Phillips / Sneddon / Stark Some Topics in Complex Analysis

1. Auflage 2014
ISBN: 978-1-4832-8272-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 86, 150 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

International Series of Monographs in Pure and Applied Mathematics, Volume 86, Some Topics in Complex Analysis deals with a variety of topics related to complex analysis. This book discusses the method of comparison, periods of an integral, generalized Joukowski transformations, and Koebe's distortion theorems. The deductions from the maximum-modulus principle, canonical products and genus of an I.F., and Weierstrass's primary factors are also reviewed. This text likewise considers Mittag-Leffler's theorem, summation of series by the calculus of residues, definition of regular functions by integrals, and Riemann zeta function. This publication is a good reference for students and specialists researching in the field of applied and pure mathematics.

Autoren/Hrsg.

Phillips, E. G.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Some Topics in Complex Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;CHAPTER 1. ELLIPTIC FUNCTIONS;10
6.1;1.1 Definition;10
6.2;1.2 Fundamental Theorems;11
6.3;1.3. The Functions p(u), s(u) and .(w);14
6.4;1.4 The Method of Comparison;16
6.5;1.5 The Double Periodicity of p(u);16
6.6;1.6 Descriptive Properties of p(u), s(u) and .(u);17
6.7;1.7 The Addition Theorem for p(u);17
6.8;1.8 The Relation Between p(u) and p'(u);18
6.9;1.9 Another Form of the Addition Theorem;21
6.10;1.10 Fundamental Expressions of E.F.;21
6.11;1.11 Some Fundamental Formulae;23
6.12;1.12 Example;25
6.13;Examples 1;26
7;CHAPTER 2. THE JACOBIAN ELLIPTIC FUNCTIONS;29
7.1;2.1 The Periods of an Integral;29
7.2;2.2 The Function sn u;30
7.3;2.3 The Constants K and K;32
7.4;2.4 The Functions cn u, dn u;33
7.5;2.5 The Addition Theorems;34
7.6;2.6 Periodicity;35
7.7;2.7 Expansions in Powers of u;36
7.8;2.8 Identities and Duplication Formulae;36
7.9;2.9 Jacobi's "Imaginary" Transformation;38
7.10;2.10 The Jacobian Functions for Values Connected with the Periods;39
7.11;2.11 Applications of the Method of Comparison to Jacobian Functions;42
7.12;2.12 The Relation Between Weierstrassian and Jacobian E.F.;43
7.13;2.13 Elliptic Integrals;44
7.14;2.14 The Functions E(u) and Z(u);46
7.15;Examples 2;48
8;CHAPTER 3. CONFORMAL TRANSFORMATION;52
8.1;3.1 Ratio of Two Quadratics;52
8.2;3.2 Generalized Joukowski Transformations;54
8.3;3.3 Boundary a Closed Polygon;57
8.4;3.4 Schwarz–Christoffel Transformation;60
8.5;3.5 Transformations Involving Elliptic Functions;62
8.6;3.6 Note on Transformations Involving E.F.;67
8.7;3.7 Schwarz's Lemma;68
8.8;3.8 Extension of Schwarz's Lemma;69
8.9;3.9 An Estimate of the Derivative of a Bounded Function;70
8.10;3.10 Functions with a Positive Real Part;71
8.11;3.11 Schwarz's Symmetry Principle;72
8.12;3.12;73
8.13;Examples 3;74
9;CHAPTER 4. SCHLICHT FUNCTIONS;77
9.1;4.1;77
9.2;4.2 Definition;78
9.3;4.3 Some Distortion Theorems;81
9.4;4.4;84
9.5;4.5 Koebe's Distortion Theorems;86
9.6;4.6 Bieberbach's Inequality;88
9.7;Examples 4;88
10;CHAPTER 5. THE MAXIMUM-MODULUS PRINCIPLE;91
10.1;5.1 The Maximum-Modulus Theorem;91
10.2;5.2 The Phragmén–Lindelöf Extension;92
10.3;5.3 Deductions from the Maximum-Modulus Principle;92
10.4;5.4;94
10.5;5.5;95
10.6;5.6;96
10.7;Examples 5;97
11;CHAPTER 6. INTEGRAL FUNCTIONS;99
11.1;6.1 Definition and Preliminaries;99
11.2;6.2 Weierstrass's Primary Factors;100
11.3;6.3 The Order of an I.F.;102
11.4;6.4 Jensen's Theorem;103
11.5;6.5 The Function n(r) for an I.F.;105
11.6;6.6 Canonical Products and Genus of an I.F.;107
11.7;6.7 Hadamard's Theorem on I.F. of Finite Order;108
11.8;6.8 The Coefficients in the Expansion of an I.F. of Finite Order;109
11.9;Examples 6;111
12;CHAPTER 7. EXPANSIONS IN INFINITE SERIES;114
12.1;7.1 Lagrange's Expansion;114
12.2;7.2 Teixeira's Theorem;116
12.3;7.3 Mittag–Leffler's Theorem;118
12.4;7.4 Weierstrass's Theorem;119
12.5;7.5 Summation of Series by the Calculus of Residues;120
12.6;7.6 On Some Meromorphic Functions;122
12.7;7.7 Some Further Summations;125
12.8;Examples 7;126
13;CHAPTER 8. CONTOUR INTEGRALS DEFINING SOME SPECIAL FUNCTIONS;129
13.1;8.1 Definition of Regular Functions by Integrals;129
13.2;8.2 Analytic Continuation by Means of an Integral;130
13.3;8.3 The Gamma Function;131
13.4;8.4 The Riemann Zeta function;133
13.5;8.5 A Contour Integral for .(s, a);135
13.6;8.6 The Legendre Function Pn(z);136
13.7;8.7 The Bessel Function Jn(z) when n is an Integer;137
13.8;8.8 Bessel's Integral when n is not an Integer;138
13.9;8.9 Hankel's Contour Integral for Jn(z);140
13.10;Examples 8;142
14;BIBLIOGRAPHY;144
15;INDEX;146
16;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;149