Handbook of Complex Variables Book

	Preface
	List of Figures
1	The Complex Plane	1
1.1	Complex Arithmetic	1
1.2	The Exponential and Applications	7
1.3	Holomorphic Functions	12
1.4	The Relationship of Holomorphic and Harmonic Functions	16
2	Complex Line Integrals	19
2.1	Real and Complex Line Integrals	19
2.2	Complex Differentiability and Conformality	23
2.3	The Cauchy Integral Thoerem and Formula	26
2.4	A Coda on the Limitations of the Cauchy Integral Formula	28
3	Applications of the Cauchy Theory	31
3.1	The Derivatives of a Holomorphic Function	31
3.2	The Zeros of Holomorphic Function	36
4	Isolated Singularities and Laurent Series	41
4.1	The Behavior of a Holomorphic Function near an Isolated Singularity	41
4.2	Expansion around Singular Points	43
4.3	Examples of Laurent Expansions	46
4.4	The Calculus of Residues	48
4.5	Applications to the Calculation of Definite Integrals and Sums	51
4.6	Meromorphic Functions and Singularities at Infinity	63
5	The Argument Principle	69
5.1	Counting Zeros and Poles	69
5.2	The Local Geometry of Holomorphic Functions	73
5.3	Further Results on the Zeros of Holomorphic Functions	74
5.4	The Maximum Principle	76
5.5	The Schwarz Lemma	77
6	The Geometric Theory of Holomorphic Functions	79
6.1	The Idea of a Conformal Mapping	79
6.2	Conformal Mappings of the Unit Disc	80
6.3	Linear Fractional Transformations	81
6.4	The Riemann Mapping Theorem	86
6.5	Conformal Mappings of Annuli	87
7	Harmonic Functions	89
7.1	Basic Properties of Harmonic Functions	89
7.2	The Maximum Principle and the Mean Value Property	91
7.3	The Poisson Integral Formula	92
7.4	Regularity of Harmonic Functions	94
7.5	The Schwarz Reflection Principle	95
7.6	Harnack's Principle	97
7.7	The Dirichlet Problem and Subharmonic Functions	97
7.8	The General Solution of the Dirichlet Problem	101
8	Infinite Series and Products	103
8.1	Basic Concepts Concerning Infinite Sums and Products	103
8.2	The Weierstrass Factorization Theorem	109
8.3	The Theorems of Weierstrass and Mittag-Leffler	110
8.4	Normal Families	113
9	Applications of Infinite Sums and Products	117
9.1	Jensen's Formula and an Introduction to Blaschke Products	117
9.2	The Hadamard Gap Theorem	119
9.3	Entire Functions of Finite Order	120
10	Analytic Continuation	123
10.1	Definition of an Analytic Function Element	123
10.2	Analytic Continuation along a Curve	130
10.3	The Monodromy Theorem	131
10.4	The Idea of Riemann Surface	135
10.5	Picard's Theorems	140
11	Rational Approximation Theory	143
11.1	Runge's Theorem	143
11.2	Mergelyan's Theorem	146
12	Special Classes of Holomorphic Functions	149
12.1	Schlicht Functions and the Bieberbach Conjecture	149
12.2	Extension to the Boundary of Conformal Mappings	151
12.3	Hardy Spaces	152
13	Special Functions	155
13.0	Introduction	155
13.1	The Gamma and Beta Functions	155
13.2	Riemann's Zeta Function	158
13.3	Some Counting Functions and a Few Technical Lemmas	162
14	Applications that Depend on Conformal Mapping	163
14.1	Conformal Mapping	163
14.2	Application of Conformal Mapping to the Dirichlet Problem	164
14.3	Physical Examples Solved by Means of Conformal Mapping	168
14.4	Numerical Techniques of Conformal Mapping	175
15	Transform Theory	195
15.0	Introductory Remarks	195
15.1	Fourier Series	195
15.2	The Fourier Transform	202
15.3	The Laplace Transform	212
15.4	The z-Transform	214
16	Computer Packages for Studying Complex Variables	219
16.0	Introductory Remarks	219
16.1	The Software Packages	219
	Glossary of Terms from Complex Variable Theory and Analysis	231
	List of Notation	269
	Table of Laplace Transforms	273
	A Guide to the Literature	275
	References	279
	Index	283