{s, t} _f, i=t

15.3 The maximal flow — minimal cut theorem: The value f* of the maximal flow in G is equal to the capacity of a minimal cut-set in G. Notes: These network flow problems are very special LP problems, and are solved with specialized, efficient algorithms. Key words: Dijkstra, Ford, Preflow-push, network simplex, out-of-kilter, scaling methods. Further reading: M. S. Bazaraa et. al., 1990, Linear Programming and Network Flows, 2nd edition, Wiley.

15.3 Integer and Combinatorial Optimization* Problem (IP)

min cTx s.t.

{Ax > b v > x 0, integer

=: X (A of type mxn)

The problem (IP) is called binary IP if v = in, since then xi E {0, l }. Instances: (a) the knapsack problem. m = 1; c, AT, b integer vectors

0; v integer vector

O.

(b) the set covering problem. aye {0, 1}; bi = 1; v = in. Modelling of (IP) Fixed costs Let x denote the level of production. A fixed cost c is associated with producing x > 0, and there is a production cost a per unit produced. Modelling of total cost: Let v E { 0, 1}. Total cost: cv + ax. Additional logical constraint: x M • v, where M is large enough for the constraint x M to be valid in the model. Note: the value of M should be kept down to a minimum, since larger values create more difficult problems; if production is capacitated, M is naturally associated with a production capacity. Effect: if x > 0, then v = 1 holds. Choosing a good formulation — uncapacitated facility location Potential depots: N= (1, ..., ri); clients: M= {1, ..., m}; fixed cost for using depot j: fi, transportation cost cy if client i's order is delivered from depot j. Problem: decide depots to open and which depots to serve each client in order to minimize total cost. Variables: vi E {0, 1}, 1 if depot j is open and 0 otherwise; XuE {0, 1}: the fraction of client i's demand which is satisfied from depot j. * Compiled by Michael Patriksson, Chalmers University of Technology, Gothenburg.

379

15.3 Model 1: (UFLP1) min I 1 ciixii + y, fiyi ie i 1 I jE N

S.t.

jE N

I xii= 1, ie M, JEN m • vj > je N, xij jeN ViE {0, 1}, iE

(demand) (link x — v)

M, jE N.

Model 2: replace the linking constraints with xij

vj, ieM, jeN

Let conv(X) be the convex hull of the points in X. If conv(X) was known, we could solve (IP) as the LP problem min cTx. An ideal problem formulation is such XE conv(X) that the convex hull of its feasible solutions is exactly conv(X). For UFL, this is true for model 2 but not for model 1; a model with more variables and/or constraints, a disaggregated model, usually is a better formulation, because then relaxations provide better solutions. (See below.) Solution approaches Problem (1P2)

z* = min cTx

S.t.

Dx d (D of type pxn) Bx .1) v > x 0, integer }=: X

This problem is solved by relaxation or restriction based methods, usually a combination of both, such that both lower and upper bounds on z* are provided. Relaxations LP relaxation: (LP)

Find z*LP = min cTx

S.t.

{Dx > d Bx b v>x>0 380

15.3 zL*p z* holds. Lagrangian relaxation: (LD)

Find k*p = max L(y), s.t. y Om,

where L(y) = min L(x, y)

cT x — yT (Dx — d) (c — DTy)Tx

a y s.t. xe X.

Convexification: (CP)

Zip = min cTx {Dx d s.t. x E conv(X)

* < * * e... ZLp — ZLD = Zcp =.• Z* holds.

: Bx b; v x 0} then kp = kD (integrality property). If conv(X) = {xE * * The difference z — zLD is the duality gap. Relaxations without the integrality property are preferable, since they will have smaller duality gaps. Solving the Lagrangian relaxation: subgradient optimization. O. Let yo Om, k : = 0 (iteration counter). 1. Let x(yk)E arg min L(x,yk) s.t. xE X.

2. Let yt + 1 :=max{0,

+ yk(d — Dx(yk))i} , i = 1,

m.

3. k k+ 1; goto 1. Step length choices: (a) yk = a/(b + ck), a, b, c > 0 or (b) yk = Yk(Z — L(yk))/Ild — Dx(Y k )Ili with yk E(0, 2), In (b), z is usually the value of a feasible solution in (IP2), if one can be found. Primal heuristics can be used to adjust x(yt) towards a feasible solution, and to update z . The class of bundle algorithms are efficient, but also more complex, algorithms for finding y*. Restrictions Greedy heuristics For the knapsack problem: sort elements cIA• J J in decreasing order; set x.J = Jaccording to the ordering, until the knapsack is (over)full. The last feasible solution is the final one. Travelling salesman problem: cif: cost of travel between cities i and J; xy E (0, 1) equals 1 if j is visited directly after i, 0 otherwise. 381

15.3

(TSP)

min

n

n

E E 1 j=1 E xy = 1, i = 1, j:j#i E xi] = 1, j = 1,

n

(assignment)

n

(assignment)

s.t. EE

1, S c N, S # 0 (subtour elimination)

x,i€ 10,11, ZEN, EN Greedy heuristic (nearest neighbour). Choose any starting node; visit the nearest (in terms of c node unless a subtour would be formed; continue until a tour involving all cities are found. Greedy heuristics must be adapted to the problem structure. Still they may perform arbitrarily badly. Local search Problem: min c(S), where S is a complete solution. s.t.

g(S) = 0 ScN I

Basis: a local neighbourhood Q(S), a goal function f(S) = c(S) + ag(S'), a O. Local search heuristic: choose an initial solution S. Search for a set S' Q(S) with f(S') f(S) holds. Genetic algorithms: works with populations (set of solutions) S1, S2, ..., Sk, which evolve from one generation to the next. Notes: Meta-heuristics come with no performance guarantees, but can give good solutions if carefully implemented and adopted to the problem at hand. For combinatorial problems which have clear structures, usually it is best to combine metaheuristics with optimization-based methods, like Lagrangean relaxation or cutting plane methods. Further key words: neural networks, constraint logic programming 382

15.4 Branch & bound — implicit enumeration Idea: Solving to optimality an (IP) by divide and conquer, where the solutions are enumerated implicitly by making good use of bounds on the optimal value z*. Example method: Land-Doig-Dakin Consider (IP2). Solve its LP-relaxation (LP), obtaining a lower bound (LBD) zi p

z*. If xL* p is feasible, then xL* p solves (IP2), otherwise it has a fractional

variable xj. Let x = xi p , z = zip , and consider branching the feasible region into the union of the two sets S1 = Snix;xi Note:

and

S2 =

Snfx;x f >_ rx j11.

