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Complex Analysis
posted by
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on 4/2/2009
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Abstract/Syllabus:
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COMPLEX ANALYSIS1
Douglas N. Arnold2
References:
John B. Conway, Functions of One Complex Variable, Springer-Verlag, 1978.
Lars V. Ahlfors, Complex Analysis, McGraw-Hill, 1966.
Raghavan Narasimhan, Complex Analysis in One Variable, Birkh¨auser, 1985.
CONTENTS
| I. The Complex Number System.. . . . . . . . . . . . . . |
2 |
| II. Elementary Properties and Examples of Analytic Fns. . . . . |
3 |
| Differentiability and analyticity.. . . . . . . . . . . . |
4 |
| The Logarithm.... . . . . . . . . . . . |
6 |
| Conformality.... . . . . . . . . . . . . . |
6 |
| Cauchy–Riemann Equations.. . . . . . . . . . . . . . . . |
7 |
| M¨obius transformations... . . . |
9 |
| III. Complex Integration and Applications to Analytic Fns. . . |
11 |
| Local results and consequences.. . . . . . . . . . . . |
12 |
| Homotopy of paths and Cauchy’s Theorem.. . . . . . . . . . . . |
14 |
| Winding numbers and Cauchy’s Integral Formula. . . . . . . . . |
15 |
| Zero counting; Open Mapping Theorem.. . . |
17 |
| Morera’s Theorem and Goursat’s Theorem. |
18 |
| IV. Singularities of Analytic Functions... . . . |
19 |
| Laurent series... . . . . . . . . . . . . |
20 |
| Residue integrals... . . . . . . . . . |
23 |
| V. Further results on analytic functions... |
26 |
| The theorems of Weierstrass, Hurwitz, and Montel. . . . . . . . |
26 |
| Schwarz’s Lemma... . . . . . . . . |
28 |
| The Riemann Mapping Theorem.. . . . . . . . . . |
29 |
| Complements on Conformal Mapping.. . . . . |
31 |
| VI. Harmonic Functions... . . . . . . |
32 |
| The Poisson kernel... . . . . . . . |
33 |
| Subharmonic functions and the solution of the Dirichlet Problem |
36 |
| The Schwarz Reflection Principle.. . . . . . . . . . |
39 |
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