An Introduction to Complex Analysis: Classical and Modern Approaches

by Tutschke; Wolfgang

• ISBN13: 9781584884781
• ISBN10: 1584884789
• eBook ISBN(s): 9781040164235
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-06-25
• Publisher: Chapman & Hall/
• More Book Details

Summary

Like real analysis, complex analysis has generated methods indispensable to mathematics and its applications. Exploring the interactions between these two branches, this book uses the results of real analysis to lay the foundations of complex analysis and presents a unified structure of mathematical analysis as a whole.To set the groundwork and mitigate the difficulties newcomers often experience, An Introduction to Complex Analysis begins with a complete review of concepts and methods from real analysis, such as metric spaces and the Green-Gauss Integral Formula. The approach leads to brief, clear proofs of basic statements - a distinct advantage for those mainly interested in applications. Alternate approaches, such as Fichera's proof of the Goursat Theorem and Estermann's proof of the Cauchy's Integral Theorem, are also presented for comparison. Discussions include holomorphic functions, the Weierstrass Convergence Theorem, analytic continuation, isolated singularities, homotopy, Residue theory, conformal mappings, special functions and boundary value problems. More than 200 examples and 150 exercises illustrate the subject matter and make this book an ideal text for university courses on complex analysis, while the comprehensive compilation of theories and succinct proofs make this an excellent volume for reference.

Author Biography

Wolfgang Tutschke is a retired professor from Graz University of Technology in Graz, Austria Harkrishan L. Vasudeva is a retired professor from Panjab University in Chandigarh, India, and had been a visiting professor at the Graz University of Technology

Table of Contents

Preliminaries	p. 1
The field of complex numbers	p. 1
The complex plane	p. 6
Metric spaces	p. 14
Mappings and functions. Continuity	p. 28
Exercises to Chapter 1	p. 82
The classical approach	p. 87
Ordinary complex differentiation	p. 87
Preliminaries of the Integral Calculus	p. 102
Complex Integral Theorems	p. 112
Exercises to Chapter 2	p. 116
An alternative approach	p. 121
Partial complex differentiations	p. 121
Complex Green-Gauss Integral Theorems	p. 123
Generalized Cauchy Integral Formula	p. 125
The classical Cauchy Integral Formula	p. 129
Comparison	p. 130
Exercises to Chapter 3	p. 131
Local properties	p. 135
Existence of higher order derivatives	p. 135
Local power series representation	p. 138
Distribution of zeros	p. 144
The Weierstrass Convergence Theorem	p. 145
Connexion with plane Potential Theory	p. 147
Complex Integral Theorems revisited *	p. 148
Exercises to Chapter 4	p. 163
Global properties	p. 169
Analytic continuation	p. 169
Maximum Modulus Principle (Maximum Principle)	p. 182
Entire functions	p. 186
Fundamental Theorem of Algebra	p. 188
Exercises to Chapter 5	p. 195
Isolated singularities	p. 199
Classification	p. 199
Laurent series	p. 202
Characterization by the principal part	p. 206
Meromorphic functions	p. 208
Behaviour at essential singularities	p. 210
Behaviour at infinity	p. 211
Partial fractions of rational functions	p. 213
Meromorphic functions on the Sphere	p. 216
Exercises to Chapter 6	p. 217
Homotopy	p. 223
Statement of the problem	p. 223
Homotopic curves	p. 225
Path independent line integrals	p. 227
Simply connected domains	p. 229
Solution of first order systems	p. 232
Conjugate solutions	p. 234
Inversion of complex differentiation	p. 235
Morera's Theorem	p. 237
Potentials of vector fields	p. 238
Exercises to Chapter 7	p. 241
Residue theory	p. 245
Statement of the problem	p. 245
Winding numbers	p. 245
The integration of principal parts	p. 246
Residue Theorem	p. 249
Calculation of residues	p. 251
Exercises to Chapter 8	p. 254
Applications of residue calculus	p. 257
Total number of zeros and poles	p. 257
Evaluation of definite integrals	p. 260
Sum of certain series	p. 277
Exercises to Chapter 9	p. 280
Mapping properties	p. 287
Continuously differentiable mappings	p. 287
Conformal mappings	p. 293
Examples of conformal mappings	p. 299
Univalent functions *	p. 316
Riemann's Mapping Theorem *	p. 321
Construction of flow lines *	p. 328
Exercises to Chapter 10	p. 332
Special functions	p. 339
Prescribed principal parts	p. 339
Prescribed zeros	p. 346
Infinite products *	p. 348
Weierstrass products *	p. 362
Gamma function *	p. 369
The Riemann zeta function *	p. 381
Elliptic functions *	p. 388
Exercises to Chapter 11	p. 416
Boundary value problems	p. 425
Preliminaries	p. 425
The Poisson Integral Formula	p. 428
Cauchy Type Integrals *	p. 436
Desired holomorphic functions *	p. 447
Exercises to Chapter 12	p. 450
References	p. 455
Index	p. 457
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