1st Edition

An Introduction to Complex Analysis and the Laplace Transform

By Vladimir Eiderman Copyright 2022
    398 Pages 112 B/W Illustrations
    by Chapman & Hall

    398 Pages 112 B/W Illustrations
    by Chapman & Hall

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    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.


    1. 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



    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.