S= Si u S2 and Si nS2=0

In order to prune the tree we need a feasible solution. The node selection is often done using depth-first search. The node Si chosen is re-optimized by the simplex method. If its value zi satisfies zi z, the node is pruned, otherwise it is further branched unless x' (the LP-solution) is feasible. If we have at hand a feasible solution x , with objective value z , we also prune a node i if zi . Efficient B&B implementations include preprocessing, possibilities for choosing special branching and node selection rules, variable fixing tests, and primal heuristics to obtain upper bounds. Further reading: L. A. Wolsey, Integer Programming, Wiley.

15.4 Nonlinear Optimization* Problem ( NLP)

min f(xi , x2, ..., xr,)

< min f(x)

gi(xi, x2, ..., xn) < 0 g(x$ Om s.t.

s.t.

g(x m 1, x2, ..., xn) 0 hi (xl , x2, ..., x,,)= 0

h(x)=01 {

hi(xi, x2, ..., xr,) = 0

* Compiled by Michael Patriksson, Chalmers University of Technology, Gothenburg.

383

15.4

Convexity A set S c Rn is convex if xl, x2E S and A.E [0,1] imply that Axi + (1 — A)x2E S. A real-valued function f is convex on a convex set S c Rn if — 2,)x2) < Af(xl ) + (1 — A)f(x2), x1, x2E S, Ae [0,1]. f(Axi If f is twice continuously differentiable on an open neighbourhood of S, then f is convex on S if and only if its Hessian (matrix of second order partial derivatives, a2foiyaxiaxi) v2f(x) is positive semidefinite for every xe S. The function f is concave if —f is convex. f(x)

A,1(xi ) + (1 -t.)1(x2)

f(Axi + (1 -iA)x2) x1

Axi + (1 -1)x 2

x2

x A convex set

A convex function

If gi is a convex function on le, then the set {xe le; gi(x) 0} is convex. The problem (NLP) is said to be convex if f is convex and the feasible set X is convex. Problem transformations between min and max problems, etc., are performed as in linear optimization. We presume that the functions f, gi, hr are continuosly differentiable at least once. Otherwise, we speak of non-differentiable optimization, for which common techniques are subgradient methods (when those are available), and direct search /pattern search (like Nelder-Mead), when only function values are available.

Optimality The point x*E le is locally optimal in (NLP) if X*E X and f(x*) f(x) holds for all xE X in an arbitrarily small neighbourhood of x*. x* is globally optimal if ) = {i = 1, ..., m : gi( ) = 0} is the set f(x*) f(x)for all xE X. At E X, the set of binding inequality constraints. For convex problems, locally optimal solutions are globally optimal.

384

15.4

Constraint qualifications (CQ) 1. All constraints are linear, i.e., the functions gi and It, are of the form aTx — b, aE Rn, be R. 2. (Linear independence CQ). The vectors Vgi(x*), ic /(x*), Vh,.(x*), r =1, l, are linearly independent. 3. (Slater CQ). Assume that l = 0 (no equality constraints). The functions gi, i = 1,...,m are convex, and there exists a vector x with gi(X) < 0, i=1 m (an interior point). The Lagrangian function associated with (NLP) is L(x, y, v) =f(x) + yTg(x) + v7h(x) =f(x) + F yigi(x) + vrhr(x). i=1 r =1

Necessary optimality conditions (Karush-Kuhn-Tucker Theorem) Consider x*E Rn, and let a CQ hold. If x* is a locally optimal solution to (NLP), then there exist vectors y*E Wn and V*E R1 such that: (a) VxL(x*, y*, v*) = On, i.e., aL(x*, y*, v*)/axj = 0, j = 1, ..., n; (b) y* Om; (c) g(x*) < O' 2; h(x*) = 01; (d) (y*)Tg(x*) = 0, i.e., yr • gi(x*) = 0, i = 1,

m.

Such points are called KKT points, and are what algorithms for (NLP) can be expected to converge to. Special case: m = l = 0, that is, unconstrained optimization: If x* is a local minimum of f over Rn, then Vf(x*) = On. If further f is twice continuously differentiable, then V2f(x*) is positive semidefinite.

Sufficient optimality conditions If (NLP) is a convex problem, then every KKT point is a global optimum. In the special case of unconstrained optimization, every point with Vf(x*) = On is a global optimum.

Lagrangian duality The Lagrangian dual function is LD(y, v) = inf L(x, y, v) . XE

385

Rn

15.4

1. (Weak duality). If x is feasible in (NLP) and y Especially, sup LD(y, v)

Om then L(y, v) f(x).

inf f(x) g(x) Om h(x) = 01

y Cr ve R1

2. (Convexity of the dual problem). The Lagrangian dual function is concave, and is differentiable at (y, v) if (g(x),h(x)) is unique on arg min L(x, y, v). xe R"

3. (Primal — dual global optimality conditions). Suppose that for an X*E Itn, y* 0m and v*E RI, the following conditions hold: (i) LD(y*, v*) = L(x*, y*, v*), that is, x* globally minimizes the Lagrangian function given (y*, v*) (ii), g(x*) Om, h(x) = 01 (primal feasibility) (iii) (y*)Tg(x*) = 0 (complementarity) Then, (i) (y*, v*) solves the Lagrangian dual problem: (LD) max LD(y, v), 37 Om

vett/

(ii) x* is a globally optimal solution to (NLP), and (iii) LD(y*, v*) =f(x*) (strong duality).

Strong duality holds for convex problems satisfying a CQ; otherwise, there is a duality gap f(x*) — LD(y*, v*) > 0, and (i)—(iii) is not satisfied for any primal — dual optimal solution (x*, y*, v*).

