Theory of Complex Functions

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Springer Science & Business Media, 1991 - Mathematics - 453 pages
1 Review
A 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.
  

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Contents

Historical Introduction
1
of continuity
9
2 Fundamental topological concepts
17
6 Connected spaces Regions in C
39
ComplexDifferential Calculus
45
3 Holomorphic functions
56
4 Partial differentiation with respect to x y z and z
65
Holomorphy and Conformality Biholomorphic Mappings
71
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
3 Holomorphic logarithms and holomorphic roots
276
6 Asymptotic power series developments
294
Isolated Singularities Meromorphic Functions
303

2 Biholomorphic mappings
80
Modes of Convergence in Function Theory
91
2 Convergence criteria
101
Chapter 4 Power Series
109
2 Examples of convergent power series
115
3 Holomorphy of power series
123
4 Logarithm functions
154
5 Discussion of logarithm functions
160
Part B The Cauchy Theory
167
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
5 Special Taylor series Bernoulli numbers
220
2 Automorphisms of punctured domains
310
Convergent Series of Meromorphic Functions
321
4 The Eisenstein theory of the trigonometric functions
335
Laurent Series and Fourier Series
343
2 Properties of Laurent series
356
4 The theta function
365
The Residue Calculus
377
2 Consequences of the residue theorem
387
Chapter 14 Definite Integrals and the Residue Calculus
395
Short Biographies oAbel Cauchy Eisenstein Euler Riemann
417
Literature
423
Symbol Index
435
Subject Index
443
Copyright

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About the author (1991)

Hans Grauert (b. 1930 in Harem /Ems, Germany) and Reinhold Remmert (b. 1930 in OsnabrA1/4ck, Germany) met at the University of MA1/4nster, where they both studied mathematics and physics from 1949 to 1954. In 1950 they were invited by Heinrich Behnke and Karl Stein to attend their "Oberseminar," which was held on Saturdays, for 2 hours from 9 a.m.

Five years after the tragic events of World War 2, Behnke's old friend Henri Cartan visited MA1/4nster. His lecture on recent developments in the theory of "Several Complex Variables" was a real eye-opener for the young students and had a strongly formative influence on them: indeed this was to determine the course of their scientific research careers from then on.

In June 1954 Grauert and Remmert received their respective doctorates from the University of MA1/4nster. In 1957 they both became lecturer (Privatdozent) there. In 1959 resp. 1960, Grauert and Remmert were appointed full professors at GAttingen resp. Erlangen.

The original German edition of "Theory of Stein Spaces" was written at a time when complex spaces, coherent analytic sheaves and the so-called Theorems A and B had already become established notions and theorems. Dedicated to Karl Stein, the book was published in 1977, and the English edition was to follow in 1979. The first announcement of the book, in Springer's promotion, consisted of the picture reproduced on the inside cover flap of this book, taken during the boat trip of the annual Bonn Arbeitstagung some time earlier, showing three men on a boat, with the minimalistic caption "Grundlehren 227.

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