Theory of Complex FunctionsThe material from function theory, up to the residue calculus, is developed in a lively and vivid style, well motivated throughout by examples and practice exercises. Additionally, there is ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations (original language together with English translation) from their classical works. Yet the book is far from being a mere history of function theory. Even experts will find here few new or long forgotten gems, like Eisenstein's novel approach to the circular functions. This book is destined to accompany many students making their way into a classical area of mathematics which represents the most fruitful example to date of the intimate connection between algebra and analysis. For exam preparation it offers quick access to the essential results and an abundance of interesting inducements. Teachers and interested mathematicians in finance, industry and science will also find reading it profitable, again and again referring to it with pleasure. |
Contents
Historical Introduction | 1 |
Part A Elements of Function Theory | 9 |
2 Fundamental topological concepts | 17 |
4 Convergent and absolutely convergent series | 26 |
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 | 65 |
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 | 253 |
Miscellany | 265 |
3 Holomorphic logarithms and holomorphic roots | 276 |
Logarithmic derivative Existence lemma 2 Homologically simply | 283 |
6 Asymptotic power series developments | 297 |
Holomorphy and Conformality Biholomorphic Mappings | 71 |
2 Biholomorphic mappings | 80 |
n for automorphisms of E4 Homogeneity of E and | 91 |
2 Convergence criteria | 98 |
3 Normal convergence of series | 104 |
2 Examples of convergent power series | 117 |
Formal termwise differentiation and integration | 123 |
4 Logarithm functions | 154 |
5 Discussion of logarithm functions | 160 |
Complex Integral Calculus | 167 |
2 Properties of complex path integrals | 179 |
The Integral Theorem Integral Formula and Power Series | 191 |
2 Cauchys Integral Formula for discs | 201 |
3 The development of holomorphic functions into power series | 208 |
4 Discussion of the representation theorem | 214 |
Isolated Singularities Meromorphic Functions | 303 |
281 | 309 |
2 Automorphisms of punctured domains | 310 |
Convergent Series of Meromorphic Functions | 321 |
2 The partial fraction development of π сot | 329 |
4 The EISENSTEIN theory of the trigonometric functions | 335 |
Laurent Series and Fourier Series | 343 |
3 Periodic holomorphic functions and Fourier series | 361 |
The Residue Calculus | 377 |
2 Consequences of the residue theorem | 387 |
2 Further evaluation of integrals | 401 |
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 bijection boundary BR(c C-algebra calculus called Cauchy integral formula Cauchy integral theorem Cauchy-Riemann equations Cauchy's compact complex numbers complex-differentiable functions concept consequently continuation theorem continuous function continuously differentiable convergent sequence convergent series converges compactly converges normally converges uniformly criterion defined denote domain equivalent EULER example Exercises Exercise exists exponential finite follows func function f function theory GAUSS Historical remarks holomorphic functions Identity Theorem infinite series integral theorem lemma Let f lim cn limit function locally constant logarithm function Math mathematical mathematician metric space neighborhood normally convergent open disc polynomial power series Proof proved R-linear radius of convergence real numbers real-differentiable region G respect RIEMANN satisfies sequence fn Show subset Taylor series tion uniform convergence unit disc WEIERSTRASS Werke zero zero-free ας მყ