Algorithms for unconstrained optimization (Gradient methods) Iterative search method 1. Find a starting point x°E R". Iteration counter k:= O. 2. Construct a search direction pk. 3. Choose a step length yk in the direction of pk. 4. Let xk+1 = Xk + yk pk 5. If xk+1 is close to stationary, then stop. Otherwise let k:= k+1 and go to step 2. level curve 386

15.4 Alternatives for step 2 (search direction): • pk = -Vf(xk )/11V(x k )112

(steepest descent method)

• pk _[V2 f(x4-1 • vf(xk)

(Newton's method)

• pk = _Qk . V f ( xk); Qk positive definite and = [02f(xk)]-1 (Quasi-Newton method) • pk coordinate direction of variable xj, j = k + 1 (mod n) (coordinate search) Alternatives for step 3: • yk approximately minimizes f(xk + ypk) over

0 (line search)

• yk minimizes a quadratic interpolation of f • 7k = Pik, where ik is the smallest integer i > 0 with f(xk -.T(xk) apt.c'f(xk)Tpk , where a, OE (0, 1) [typical choices: 0= 0.5, a= 0.011 • 7k

(Armijo rule)

a a,b, c > 0 (if Vf(xk) is calculated with large errors) b + ck'

Alternatives for step 5: terminate if (here p = 2 if n small, p = oo otherwise) • Ilvflxk Nip El ( + Ifixk 1 )1); .f(xk) _j(xk+1) < E2 1+ Iflxk + 1)1 ; •11xk _ xk +111 p _< 53(1 +11Xk + 1 11 p • V2f(xk+1) + 54In is positive semidefinite (if Hessian available). The value of Ei depends on the accuracy in computer calculations. Suggested rule: Ei

E3 = ATE2 , E4= E211Vf(xk + 1)112 ; E2 = machine epsilon.

Algorithms for constrained optimization Penalty methods Idea: replace the constraints defining the feasible set X with a function which mimicks the indicator function

0, ix(x):=I 00, 387

xe X x0 X

, and is easier to minimize.

15.4 Exterior penalty methods Penalty function: a continuous function V' with `F(x)

= 0, >0

X E X, xe X.

For equality constraints: 1 7/(x) = - 1hr(x)1s, s

1, r = 1, 2, ..., /

For - constraints: 1 '11(x) = - [max{ 0, gi(x)}1s, s

1, i = 1, 2, ..., m

For k = 0, 1, ..., do: starting from xk, solve the following problem to obtain Xk+1: min f(x) + ak 'F(x); {ok} is an increasing sequence of positive values tending to c Rn

infinity. (For numerical reasons, this is better than solving one problem for a very large a.) Interior penalty (barrier) methods Barrier function: a function 0 which is continuous on the interior of X and satisfies that 0(x) tends to +oo as x approches the boundary of X.

pcP(x)

For inequality constraints: 41)(x) equals -log(-gi(x)) or -1/gi(x) (note: both functions are convex if g• is convex), i = 1, 2,. ..,m. For k = 0, 1, ..., do: starting from xk, solve the following problem to obtain xk+1: min f(x) + pk I(x); {Pk } is a

X = [a, b]

c 12"

decreasing sequence of positive values tending to zero. Note: numerical instabilities are handled by solving each problem with a specially designed Newton method. Barrier methods are rivals to the simplex method for LP, especially the path following primal-dual versions, and most LP solvers have both simplex and interior point codes built in. Sequental quadratic programming (SQP) The Lagrangian function for (NLP) with m = 0 (no inequality constraints) is L(x, v) =f(x) + vTh(x), and the KKT conditions are VL(x, v) = Vf(x) + vTVh(x) = On VL(x, v) =

h(x) = 01

388

15.5 SQP generalizes Newton's method for unconstrained optimization. Given an iterate (xk, vk), Newton's method applied to the KKT system of size (n + 1)x(n + 1) is to solve xk + 1 _ xk V 2 L(X k , Vk)

— — V L(X k , Vic)

vk + 1— vk) —

V LL(X k ,V k ) Oh(xk ) Vh(xk ) T

(xk

xki

vk + 1 _ vk

0

xi.(xk , Vk )) —h(xk )

With px k+1 — xk, this is the KKT system for the quadratic program min pTV xL(xk, vk) +

Vk) p

s.t. h(xk) + Vh(xk)T p = O. (1,k+1 — vk is a vector of Lagrange multipliers for the linearized constraints.) As the classic Newton method, convergence is usually not guaranteed for unit steps, so p is usually taken as a search direction for use in a line search. Typically, this is performed in an exterior penalty function, like f(x) + ak ilf(x) (see above). If Vx2x L(xk, vk) is not positive definite, such approximations are then used, as in quasi-Newton methods for unconstrained optimization. Solvers: NEOS, Matlab, Excel, Lancelot, MINOS, CONOPT, GAMS, SNOPT, LINGO, NAG, filter-SQP. Further reading: S.G. Nash, A. Sofer, Linear and Nonlinear Programming, MacGraw-Hill.

15.5 Dynamic Optimization t =N, N-1, ..., 1, 0 recursive variable, x(t) = state variable, u(t) = decision variable (x and u may be vectors).

Discrete deterministic dynamic programming Problem N

Find min x, [fdx(0))+

if

t=1

ft (x(t), u(t))]

x(t)EX(t), 0 t N u(t)e U(t, x(t)), 15_ t N x(t — 1) = TAW, u(0), 1 xn+i by solving the linear system M(xn)(xn+i - xn) + f(xn) = 0 xn+i = xn -M(xn)-1f(xn), where

moo= (— ,afi ) is the functional matrix [cf. sec. 10.6]. Systems of linear equations Gauss' elimination method Consider the system aiixi +a,2x2 + +ainxn = bi, i = 1, 2, ..., ti or Ax = b with bi xl ail. ain x=

A= ani• • - ann

b= xn

bn L

The system is assumed to be non-singular, i.e. det A #0. When using the Gauss elimination method (see also sec. 4.3) the system is brought to the following triangular form by successive elimination allxi +al2x2+ai3x3+...+ainxn=bi () () (I) LP) a22x2+ a23x3 + + a2nxn = u2 (2) 2) (2) a33x3+ + a3nXn = u3 (n-1) L(n-1) an,n xn= un 394

16.2 This system is then solved by back substitution, computing xn from the nth equation, xn_ 1 from the (n — 1)st equation and so on in that order. The numbers (1) (2) all , a 22 , a 33 ... are called pivot elements. Partial pivoting: At the rth elimination step use that equation of the (n — r + 1) last ones for the elimination where xr has the largest coefficient in absolute value. Complete pivoting: At the rth elimination step the largest element 4) , i, j_ r+1 is determined. If this is a° then the pth equation is used to eliminate xqfrom the remaining n — r — 1 equations. pq 9

