Basic Complex Analysis

Basic Complex Analysis skillfully combines a clear exposition of core theory with a rich variety of applications. Designed for undergraduates in mathematics, the physical sciences, and engineering who have completed two years of calculus and are taking complex analysis for the first time. . Booknews This textbook introduces complex numbers, analytic functions, Cauchy's Theorem, the calculus of residues, conformal mappings, asymptotic methods, and Laplace transforms. For the third edition, the evaluation of integrals has been rewritten to make the treatment less encyclopedic. Annotation c. Book News, Inc., Portland, OR (booknews.com) Contents Preface Chapter 1: Analytic Functions 1.1 Introduction to Complex Numbers 1.2 Properties of Complex Numbers 1.3 Some Elementary Functions 1.4 Continous Functions 1.5 Basic Properties of Analytic Functions 1.6 Differentiation of the Elementary Functions Review Excersices for Chapter 1 Chapter 2: Cauchy's Theorem 2.1 Contour Integrals 2.2 Cauchy's Theorem - A First Look 2.3 A Closer Look at Cauchy's Theorem 2.4 Cauchy's Integral Formula 2.5 Maximum Modulus Theorem and Harmonic Functions Review Excersices for Chapter 2 Chapter 3: Series Representation of Analytic Functions 3.1 Convergent Series of Analytic Functions 3.2 Power Series and Taylor's Theorem 3.3 Laurent Series and Classification of Singularities Review Excersices for Chapter 3 Chapter 4: Calculus of Residues 4.1 Calculation of Residues 4.2 Residue Theorem 4.3 Evaluation of Definite Integrals 4.4 Evaluation of Infinite Series and Partial-Fraction Expansions Review Excersices for Chapter 4 Chapter 5: Conformal Mappings 5.1 Basic Theory of Conformal Mappings 5.2 Fractional Linear and Schwarz-Cristoffel Transformations 5.3 Applications of Conformal Mappings to Laplace's Equation, Heat Conduction, Electrostatics, and Hydrodynamics Review Excersices for Chapter 5 Chapter 6: Further Development of the Theory 6.1 Analytic Continuation and Elementary Riemann Surfaces 6.2 Rouché's Theorem and Principle of the Argument 6.3 Mapping Properties of Analytic Functions Review Excersices for Chapter 6 Chapter 7: Asymptotic Methods 7.1 Infinite Products and the Gamma Function 7.2 Asymptotic Expansions and the Method of Steepest Descent 7.3 Stirling's Formula and Bessel Functions Review Excersices for Chapter 7 Chapter 8: Laplace Transform and Applications 8.1 Basic Properties of Laplace Transforms 8.2 Complex Inversion Formula 8.3 Application of Laplace Transforms to Ordinary Differential Equations Review Excersices for Chapter 8 Answers to Odd-Numbered Excercises Index ""Basic Complex Analysis" skillfully combines a clear exposition of core theory with a rich variety of applications. Designed for undergraduates in mathematics, the physical sciences, and engineering who have completed two years of calculus and are taking complex analysis for the first time"--Amazon.com