Introduction to Complex Analysis

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OUP Oxford, 28.08.2003 - 344 Seiten
Complex analysis is a classic and central area of mathematics, which is studied and exploited in a range of important fields, from number theory to engineering. Introduction to Complex Analysis was first published in 1985, and for this much awaited second edition the text has been considerably expanded, while retaining the style of the original. More detailed presentation is given of elementary topics, to reflect the knowledge base of current students. Exercise setshave been substantially revised and enlarged, with carefully graded exercises at the end of each chapter.This is the latest addition to the growing list of Oxford undergraduate textbooks in mathematics, which includes: Biggs: Discrete Mathematics 2nd Edition, Cameron: Introduction to Algebra, Needham: Visual Complex Analysis, Kaye and Wilson: Linear Algebra, Acheson: Elementary Fluid Dynamics, Jordan and Smith: Nonlinear Ordinary Differential Equations, Smith: Numerical Solution of Partial Differential Equations, Wilson: Graphs, Colourings and the Four-Colour Theorem, Bishop: Neural Networks forPattern Recognition, Gelman and Nolan: Teaching Statistics.
 

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Inhalt

1 The complex plane
1
2 Geometry in the complex plane
12
3 Topology and analysis in the complex plane
30
4 Paths
47
5 Holomorphic functions
56
6 Complex series and power series
67
7 A cornucopia of holomorphic functions
78
8 Conformal mapping
91
15 Zeros of holomorphic functions
176
further theory
188
17 Singularities
194
18 Cauchys residue theorem
211
19 A technical toolkit for contour integration
221
20 Applications of contour integration
234
21 The Laplace transform
256
22 The Fourier transform
278

9 Multifunctions
107
10 Integration in the complex plane
119
basic track
128
advanced track
142
13 Cauchys formulae
151
14 Power series representation
161
23 Harmonic functions and conformal mapping
289
new perspectives
309
Bibliography
319
Notation index
321
Index
323
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