Scaling: Multiply equations by numbers = 0 and/or make substitutions xi =ci yi, ci # O. For an example of Gauss-elimination see sec. 4.3. LU-decomposition For any nx n-matrix A there is a decomposition PA= LU where P is obtained from the identity nx n matrix by row interchanges, L is an nx n lower triangular matrix with diagonal elements =1 U is an nx n upper triangular matrix. Remark. The system Ax = b may be solved iteratively by Ly = Pb and Ux = y. Determination of L and U Explanation by an example: By elementary row operations (cf, sec. .1), 2 4 6 A= 4 11 17 6 24 39

2 4 6 0 3 5 0 12 21

2 46 0 3 5 = U; L= 001

L has entries 'I" in the diagonal and, as indicated with circles, in the lower part of the matrix the "transformation factors" appear with signs changed. Since no row interchanges are neccessary, the "permutation matrix" P = I. QR-decomposition Any mx n-matrix A with and with linearly independent columns can be factored into A = QR where Q is an mx n matrix with orthonormal column vectors, i.e. QTQ = I R is an nx n upper triangular invertible matrix, i.e. rij # 0, all i Determination of Q and R Using the Gram-Schmidt orthogonalization method (see Sec. 4.7), the orthonormal columns q i , q2, qn of Q are determined by orthonormalizing the column vectors a l, a2, ..., an of A. The entries rid of R are obtained when expressing the ai in the q„ i.e. a i a2 = ruq i +r22q2, ..., = rinq i + r2,q2+ + rnnqn . 395

16.2 Iterative methods Write A = D - L - R, where D is a diagonal matrix and L and R are left and right triangular matrises. Start with an arbitrary initial vector x". The Jacobi method X(k) = D-1(L+ R)X(k-1) + 1)- lb The Gauss-Seidel method 1 Rx(k-1) + (D - L)- lb x(k) = (D Example 10x1 + x2 + 7x3 = 33 xi + 10x2 + 6x3 = 39 3x1 + 5x2 + 10x3 = 43 In this case the Jacobi method will use the following recursive formulas. -0.7 x(k-1) +3.3 =-0.1 =-0.1 4-1) -0.6 4-11 +3.9 xgo =-0.3 x11-1) -0.5 4-11 +4.3 While the Gauss-Seidel method will make use of the following recursive formulas. = - 0.1 4-1) -0.7 4- 1) +3.3 4) = - 0.1

40 _0.6 x('

xgo =-0.3 4) -0.541

+ +4.3

A simple sufficient condition for the convergence of the Jacobi and Gauss-Seidel methods is that the absolute values of each diagonal element in A is larger than the sum of the absolute values of the other elements in the same row of A.

Iterative methods to find eigenvalues and eigenvectors The power method Assume that R1I> ICI ... where Al are the eigenvalues of a square matrix A. Starting with a column vector v1 (almost arbitrarily chosen), then vk+1 =AvkilAvk I (if k is big enough) can be used as an approximation of an eigenvector of length 1 corresponding to Al, and At--- vjAvk. The QR method Let A = Q0R0 be a QR-decomposition of a real matrix A. Set A1 = R0Q0 and inductively (if An_1 = Qn_1Rn_1 is a QR-decomposition) An =Rn_10,_1. Then An converges to a "quasi-triangular" matrix whose diagonal blocks contain the eigenvalues. (The convergence of the QR algorithm may be accelerated using shift techniques.) 396

16.3

16.3 Perturbation analysis Vector norms and matrix norms Let x = (x1, ..., Xn )T and A = (a d be of order nxn. In any vector norm VII the corresponding matrix norm is II A x11

11A II =

II A x11

=

IIA B11- u as h, r —> 0 if r/h2= constant 1/4.

Further examples of the finite difference method Notation.

U = U(ih, nk) 5,U =U

n 1 — Ui

f n = f (nk)

fi = f (ih)

OxUin=Uin— Ur-i

SxSx U =UP+1 — 2Uin +ur_i

Finite difference methods for the equation au —

at

2 2 au a —

x2

(a = constant)

418

16.6

1. Forward difference scheme (explicit), q= ka: = constant hz.

a

n+1

2

lc— 'Win= h2 ax 8x Uin i-1

Ur + I = Uin+ q(Ur+1-2Uin+Ur_i)

i

i+1

Local truncation error= 0(k+ h2). Stable if q 5 . 2 Special cases: . 1 Ur +1 = (Utiz.,1 +U7-1 ) (1) q= 2 1 (ii) q= 6 : Local truncation error= 0(k2 +124). 2. Backward difference scheme (implicit), q= 1 k

ka2 = constant h` n+1

IIP= 62 8 V +1 h2 x x

n

+ (1 +2q)UP +1 -qU7+11 = Uin

i+1

Local truncation error= 0(k+ h2). Always stable. ka2 3. Crank-Nicolson scheme (implicit), q= — = constant 2 a 2 1 3 (U•n +U"1) —3 UP= k 2h2 x x " - qU'iV +2(1 + 2q)UP +1

n+1

=

=q1Ir+1 + 2(1 -2q)Uin + qUfl_

i-1

i

i+1

Local truncation error= 0(k2 + h2). Always stable. Initial - Boundary - Value Problem. (PDE) ut=a2uxx , 00 (IC) u(x, 0)= p(x), 0 E)= 0 for each e>0 n -*oo

n—>co

Convergence almost surely plim X„ =X P( lim Xn = X) =1 n->oo

12—>00

Convergence in distribution Xn —> X lim P(Xn x)= P(X.x) for each x such that P(X.x)

is continuous in x Convergence in mean 1.i.m. Xn = X lim E[ IX, -X12] =0 n->00

For these kinds of convergence the following holds. plim X„= X n limp X„ Xn = X Xn -> X in distribution Xn = X

Twodimensional (bivariate) random variable Continuous variable (X, Y)

Discrete variable (X, Y)

x P((X, Y)E A) = EE

(x, y) A

►X

P((X, Y)E A) =if f (x, y)dxdy

f (x, y)

A

f= twodimensional probability function

f= twodimensional probability density

Distribution function F

Distribution function F xy

F(x, y)=P(X

E E f (u, v)

F(x,y)= P(X.x,11 .y)= f f f (u,v)dudv

u5.x, vy

429

17.1 Marginal distributions

Marginal distributions

fx(x)=1,f (x, y)

fx(x) = f f(x, y)dy

fy(Y)

fy(y) =

(x,

J f (x, y)dx _00

Expectation E[g(X, Y)]

Expectation E[g(X, Y)]

