Basic Complex Analysis Book

	Preface	vii
1	Analytic Functions	1
1.1	Introduction to Complex Numbers	1
1.2	Properties of Complex Numbers	12
1.3	Some Elementary Functions	25
1.4	Continuous Functions	41
1.5	Basic Properties of Analytic Functions	60
1.6	Differentiation of the Elementary Functions	81
2	Cauchy's Theorem	95
2.1	Contour Integrals	95
2.2	Cauchy's Theorem--A First Look	111
2.3	A Closer Look at Cauchy's Theorem	123
2.4	Cauchy's Integral Formula	144
2.5	Maximum Modulus Theorem and Harmonic Functions	164
3	Series Representation of Analytic Functions	183
3.1	Convergent Series of Analytic Functions	184
3.2	Power Series and Taylor's Theorem	203
3.3	Laurent Series and Classification of Singularities	222
4	Calculus of Residues	243
4.1	Calculation of Residues	243
4.2	Residue Theorem	256
4.3	Evaluation of Definite Integrals	269
4.4	Evaluation of Infinite Series and Partial-Fraction Expansions	304
5	Conformal Mappings	319
5.1	Basic Theory of Conformal Mappings	319
5.2	Fractional Linear and Schwarz-Christoffel Transformations	327
5.3	Applications of Conformal Mappings to Laplace's Equation, Heat Conduction, Electrostatics, and Hydrodynamics	345
6	Further Development of the Theory	365
6.1	Analytic Continuation and Elementary Riemann Surfaces	365
6.2	Rouche's Theorem and Principle of the Argument	384
6.3	Mapping Properties of Analytic Functions	398
7	Asymptotic Methods	409
7.1	Infinite Products and the Gamma Function	409
7.2	Asymptotic Expansions and the Method of Steepest Descent	427
7.3	Stirling's Formula and Bessel Functions	446
8	Laplace Transform and Applications	457
8.1	Basic Properties of Laplace Transforms	457
8.2	Complex Inversion Formula	471
8.3	Application of Laplace Transforms to Ordinary Differential Equations	476
	Answers to Odd-Numbered Exercises	481
	Index	506