Introduction to Complex AnalysisComplex analysis is a classic and central area of mathematics, which is studies 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 sets have been substantially revised and enlarged, with carefully graded exercises at the end of each chapter. |
Contents
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 |
| 319 | |
| 321 | |
| 323 | |
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Common terms and phrases
argz bounded branch points Cauchy-Riemann equations Cauchy's integral formula Cauchy's residue theorem Cauchy's theorem Chapter circline circular arcs closed path complex analysis complex numbers conformal mapping consider constant continuous function defined definition derivatives evaluate example Exercise exists f be holomorphic finite Fourier transform function ƒ ƒ is holomorphic G₁ geometric given half-plane Hence holomorphic branch holomorphic function holomorphic inside integral round Jordan's inequality Laplace transform Laurent expansion lemma Let f Let ƒ Let G limit point line segments logarithm Möbius transformation multibranches multifunction obtain open disc open set parameter interval pole of order polygonal polynomial power series proof Prove that ƒ radius of convergence real axis region G reiº residue theorem result Riemann sequence simple pole singularity subset Suppose that ƒ Tactical tip uniform convergence