E[g(X, Y)]=IIg(x, y)f(x, y)

E[g(X, Y)]= f f g(x, y)f(x, y)dxdy

(E[X17])2 < E[X2]E[Y2]

(Schwarz' inequality)

xy

For independent random variables X and Y f (x, y)=fx(x) fy(Y) F(x, y) = Fx(x) Fy(y) E[XY] = E[X] E[Y]

Var[X+ Y] = Var[X]+ Var[Y] For independent continuous random variables X and Y 00

fx+ y (x) = J fx(t) fy(x — t)dt -00

For independent non-negative random variables X and Y

0,

0)

fx+y(x)= J fx(t) fy(x — t)dt 0 Covariance Cov[X, Y] Cov[X, Y] = ERX — p 1)(1 11. 2)1, E[X] =p 1 , E[ 11=1-1 2

Cov[X, X] = Var[X] Cov[X, Y] =E[XY] — EME[Y]=Cov[Y, X] X and Y independent Cov[X, Y] = 0

Var[X+ Y] = Var[X]+ Var[Y] +2 Cov[X, Y] Correlation coefficient p

Cov[X, Y] P=

VVar[X]Var[Y] p =0 X and Y independent Var[X+ Y] = Var[X]+ Var[Y] +2p,,/Var [X] Var For random variables X1 , X2, ..., Xn nn n I Cov[Xi , X1]= I Var[Xi ]+E ICov[Xi , XJ ]= Var[ Xil = i =1 =1 J i=1j=1 =

=1

Var[Xi 2E 1,Cov[Xi , X1] it)=e-lt,t>0.

Birth and death processes Birth and death process with two states

P(X(t + h) = 0 I X(t) = 1) = h + o(h)

P(X(t + h)= 1 I X(t) = 0) = A h + o(h) P0(t) = P1(t) =

(1 - e-(A+ P)t) + Po(0)e-(x+

- e-(a,+P)t)+ pine-w-ot = lim Pi(t)=

go= lim Po(t)= A to r->os

—>00

440

A +µ

17.3 General birth and death process Al

1-12

P1

P4

113

P(X(t+ h)= i-11X(t)=- i)=µ ih + o(h)

P(X(t + h)= i +1 IX(t)=i)= i h+ o(h)

A 0, A 1, 2, ... birth intensities of the process Pi , P2, P3, ... death intensities of the process = stationary (and asymptotic for t —> 00) probability of state i AoAi•••Ai-i

Pi

iri= 1

Y1Y2-44

The lengths of time To, T1 , T2, ... in states 0, 1, 2, ... are independent and Ti is E(A i + i), 11 0 =0.

Stationary processes Basic definitions Expectation function kt(t)= E[X(t)] Variance function a2(t) = Var[X(t)] Covariance kernel K(s, t) = Cov[X(s), X(t)] X(tn)} and A stochastic process {X(t)} is (strictly) stationary if {X(t1), X(t2), have the same probability distribution for all X(tn+ h)} {X(t1+h), X(t2+ h), to and h>0. t1, t2,

A stochastic process {X(t) } is weakly (second order) stationary, if Cov[X(s), X(s+t)] is independent of s and thus a function of t and E[X(s)] and E[X2(s)] are independent of s. Weakly stationary processes Autocorrelation function Rx(t)=Cov[X(s), X(s+t)] E[X(t)] = 0 Rx(t)= E[X(s)X(s+t)] Power spectral density q> (co )= J Rx(s)e-i"sds= 2 $ Rx(s) cos as ds 0 R(- t)=R(t) (P(-(0 )=49 ((o)

R(0) 49(P)) 0 IR(t)I

1

Rx(t)= 27r f (px(co)e iwtdo)

Cross-correlation function Rx,y(t)= Cov[X(s), Y(s+t)]

441

17.3

00

Cross-spectral density yo x y(t)= f Rx,y(s)e-i'ds , Rx, )4. 0= 77r 5

y(w)e

i co tdco

2 _co

X(t)= a U(t) Rx, y(t)=a Ru,y(t)

Rx(t)=a 2Ru(t)

Ry, x(t)=aRy,u(t)

x,y(w)= a Cou,Y(w)

49 x(w)= a 2 49 u(0) )

Y,x(co)= a TY,u(w)

X(t)= U(t)+V(t) Rx(t) = Ru(t) + Rv(t) + Ruv(0+ Rvu(t) x(w)=Tu(c0 ) -1- 49v(c0 )+Tu,v(0))+Vv,u(0)) Rx,y(t)=Ru,y(t)+Rv y(t) x,y(c0 )=Tu,y(c0 )+ Tv,y(w) Ergodic processes 1 Rx(t)= lim — f X(u)X(u + t)du 27' _ T T Rx,y(0= lim J X(u)Y(u+ t)du T

—T

Linear filters L(a 1 X1 (t)+ a2X2(0)=a1 L(X1 (t))+ a2L(X2(0) 00

X(t)

Y(0= $ h(r)X(t—r)d-c h= impulse response (weighting function) 00

Frequency-response function G(i0))= f e-iwth(t)dt 0 E[Y]= E[X] f h(r)dr o CO 00

Ry(t) = f Rx(t + u - v)h(u)h(v)dudv 00 tP y(w)=1G(ico)12Sx(o) MA-, AR- and ARMA-processes Here fed is a white-noise process. E[ej= 0, E[et2]=o- 2, E[ese] = 0, s # t. Moving Average Process, MA(q) Xt =et+bi et _ i +...+Net_q 442

L

Y(t)=L(X(t))

17.4

Autoregressive Process AR(p) Xt = et - ai X,_ 1 - ...-apX,_ p Autoregressive-Moving Average Process ARMA(p, q) Xt= et + biet_ i + ...+Net_q-ai Xt_i - ...-apXt_p Random sine signal process X(t)= A sin(co t +a) where a is U(0, 2g)

E[X(t)] = 0

- a/co

A2 R(t)=-27 coswt Random telegraph signal process X(t)= (- 1)a +Y(1), t>_ 0 where P(a =0)= P(a =1)=0.5 and Y(t) is a Poisson process with intensity A . 1

E[X(t)1 = 0

t

R(t) = e-2A111 -1

4A

49(0))= 42,2+0)2 Filtered Poisson process (shot noise)

