An Introduction to Complex Analysis and the Laplace Transform

Vladimir Eiderman

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000511123
• Edition: 1
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

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.

1Complex Numbers and Their Arithmetic

DOI: 10.1201/​9780367810283-1

1.1 Complex Numbers

A complex numberz is an ordered pair $(x,y)$ of real numbers x and y. That $(x,y)$ is an ordered pair means that if $x\cancel{=}y$ then $(x,y)$ is distinct from $(y,x)$; in other words not just the values of x and y are important in identifying the pair, but also the order in which they are listed.

Complex numbers have a simple geometric meaning. Introduce a system of rectangular Cartesian coordinates into the plane (Fig. 1). Then every pair $z=(x,y)$ of real numbers corresponds to a point in the plane with coordinates $(x,y)$, and conversely every point in the plane corresponds to a complex number $z=(x,y)$.

Fig. 1

The first component x of the complex number $z=(x,y)$ is called the real part of z and is denoted by $x=\text{Rez}$; the second component y is called the imaginary part of z and is denoted by $y=\text{Imz}$. The terms “real” and “imaginary” harken back to the uncomfortable early history of the theory of complex numbers; it is plain to us nowadays that the first coordinate of the point z is no more real than the second.

When the points are...