Complex Analysis - Maths KU

x Preface
is guided through the starting steps or a strong hint on how to proceed is
provided.
The writing herein is straightforward and reflects the no-nonsense style
of Advanced Engineering Mathematics.
Content We have purposely limited the number of chapters in this text
to seven. This was done for two “reasons”: to provide an appropriate quantity
of material so that most of it can reasonably be covered in a one-term course,
and at the same time to keep the cost of the text within reason.
Here is a brief description of the topics covered in the seven chapters.
• Chapter 1 The complex number system and the complex plane are
examined in detail.
• Chapter 2 Functions of a complex variable, limits, continuity, and
mappings are introduced.
• Chapter 3 The all-important concepts of the derivative of a complex
function and analyticity of a function are presented.
• Chapter 4 The trigonometric, exponential, hyperbolic, and logarithmic
functions are covered. The subtle notions of multiple-valued functions
and branches are also discussed.
• Chapter 5 The chapter begins with a review of real integrals (including
line integrals). The definitions of real line integrals are used to
motivate the definition of the complex integral. The famous Cauchy-
Goursat theorem and the Cauchy integral formulas are introduced in
this chapter. Although we use Green’s theorem to prove Cauchy’s theorem,
a sketch of the proof of Goursat’s version of this same theorem is
given in an appendix.
• Chapter 6 This chapter introduces the concepts of complex sequences
and infinite series. The focus of the chapter is on Laurent series, residues,
and the residue theorem. Evaluation of complex as well as real integrals,
summation of infinite series, and calculation of inverse Laplace and inverse
Fourier transforms are some of the applications of residue theory
that are covered.
• Chapter 7 Complex mappings that are conformal are defined and
used to solve certain problems involving Laplace’s partial differential
equation.
Features Each chapter begins with its own opening page that includes a
table of contents and a brief introduction describing the material to be covered
in the chapter. Moreover, each section in a chapter starts with introductory
comments on the specifics covered in that section. Almost every section
ends with a feature called Remarks in which we talk to the students about
areas where real and complex calculus differ or discuss additional interesting
topics (such as the Riemann sphere and Riemann surfaces) that are related