Advanced Engineering Mathematics

A Second Course with MatLab

Dean G. Duffy

• Publisher: Chapman and Hall/CRC
• Year: 2022
• ISBN:
  9781000551303
• Edition: 1
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Through four previous editions of Advanced Engineering Mathematics with MATLAB, the author presented a wide variety of topics needed by today's engineers. The fifth edition of that book, available now, has been broken into two parts: topics currently needed in mathematics courses and a new stand-alone volume presenting topics not often included in these courses and consequently unknown to engineering students and many professionals.

The overall structure of this new book consists of two parts: transform methods and random processes. Built upon a foundation of applied complex variables, the first part covers advanced transform methods, as well as z-transforms and Hilbert transforms--transforms of particular interest to systems, communication, and electrical engineers. This portion concludes with Green's function, a powerful method of analyzing systems.

The second portion presents random processes--processes that more accurately model physical and biological engineering. Of particular interest is the inclusion of stochastic calculus.

The author continues to offer a wealth of examples and applications from the scientific and engineering literature, a highlight of his previous books. As before, theory is presented first, then examples, and then drill problems. Answers are given in the back of the book.

This book is all about the future: The purpose of this book is not only to educate the present generation of engineers but also the next.

"The main strength is the text is written from an engineering perspective. The majority of my students are engineers. The physical examples are related to problems of interest to the engineering students." -- Lea Jenkins, Clemson University

Chapter 1 Complex Variables

The theory of complex variables was originally developed by mathematicians as an aid in understanding functions. Functions of a complex variable enjoy many powerful properties that their real counterparts do not. That is not why we will study them. For us they provide the keys for the complete mastery of transform methods and differential equations.

In this chapter all of our work points to one objective: integration on the complex plane by the method of residues. For this reason we minimize discussions of limits and continuity, which play such an important role in conventional complex variables, in favor of the computational aspects. We begin by introducing some simple facts about complex variables. Then we progress to differential and integral calculus on the complex plane.

1.1 Complex Numbers

A complex number is any number of the form a + bi, where a and b are real and $i=\sqrt{-1}.$ We denote any member of a set of complex numbers by the complex variable z = x + iy. The real part of z, usually denoted by ℜ(z), is x while the imaginary part of z, ℑ(z), is y. The complex conjugate, z̅ or z*,...