Functions of a Complex Variable: Theory and Technique

454 Pages·2005·13.551 MB·English

by  George F. Carrier, Max Krook, Carl E. Pearson

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P.
Functions of
a
Complex Variable
Theory and Technique
George F Carrier
Max Krook
Carl E. Pearson
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In Applied Mathematics
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Functions of a
Complex Variable
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/'/i/s page in/enlionally
P
Functions of a
Complex Variable
Theory and Technique
George F. Carrier
Max Krook
Carl E. Pearson
Society lotIndustrial anti Applied Mathematics
Philadelphia
Copyright ©2005 by the Society forindustrial and Applied Mathematics
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10987654321
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contents
Preface xi
I
complex numbers and their elementary
properties 1
14 Origin and Definition I
1-2 Sequences and Series 5
1-3 Power Series 8
1-4 Powers and Logarithms 13
1-5 Geometric Properties of Simple Functions 18
2
analyticfunctions 25
2-1 Differentiation in the Complex Plane 25
2-2 Integration in the Complex Plane 30
2-3 Cauchy's Integral Formula 37
2-4 Maximum Modulus Theorem 42
2-5 Harmonic Functions 44
2-6 Taylor Series 49
2-7 Laurent Series 54
2.8 Analytic Continuation 63
2-9 Entire and Meromorphic Functions 67
240 Results Concerning the Modulus of /(z) 73
3
contour integration 77
3-1 Illustrative Examples 77
3-2 Series and Product Expansions 95
3-3 Integral Representations of Functions 101
vu
4
conformal ma/ping 111
4.1 Two-dimensional Potential Problem 111
4-2 Conformal Transformation 121
4-3 Bilinear Transformations 126
4-4 TheSchwarz-Christoffel Transformation 136
4-5 The Joukowsky Transformation 157
4-6 The Hodograph 162
4-7 Periodic Domains and Fields 166
4-8 Integral Equations and Approximation Techniques 174
4-9 The Biharmonic Equation 180
5
specialfunctions 183
5-1 The Gamma Function 183
5-2 Differential Equations 194
5-3 Hypergeometric Functions 202
5-4 Legendre Functions 210
5-5 Bessel Functions 220
6
asymptotic nwtliods 241
6-1 The Nature of an Asymptotic Expansion 241
6-2 Laplace's Method 249
6.3 Method of Steepest Descents 257
6-4 Method of Stationary Phase 272
6-5 Phase, Group, and Signal Velocities 275
6-6 Differential-equation Methods 283
6-7 WKB Method 291
7
transform met/wds 301
7-1 Fourier Transforms 301
7-2 The Application of Fourier Transforms to
Boundary-value Problems 332
7-3 The Laplace Transform 347
7-4 Hankel Transforms 366
mu
9
special techniques 376
81 The
Method 376
8.2 The Kernel Decomposition 382
8-3 Integral Equations with Displacement Kernels 386
8-4 The Use of Approximate Kernels 393
8-5 Dual Integral Equations 390
8-6 Singular Integral Equations 408
Index 433
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