Fuchs / Shabat / Sneddon Functions of a Complex Variable and Some of Their Applications

1. Auflage 2014
ISBN: 978-1-4831-5505-0
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 458 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Functions of a Complex Variable and Some of Their Applications, Volume 1, discusses the fundamental ideas of the theory of functions of a complex variable. The book is the result of a complete rewriting and revision of a translation of the second (1957) Russian edition. Numerous changes and additions have been made, both in the text and in the solutions of the Exercises. The book begins with a review of arithmetical operations with complex numbers. Separate chapters discuss the fundamentals of complex analysis; the concept of conformal transformations; the most important of the elementary functions; and the complex potential for a plane vector field and the application of the simplest methods of function theory to the analysis of such a field. Subsequent chapters cover the fundamental apparatus of the theory of regular functions, i.e. basic integral theorems and expansions in series; the general concept of an analytic function; applications of the theory of residues; and polygonal domain mapping. This book is intended for undergraduate and postgraduate students of higher technical institutes and for engineers wishing to increase their knowledge of theory.

Autoren/Hrsg.

Fuchs, B. A.

Shabat, B. V.

Sneddon, I. N.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Functions of a Complex Variable and some of their applications;6
3;Copyright Page;7
4;Table of Contents;8
5;From the foreword to the first edition;12
6;Foreword to the English edition;16
7;INTRODUCTION;18
7.1;1. Complex numbers;18
7.2;2. The simplest operations;19
7.3;3. Multiplication, division, integral powers and roots;25
7.4;4. Complex powers. Logarithms;31
7.5;Exercises;34
8;CHAPTER 1. THE FUNDAMENTAL IDEAS OF COMPLEX ANALYSIS;37
8.1;5. The sphere of complex numbers;37
8.2;6. Domains and their boundaries;39
8.3;7. The limit of a sequence;42
8.4;8. Complex functions of a real variable;45
8.5;9. The complex form of an oscillation;49
8.6;10. Functions of a complex variable;52
8.7;11. Examples;53
8.8;12. The limit of a function;58
8.9;13. Continuity;59
8.10;14. The Cauchy–Riemann conditions;62
8.11;Exercises;66
9;CHAPTER 2. CONFORMAL MAPPINGS;69
9.1;15. Conformal mappings;69
9.2;16. Conformal mapping of domains;73
9.3;17. Geometric significance of the differential dw;75
9.4;18. Bilinear mappings;77
9.5;19. The circle property;81
9.6;20. Invariance of the conjugate points;82
9.7;21. Conditions determining bilinear mappings;86
9.8;22. Particular examples;88
9.9;23. General principles of the theory of conformal mapping;91
9.10;Exercises;95
10;CHAPTER 3. ELEMENTARY FUNCTIONS;97
10.1;24. The functions w = zn and their Riemann surfaces;97
10.2;25. The concept of a regular branch. The functions w = n/z;102
10.3;26. The function w = 1/2[z+(1/z)] and its Riemann surface;106
10.4;27. Examples;109
10.5;28. The Joukowski profile;115
10.6;29. The exponential function and its Riemann surface;118
10.7;30. The logarithmic function;120
10.8;31. Trigonometrical and hyperbolic functions;122
10.9;32. The general power;127
10.10;33. Examples;129
10.11;Exercises;133
11;CHAPTER 4. APPLICATIONS TO THE THEORY OF PLANE FIELDS;136
11.1;34. Plane vector fields;136
11.2;35. Examples of plane fields;138
11.3;36. Properties of plane vector fields;142
11.4;37. The force function and potential function;147
11.5;38. The complex potential in electrostatics;156
11.6;39. The complex potential in hydrodynamics and heat conduction;162
11.7;40. The method of conformal mapping;167
11.8;41. The field in a strip;169
11.9;42. The field in a ring domain;172
11.10;43. Streamlining an infinite curve;176
11.11;44. The problem of complete streamlining. Chaplygin's condition;179
11.12;45. Other methods;186
11.13;Exercises;191
12;CHAPTER 5. THE INTEGRAL REPRESENTATION OF A REGULAR FUNCTION. HARMONIC FUNCTIONS;193
12.1;46. The integral of a function of a complex variable;193
12.2;47. Cauchy's integral theorem;195
12.3;48. Cauchy's residue theorem. Chaplygin's formula;200
12.4;49. The indefinite integral;205
12.5;50. Integration of powers of (z—a);208
12.6;51. Cauchy's integral formula;212
12.7;52. The existence of higher derivatives;214
12.8;53. Properties of regular functions;217
12.9;54. Harmonic functions;222
12.10;55. Dirichlet's problem;226
12.11;56. The integrals of Poisson and Schwarz;232
12.12;57. Applications to the theory of plane fields;235
12.13;Exercises;241
13;CHAPTER 6. REPRESENTATION OF REGULAR FUNCTIONS BY SERIES;244
13.1;58. Series in the complex domain;244
13.2;59. Weierstrass's theorem;247
13.3;60. Power series;250
13.4;61. Representation of regular functions by Taylor series;254
13.5;62. The zeros of a regular function. The uniqueness theorem;258
13.6;63. Analytic continuation. Analytic functions;260
13.7;64. Laurent series;266
13.8;65. Isolated singularities;275
13.9;66. Removable singularities;276
13.10;67. Poles;278
13.11;68. Essential singularities;283
13.12;69. Behaviour of a function at infinity;287
13.13;70. Joukowski's theorem on the thrust on an aerofoil;291
13.14;71. The simplest classes of analytic functions;297
13.15;Exercises;300
14;CHAPTER 7. APPLICATIONS OF THE THEORY OF RESIDUES;303
14.1;72. Evaluation of integrals of the form 2...;303
14.2;73. Integrals of the form ....;306
14.3;74. Other integrals;313
14.4;75. Integrals involving multi-valued functions;322
14.5;76. The representation of functions by integrals;331
14.6;77. The logarithmic residue;337
14.7;78. Expansion of cot z in simple fractions. Mittag-Leffler's theorem;344
14.8;79. Expansion of sin z as an infinite product. Weierstrass's theorem;350
14.9;80. Euler's gamma function G(z);357
14.10;81. Integral representations of the G-function;362
14.11;Exercises;368
15;CHAPTER 8. MAPPING OF POLYGONAL DOMAINS;371
15.1;82. The symmetry principle;371
15.2;83. Illustrative examples;376
15.3;84. The Schwarz–Christoffel integral;385
15.4;85. Degenerate cases;392
15.5;86. Illustrative examples;398
15.6;87. Determination of the field at the edges of a condenser. Rogowski's condenser;405
15.7;88. The field of angular electrodes;410
15.8;89. The mapping of rectangular domains. Introduction to elliptic integrals;413
15.9;90. Introduction to Jacobian elliptic functions;417
15.10;Exercises;421
16;ANSWERS AND HINTS FOR SOLUTION OF EXERCISES;425
17;INDEX;446