An Introduction to Complex Analysis and the Laplace Transform  book cover
1st Edition

An Introduction to Complex Analysis and the Laplace Transform

ISBN 9780367409784
Published December 21, 2021 by Chapman and Hall/CRC
398 Pages 112 B/W Illustrations

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Book Description

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.

Table of Contents


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


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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.