X(t)= I g(t- tk , Yk), where itk roo are time epochs of a Poisson process with

tk t

E[X(t)]=A f E[g(u, Y)]du _o R(t)=A f E[g(t + u, Y)g(u, Y)]du o

17.4 Algorithms for Calculation of Probability Distributions The binomial distribution The probability function f(x) and the distribution function F(x),

f(x)= ('x')px(1 -p)'-x

and

F(x)_

k=0

(

n()pko _ pr-k

can be calculated for given values of n, p and x according to the formulas 443

17.4

f (x)=

1

-pp n - x + 1 f (x - 1), f (0) = (1 - pr x

Test values

n = 10, p = 0.5 and x=5 gives f (x) = 0.24609375 and F(x) = 0.623046875 n = 20, p =0.7 and x=16 gives f (x)= 0.130420974 and F(x) = 0.8929131955

The Poisson distribution The probability function f (x) = e- A - A. x/x! and the distribution function

F(x)= I e-'1 Ak /lc! can be calculated for given values of A and x according to the k=0

formulas

f (x)= -A. •f (x - 1), f (0) = e- A. . x Test values A =4 and x = 5 gives f (x)= 0.1562934518 and F(x)= 0.785130387 A=10 and x =10 gives f (x)= 0.1251100357 and F(x)= 0.5830397502

The hypergeometric distribution The binomial coefficient (x) for given values of n and x can be calculated according to the formulas 1 ), (n o)= 1

(n x) n - x +1 +1 n This can be used to calculate )

x n) f(x)= rxprn-Nxp)/(N and F(x) = I f (k) k =0

Test values

Np = 200, N- Np = 300, n = 50 and x = 30 gives f (x) = 0.0013276703 Np = 50 , N-Np = 450, n = 40 and x = 6 gives f (x)= 0.1080810796

Calculation of the normal, x2, t-and F-distributions Programs for the calculation of the distribution functions for the normal, t-, 2( 2- and F-distributions can be based on the following approximations. 444

17.5 Normal N(0, 1) distribution (x) -- 1/(1 + CP(x)), p(x) =x(1.59145 + 0.01095x+ 0.06651x2),

0

2(r) H(AIB), A = (x/r)113 + (2/9r)— 1, B= „/2/(9r)

2-distribution

H(x) = 1/(1+ CP(x)). For p(x) see above t-distribution t(r) H(Al ,11-3), A= x213 + 2/9 —2x-2/3/(9r) —1, B = 2(1 +x4/3/09, F(x) = (1+ G(x))/2 For H(x), see above. F-distribution F(rl , r2) F(x)

IH(z) if r2 4

H(z+ 0.08 z5/r23) if r2 xi _ od2.

Analysis of variance (ANOVA) Completely randomized, one factor design

µ+a4 11+04

il+a2

p+a3

Mathematical model Yy for i =1, 2, ..., p and j =1, 2, ..., ni are independent and Yu is N(µ+ai , o-) with

I ai = 0 i =1

UMVU-estimates ai = Yi.— p= aA2 S2 =

ni

P

Y i.lp with Yi. = I Kilni i=1 j=1

1

v ni

E n—p i=i j=i

(v k

v\

- i ./ 2,

n= ni i=1

Confidence intervals The pivot variable (K. —µ-- ai ),,Fi/s is t(n—p), which gives the confidence interval p-Fai =Yi .±xs/jizi

509

18.6 ai pi with

Simultaneous confidence intervals for all possible contrasts i=1

p

ai = 0 with simultaneous confidence level a are given by

pi= p+ ai and i =1

E

ai pi =

2 ai I ni ,

aiYi .±As i=1

i=1

where A = j(p — 1}xa and xa is the a-fractile of the F(p — 1, n —p) distribution.

ANOVA table Test of the hypothesis H: al = a2 = ... = ap = 0 can be organized in an ANOVA table. Source

Df

Sum of squares

Mean squares

Expectation

Treatment

p-1

n SST= 1 ni(K.-Y..)2

MST=SST I(p-1)

o2+

MSE=SSE I(n-p)

62

,=I

Error

p ni n-p SSE= 1 1 (Yii-Yi ' i=i 1=1

Total

n-1 SST = 1 E (Y11 - Y .)2

P

F

1 , P 2 L l' liai P -I i =1

MST MSE

ni

i=i i=i

p ni

n=

Y..=

I

ni

j=1

=1 i=1

Completely randomized two factor design with equal number of observations per cell Mathematical model c are independent and Yu', is Mit+ ai +A.j, o-) j =1, b, k =1, Yijk for i=1, with E cc= E = O. Instead of this additive model, a more general model with interactions Oij can be used. Then Yijk is p

N(µ+ ai + +0ii , o-) where

=1

b

()if =

I Oij =O. j=1

UMVU-estimates P = Y...

bei p b c YakipbC Y • =

= =1 j=1 k=1

h c E YijklbC j=1 k =1

510

P c

Y ••= E

i=1 k=l

YijkIPC

18.6 ANOVA table Tests of the hypotheses Hi : a1 = a2 = organized in an ANOVA table. Source

First factor

Df

=ap= 0 and H2: 21 = 2/2= =

Sum of squares

Mean squares

SSi =cb i (Yi ..-Y i =1

p -1

MS1=

.)2

SSI

=0 can be Expectation

2+

p —1

F

P cb 1 ot7' i=1 p-1

MS1 MSE

b

Second b- I factor

SS2=cp

b

y

n-p-b+1 SSE=

Total

n -1

b

c

E E E (yijk - Yi ..- Y.j y...)2 i=1 j=1 k=1 p

SST=

(Y .1 .-Y ...)2

=i

p

Error

, SS2 M 2 -— b b -1

b

m, aE=

SSE n- p-b +1

cp o2 +

02

c

i=1 j=1 k=1

j= 1,

p b

b can be tested using the

2

(n- pb)c

F (Yii .-Yi .. -Y. j . +Y... ) i =1 j =1 b c 2 (b - 1)(p - 1 ) (Yip, - .) i=1 j=1 k=1

which, under the hypotheses, is F((b -1)(p -1), n- pb).

