Complex Variables: Introduction and Applications

Cover......Page 1
Half-title......Page 3
Series-title......Page 6
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 13
Part I Fundamentals and Techniques of Complex Function Theory......Page 15
1.1 Complex Numbers and Their Properties......Page 17
1.2.1 Elementary Functions......Page 22
1.2.2 Stereographic Projection......Page 29
1.3 Limits, Continuity, and Complex Differentiation......Page 34
1.3.1 Elementary Applications to Ordinary Differential Equations......Page 40
2.1.1 The Cauchy–Riemann Equations......Page 46
2.1.2 Ideal Fluid Flow......Page 54
2.2 Multivalued Functions......Page 60
2.3 More Complicated Multivalued Functions and Riemann Surfaces......Page 75
2.4 Complex Integration......Page 84
2.5 Cauchy’s Theorem......Page 95
2.6.1 Cauchy’s Integral Formula and Its Derivatives......Page 105
2.6.2 Liouville, Morera, and Maximum-Modulus Theorems......Page 109
2.6.3 Generalized Cauchy Formula and…Derivatives......Page 112
2.7 Theoretical Developments......Page 119
3.1 Definitions of Complex Sequences, Series and Their Basic Properties......Page 123
3.2 Taylor Series......Page 128
3.3 Laurent Series......Page 141
3.4 Theoretical Results for Sequences and Series......Page 151
3.5 Singularities of Complex Functions......Page 158
3.5.1 Analytic Continuation and Natural Barriers......Page 166
3.6 Infinite Products and Mittag–Leffler Expansions......Page 172
3.7 Differential Equations in the Complex Plane: Painlevé Equations......Page 188
3.8.1 Laurent Series......Page 210
3.8.2 Differential Equations......Page 212
4.1 Cauchy Residue Theorem......Page 220
4.2 Evaluation of Certain Definite Integrals......Page 231
4.3.1 Principal Value Integrals......Page 251
4.3.2 Integrals with Branch Points......Page 259
4.4 The Argument Principle, Rouché’s Theorem......Page 273
4.5 Fourier and Laplace Transforms......Page 281
4.6 Applications of Transforms to Differential Equations......Page 299
Part II Applications of Complex Function Theory......Page 323
5.1 Introduction......Page 325
5.2 Conformal Transformations......Page 326
5.3 Critical Points and Inverse Mappings......Page 331
5.4 Physical Applications......Page 336
5.5 Theoretical Considerations – Mapping Theorems......Page 355
5.6 The Schwarz–Christoffel Transformation......Page 359
5.7 Bilinear Transformations......Page 380
5.8 Mappings Involving Circular Arcs......Page 396
5.9.1 Rational Functions of the Second Degree......Page 414
5.9.2 The Modulus of a Quadrilateral......Page 419
5.9.3 Computational Issues......Page 422
6.1 Introduction......Page 425
6.1.1 Fundamental Concepts......Page 426
6.1.2 Elementary Examples......Page 432
6.2 Laplace Type Integrals......Page 436
6.2.1 Integration by Parts......Page 437
6.2.2 Watson’s Lemma......Page 440
6.2.3 Laplace’s Method......Page 444
6.3 Fourier Type Integrals......Page 453
6.3.1 Integration by Parts......Page 454
6.3.2 Analog of Watson’s Lemma......Page 455
6.3.3 The Stationary Phase Method......Page 457
6.4 The Method of Steepest Descent......Page 462
6.4.1 Laplace’s Method for Complex Contours......Page 467
6.5 Applications......Page 488
6.6 The Stokes Phenomenon......Page 502
6.6.1 Smoothing of Stokes Discontinuities......Page 508
6.7.1 WKB Method......Page 514
6.7.2 The Mellin Transform Method......Page 518
7.1 Introduction......Page 528
7.2 Cauchy Type Integrals......Page 531
7.3 Scalar Riemann–Hilbert Problems......Page 541
7.3.1 Closed Contours......Page 543
7.3.2 Open Contours......Page 547
7.3.3 Singular Integral Equations......Page 552
7.4 Applications of Scalar Riemann–Hilbert Problems......Page 560
7.4.1 Riemann–Hilbert Problems on the Real Axis......Page 572
7.4.2 The Fourier Transform......Page 580
7.4.3 The Radon Transform......Page 581
7.5 Matrix Riemann–Hilbert Problems......Page 593
7.5.1 The Riemann–Hilbert Problem for Rational Matrices......Page 598
7.5.2 Inhomogeneous Riemann–Hilbert Problems and Singular Equations......Page 600
7.5.3 The Riemann–Hilbert Problem for Triangular Matrices......Page 601
7.5.4 Some Results on Zero Indices......Page 603
7.6 The DBAR Problem......Page 612
7.6.1 Generalized Analytic Functions......Page 615
7.7 Applications of Matrix Riemann–Hilbert Problems and Problems......Page 618
Section 1.2......Page 641
Section 2.1......Page 642
Section 2.5......Page 643
Section 3.3......Page 644
Section 3.7......Page 645
Section 4.5......Page 646
Section 4.6......Page 647
Section 5.6......Page 648
Section 6.2......Page 649
Section 6.4......Page 650
Bibliography......Page 651
Index......Page 654