Theory of Complex FunctionsA lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations - in the original language with their English translation - from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure. |
Contents
| 1 | |
| 9 | |
3 Convergent sequences of complex numbers | 22 |
6 Connected spaces Regions in C | 39 |
ComplexDifferential Calculus | 45 |
3 Holomorphic functions | 56 |
4 Partial differentiation with respect to x y z and ž | 63 |
Holomorphy and Conformality Biholomorphic Mappings | 71 |
3 The development of holomorphic functions into power series | 208 |
4 Discussion of the representation theorem | 214 |
5 Special Taylor series Bernoulli numbers | 220 |
CauchyWeierstrassRiemann Function Theory | 227 |
3 The Cauchy estimates and inequalities for Taylor coefficients | 241 |
4 Convergence theorems of WEIERSTRASS | 248 |
5 The open mapping theorem and the maximum principle | 256 |
Miscellany | 265 |
2 Biholomorphic mappings | 78 |
Modes of Convergence in Function Theory | 91 |
2 Convergence criteria | 101 |
Power Series | 109 |
2 Examples of convergent power series | 115 |
sinn | 118 |
3 Holomorphy of power series | 123 |
2 The epimorphism theorem for expz and its consequences | 141 |
3 Polar coordinates roots of unity and natural boundaries | 148 |
4 Logarithm functions | 154 |
5 Discussion of logarithm functions | 165 |
2 Properties of complex path integrals | 178 |
The Integral Theorem Integral Formula and Power Series | 191 |
2 Cauchys Integral Formula for discs | 201 |
3 Holomorphic logarithms and holomorphic roots | 276 |
Isolated Singularities Meromorphic Functions | 303 |
2 Automorphisms of punctured domains | 310 |
Convergent Series of Meromorphic Functions | 321 |
4 The EISENSTEIN theory of the trigonometric functions | 335 |
409 | 339 |
Laurent Series and Fourier Series | 343 |
2 Properties of Laurent series | 356 |
The Fourier development in strips 4 Examples 5 Historical | 364 |
The Residue Calculus | 377 |
2 Consequences of the residue theorem | 387 |
Short Biographies of ABEL CAUCHY EISENSTEIN EULER RIEMANN | 417 |
| 435 | |
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Common terms and phrases
absolutely convergent addition theorem algebra angle-preserving automorphisms biholomorphic mapping boundary Br(c calculus called Cauchy integral formula Cauchy integral theorem Cauchy-Riemann Cauchy's closed path compact complex numbers concept continuous function continuously differentiable convergent sequence convergent series converges compactly converges normally converges uniformly criterion defined derivatives domain equivalent EULER example Exercises Exercise exists f is holomorphic finite follows func function f function theory GAUSS holomorphic functions Identity Theorem infinite injective integral theorem Laurent series lemma Let f lim cn limit function locally constant logarithm function Math meromorphic functions metric space neighborhood normally convergent open disc pole polynomial power series Proof proved R-linear radius of convergence real numbers real-differentiable RIEMANN satisfies sequence fn Show singularity subset Taylor series uniform convergence unit disc WEIERSTRASS Werke zero zero-free ας Σαν Σπί