The general linear model Y1 Y2

E12 C1p C21 E22 •• • c2p

132

E2

Yn

cnl Cn2

cnp

Pp

En

E1 C11

or y=

A.2. j

'b -1

E I, 1 (Yiik - Y...)2

For c> 1 the hypotehsis H. Ou = 0, i=1, testvariable

-

E

• 1

E

where C is a known design matrix and el , E2 , ..., en are independent and N(0, o-). If Rank (C)=p, UMVU estimates of 01,02,...,0, are given by p = (CTC)—IC Ty

511

MS2 MSE

18.7

18.7 Distribution-free Methods The Kohnogorov-Smirnov statistics Dn D„. mxax IS,(x)- Fo(x)I S„. empirical distribution function F0 = distribution function under null hypothesis F= distribution function of D„, F(x)=P(Dn x) F(x) =0.80

0.85

0.90

0.95

0.99

1 2 3 4 5

0.900 0.684 0.565 0.493 0.447

0.925 0.726 0.597 0.525 0.474

0.950 0.776 0.636 0.565 0.510

0.975 0.842 0.708 0.624 0.563

0.995 0.929 0.829 0.734 0.669

6 7 8 9 10

0.410 0.381 0.358 0.339 0.323

0.436 0.405 0.381 0.360 0.342

0.468 0.436 0.410 0.387 0.369

0.520 0.483 0.454 0.430 0.409

0.617 0.576 0.542 0.513 0.489

11 12 13 14 15

0.308 0.296 0.285 0.275 0.266

0.326 0.313 0.302 0.292 0.283

0.352 0.338 0.325 0.314 0.304

0.391 0.375 0.361 0.349 0.338

0.468 0.450 0.432 0.418 0.404

16 17 18 19 20

0.258 0.250 0.244 0.237 0.232

0.274 0.266 0.259 0.252 0.246

0.295 0.286 0.279 0.271 0.265

0.327 0.318 0.309 0.301 0.294

0.392 0.381 0.371 0.361 0.352

25 30 35

0.208 0.190 0.177

0.228 0.208 0.193

0.238 0.218 0.202

0.264 0.242 0.224

0.317 0.290 0.269

Formula for large n

1.07

1.14

1.22

1.36

1.63

,iii

VITI

„In

AFI

,In

n

512

18.7

The Wilcoxon statistic U (or W) n= number of X-observations m= number of Y-observations U= number of pairs (Xi , yi ) with Xi < W= rank sum of the Y-observations U=

W m(m+1) 2

E[U] =

Var[U] =

nm(n+m+1) 12

The table gives P(U x)= P(U nm - x) r1 = min(n, m)

r2 = max(n, m)

r1

x

r2 = 3

r2 = 4

r2 = 5

r2 = 6

r2 = 7

r2 = 8

3

0 1 2 3 4 5 6 7 8 9 10 11

0.0500 0.1000 0.2000 0.3500 0.5000 0.6500 0.8000 0.9000 0.9500 1.0000

0.0286 0.0571 0.1143 0.2000 0.3143 0.4286 0.5714 0.6857 0.8000 0.8857 0.9429 0.9714

0.0179 0.0357 0.0714 0.1250 0.1964 0.2857 0.3929 0.5000 0.6071 0.7143 0.8036 0.8750

0.0119 0.0238 0.0476 0.0833 0.1310 0.1905 0.2738 0.3571 0.4524 0.5476 0.6429 0.7262

0.0083 0.0167 0.0333 0.0583 0.0917 0.1333 0.1917 0.2583 0.3333 0.4167 0.5000 0.5833

0.0061 0.0121 0.0242 0.0424 0.0667 0.0970 0.1394 0.1879 0.2485 0.3152 0.3879 0.4606

4

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.0143 0.0286 0.0571 0.1000 0.1714 0.2429 0.3429 0.4429 0.5571 0.6571 0.7571 0.8286 0.9000 0.9429 0.9714 0.9857

0.0079 0.0159 0.0317 0.0556 0.0952 0.1429 0.2063 0.2778 0.3651 0.4524 0.5476 0.6349 0.7222 0.7937 0.8571 0.9048

0.0048 0.0095 0.0190 0.0333 0.0571 0.0857 0.1286 0.1762 0.2381 0.3048 0.3810 0.4571 0.5429 0.6190 0.6952 0.7619

0.0030 0.0061 0.0121 0.0212 0.0364 0.0545 0.0818 0.1152 0.1576 0.2061 0.2636 0.3242 0.3939 0.4636 0.5364 0.6061

0.0020 0.0040 0.0081 0.0141 0.0242 0.0364 0.0545 0.0768 0.1071 0.1414 0.1838 0.2303 0.2848 0.3414 0.4040 0.4667

513

18.7

r2 = 5

r2 = 6

r2 = 7

r2 = 8

r1

x

5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.0040 0.0022 0.0013 0.0008 0.0079 0.0043 0.0025 0.0016 0.0159 0.0087 0.0051 0.0031 0.0278 0.0152 0.0088 0.0054 0.0476 0.0260 0.0152 0.0093 0.0754 0.0411 0.0240 0.0148 0.1111 0.0628 0.0366 0.0225 0.1548 0.0887 0.0530 0.0326 0.2103 0.1234 0.0745 0.0466 0.2738 0.1645 0.1010 0.0637 0.3452 0.2143 0.1338 0.0855 0.4206 0.2684 0.1717 0.1111 0.5000 0.3312 0.2159 0.1422 0.5794 0.3961 0.2652 0.1772 0.6548 0.4654 0.3194 0.2176 0.7262 0.5346 0.3775 0.2618 0.7897 0.6039 0.4381 0.3108 0.8452 0.6688 0.5000 0.3621 0.8889 0.7316 0.5619 0.4165 0.9246 0.7857 0.6225 0.4716

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0.0011 0.0006 0.0003 0.0022 0.0012 0.0007 0.0043 0.0023 0.0013 0.0076 0.0041 0.0023 0.0130 0.0070 0.0040 0.0206 0.0111 0.0063 0.0325 0.0175 0.0100 0.0465 0.0256 0.0147 0.0660 0.0367 0.0213 0.0898 0.0507 0.0296 0.1201 0.0688 0.0406 0.1548 0.0903 0.0539 0.1970 0.1171 0.0709 0.2424 0.1474 0.0906 0.2944 0.1830 0.1142 0.3496 0.2226 0.1412 0.4091 0.2669 0.1725 0.4686 0.3141 0.2068 0.5314 0.3654 0.2454 0.5909 0.4178 0.2864 0.6504 0.4726 0.3310 0.7056 0.5274 0.3773 0.7576 0.5822 0.4259 0.8030 0.6346 0.4749

