An Introduction to Complex Analysis and the Laplace Transform

The aim of this comparatively short textbook is a sufficiently full exposition of the fundamentals of the theory of functions of a complex variable to prepare the student for various applications. Several important applications in physics and engineering are considered in the book. This thorough presentation includes all theorems (with a few exceptions) presented with proofs. No previous exposure to complex numbers is assumed. The textbook can be used in one-semester or two-semester courses. In one respect this book is larger than usual, namely in the number of detailed solutions of typical problems. This, together with various problems, makes the book useful both for self- study and for the instructor as well. A specific point of the book is the inclusion of the Laplace transform. These two topics are closely related. Concepts in complex analysis are needed to formulate and prove basic theorems in Laplace transforms, such as the inverse Laplace transform formula. Methods of complex analysis provide solutions for problems involving Laplace transforms. Complex numbers lend clarity and completion to some areas of classical analysis. These numbers found important applications not only in the mathematical theory, but in the mathematical descriptions of processes in physics and engineering.

Preface Introduction1. Complex Numbers and Their Arithmetic Operations with Complex Numbers 2. Functions of a Complex Variable Complex Plane Sequences of Complex Numbers and Their Limits Functions of Complex Variable. Limit and Continuity 3. Differentiation of Functions of a Complex Variable Derivative and Differential The Cauchy-Riemann Equations Analytic Functions Relations between Analytic and Harmonic Functions Geometric Interpretation of a Derivative of a Function of Complex Variable Notion of a Conformal Mapping 4. Conformal Mappings Linear and Linear Fractional Functions Power Function Notion of a Riemann Surface Exponential and Logarithmic Functions General Power Function and Trigonometric Functions Zukowski Function General Properties of Conformal Mappings An Application of Functions of Complex Variable in Hydrodynamics 5. Integration of Functions of a Complex Variable Integral of a Function of Complex Variable Cauchy Theorem Indefinite Integral Fundamental Theorem of Calculus Cauchy’s Integral Formula and Their Applications 6. Series Number Series Functional Series Uniform Convergence Power series Expansion of Functions in Power Series Taylor Series Uniqueness Property Analytic Continuation Complete Analytic Function Loran Series 7. Isolated Singularities and Residues Classification of Isolated Singular Points Residue of a Function in an Isolated Singular Point Evaluation of Integrals by Means of Residues Logarithmic Residue and Argument Principle 8. The Laplace Transform Laplace Transform Basic Properties of the Laplace Transform Application of the Laplace Transform to Solving Ordinary Differential Equations 9. Practicum Solving of Typical Problems Tasks for Self-Study Literature Index

Vladimir Eiderman holds a Ph.D. from Mathematical Institute of Academy of Sciences, Armenian SSR. He is Rothrock Lecturer of Indiana University. He has been Professor, Moscow State University of Civil Engineering, Visiting Professor of University of Kentucky, University of Wisconsin-Madison, and Indiana University. Dr. Eiderman has more than 30 research publications.