The Elements of Complex Analysis, 9780471225652, 0471225657

Table of contents :
Preface
Contents
1 METRIC SPACE PRELIMINARIES
1.1 Set theoretic notation and terminology
1.2 Elementary properties of metric spaces
1.3 Continuous functions on metric spaces
1.4 Compactness
1.5 Completeness
1.6 Connectedness
2 THE COMPLEX NUMBERS
2.1 Definitions and notation
2.2 Domains in the complex plane
2.3 The extended complex plane
3 CONTINUOUS AND DIFFERENTIABLE COMPLEX FUNCTIONS
3.1 Continuous complex functions
3.2 Differentiable complex functions
3.3 The Cauchy-Riemann equations
3.4 Harmonic functions of two real variables
4 POWER SERIES FUNCTIONS
4.1 Infinite series of complex numbers
4.2 Double sequences of complex numbers
4.3 Power series functions
4.4 The exponential function
4.5 Branches-of-log
5 ARCS, CONTOURS, AND INTEGRATION
5.1 Arcs
5.2 Oriented arcs
5.3 Simple closed curves
5.4 Oriented simple closed curves
5.5 The Jordan curve theorem
5.6 Contour integration
6 CAUCHY'S THEOREM FOR STARLIKE DOMAINS
6.1 Cauchy's theorem for triangular contours
6.2 Cauchy's theorem for starlike domains
6.3 Applications
7 LOCAL ANALYSIS
7.1 Cauchy's integral formulae
7.2 Taylor expansions
7.3 The Laurent expansion
7.4 Isolated singularities
8 GLOBAL ANALYSIS
8.1 Taylor expansions revisited
8.2 Properties of zeros
8.3 Entire functions
8.4 Meromorphic functions
8.5 Convergence in d(D)
8.6 Weierstrass expansions
8.7 Topological index
8.8 Cauchy's residue theorem
8.9 Mittag-Leffler expansions
8.10 Zeros and poles revisited
8.11 The open mapping theorem
8.12 The maximum modulus principle
9 CONFORMAL MAPPING
9.1 Discussion of the Riemann mapping theorem
9.2 The automorphisms of a domain
9.3 Mappings of the boundary
9.4 Some illustrative mappings
10 ANALYTIC CONTINUATION
10.1 Direct analytic continuations
10.2 General analytic functions
10.3 Complex analytic manifolds
10.4 The gamma and zeta functions
APPENDIX: RIEMANN-STIELTJES INTEGRATION
SUGGESTIONS FOR FURTHER STUDY
BIBLIOGRAPHY
INDEX OF SPECIAL SYMBOLS
SUBJECT INDEX