6

rl

x

7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

514

r2 =7

r2 =8

0.0003 0.0002 0.0006 0.0003 0.0012 0.0006 0.0020 0.0011 0.0035 0.0019 0.0055 0.0030 0.0087 0.0047 0.0131 0.0070 0.0189 0.0103 0.0265 0.0145 0.0364 0.0200 0.0487 0.0270 0.0641 0.0361 0.0825 0.0469 0.1043 0.0603 0.1297 0.0760 0.1588 0.0946 0.1914 0.1159 0.2279 0.1405 0.2675 0.1678 0.3100 0.1984 0.3552 0.2317 0.4024 0.2679 0.4508 0.3063 0.5000 0.3472 0.5492 0.3894 0.5976 0.4333 0.6448 0.4775

r1

8

x

r2 = 8

0 0.0001 i 0.0002

2 0.0003 3 0.0005 4 0.0009 5 0.0015 6 0.0023 7 0.0035 8 0.0052 9 0.0074 10 0.0103 11 0.0141 12 0.0190 13 0.0249 14 0.0325 15 0.0415 16 0.0524 17 0.0652 18 0.0803 19 0.0974 20 0.1172 21 0.1393 22 0.1641 23 0.1911 24 0.2209 25 0.2527 26 0.2869 27 0.3227 28 0.3605 29 0.3992 30 0.4392 31 0.4796

18.7

The Wilcoxon sign rank statistic n= number of differences Ri = ± rank of 141 sign of Ri= sign of 4 W=

1 n n(n + 1) R• + 2 i= 4

The table gives poy

n(n + 1) x)

x

n=1

n=2

n=3

n=4

n=5

n=6

n=7

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.5000

0.2500 0.5000

0.1250 0.2500 0.3750

0.0625 0.1250 0.1875 0.3125 0.4375

0.0313 0.0625 0.0938 0.1563 0.2188 0.3125 0.4063 0.5000

0.0156 0.0313 0.0469 0.0781 0.1094 0.1563 0.2188 0.2813 0.3438 0.4219 0.5000

0.0078 0.0156 0.0234 0.0391 0.0547 0.0781 0.1094 0.1484 0.1875 0.2344 0.2891 0.3438 0.4063 0.4688

x

n=8

n=9

n=10

n=11

n=12

n=13

n=14

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.0039 0.0078 0.0117 0.0195 0.0273 0.0391 0.0547 0.0742 0.0977 0.1250 0.1563 0.1914 0.2305 0.2734

0.0020 0.0039 0.0059 0.0098 0.0137 0.0195 0.0273 0.0371 0.0488 0.0645 0.0820 0.1016 0.1250 0.1504

0.0010 0.0020 0.0029 0.0049 0.0068 0.0098 0.0137 0.0186 0.0244 0.0322 0.0420 0.0527 0.0654 0.0801

0.0005 0.0010 0.0015 0.0024 0.0034 0.0049 0.0068 0.0093 0.0122 0.0161 0.0210 0.0269 0.0337 0.0415

0.0002 0.0005 0.0007 0.0012 0.0017 0.0024 0.0034 0.0046 0.0061 0.0081 0.0105 0.0134 0.0171 0.0212

0.0001 0.0002 0.0004 0.0006 0.0009 0.0012 0.0017 0.0023 0.0031 0.0040 0.0052 0.0067 0.0085 0.0107

0.0001 0.0001 0.0002 0.0003 0.0004 0.0006 0.0009 0.0012 0.0015 0.0020 0.0026 0.0034 0.0043 0.0054

515

18.7

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

0.3203 0.3711 0.4219 0.4727

0.1797 0.2129 0.2480 0.2852 0.3262 0.3672 0.4102 0.4551 0.5000

0.0967 0.1162 0.1377 0.1611 0.1875 0.2158 0.2461 0.2783 0.3125 0.3477 0.3848 0.4229 0.4609 0.5000

0.0508 0.0615 0.0737 0.0874 0.1030 0.1201 0.1392 0.1602 0.1826 0.2065 0.2324 0.2598 0.2886 0.3188 0.3501 0.3823 0.4155 0.4492 0.4829

516

0.0261 0.0320 0.0386 0.0461 0.0549 0.0647 0.0757 0.0881 0.1018 0.1167 0.1331 0.1506 0.1697 0.1902 0.2119 0.2349 0.2593 0.2847 0.3110 0.3386 0.3667 0.3955 0.4250 0.4548 0.4849

0.0133 0.0164 0.0199 0.0239 0.0287 0.0341 0.0402 0.0471 0.0549 0.0636 0.0732 0.0839 0.0955 0.1082 0.1219 0.1367 0.1527 0.1698 0.1879 0.2072 0.2274 0.2487 0.2709 0.2939 0.3177 0.3424 0.3677 0.3934 0.4197 0.4463 0.4730 0.5000

0.0067 0.0083 0.0101 0.0123 0.0148 0.0176 0.0209 0.0247 0.0290 0.0338 0.0392 0.0453 0.0520 0.0594 0.0676 0.0765 0.0863 0.0969 0.1083 0.1206 0.1338 0.1479 0.1629 0.1788 0.1955 0.2131 0.2316 0.2508 0.2708 0.2915 0.3129 0.3349 0.3574 0.3804 0.4039 0.4276 0.4516 0.4758 0.5000

18.7

Confidence inter.xval for th 0

0 be an oerdm dia ereed sanmple from Let x(1)5._x(2 a continuou )s ... 5.. s distributio median in. T n with hen .X (k ) is a confiden level of con ce interval for m. T fidence is cl k such that the ose to 0.9h5e.following table giv es values of The exact co nfidence lev el is also giv table. en in the Confidence

Level of confidence

interval

x(1)5...rn 5_x(5) x(1)5-In rn x

5 6 7 8 9

2 2

10 11 12 13 14

2 3 3 3 4

15 16 1'7 18 19

4 4 5 5 6

m.•Cx(-13) x0).5 in5..x0.4) x(6)C.- 1" -x(14)

20 21 22 23 24

6 6 7 7

x(0 5-rn 5-4 (16) x(7) .Srn 5..x(16) xco 5rn5...x(17)

25 26 27 28 29

8 8 9 9 10

30 31 32 33 34

10 10 11 11 12

35 36 37 38 39

12 13 13 13 14

40

14

0.938 0.969 0.984 0.930 0.961 0.979 0.935 0.961 0.9'78 0.943 0.965 0.979 0.951 0.969 0.936 0.959 0.973 0.948 0.965 0.936

x(2)5..rn 5.x(g) x(2)