Complex Analysis

 Complex Analysis

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Complex Analysis

This book is a revision of the seventh edition, which was published in 2004. That

edition has served, just as the earlier ones did, as a textbook for a one-term intro

ductory course in the theory and application of functions of a complex variable.

This new edition preserves the basic content and style of the earlier editions, the

first two of which were written by the late Ruel V. Churchill alone.

The first objective of the book is to develop those parts of the theory that are

prominent in applications of the subject. The second objective is to furnish an intro

duction to applications of residues and conformal mapping. With regard to residues.

special emphasis is given to their use in evaluating real improper integrals, finding

inverse Laplace transforms, and locating zeros of functions. As for conformal map

ping, considerable attention is paid to its use in solving boundary value problems

that arise in studies of heat conduction and fluid flow. Hence the book may be

considered as a companion volume to the authors' text "Fourier Series and Bound

ary Value Problems," where another classical method for solving boundary value

problems in partial differential equations is developed.

The first nine chapters of this book have for many years formed the basis of a

three-hour course given each term at The University of Michigan. The classes have

consisted mainly of seniors and graduate students concentrating in mathematics,

engineering, or one of the physical sciences. Before taking the course, the students

have completed at least a three-term calculus sequence and a first course in ordinary

differential equations. Much of the material in the book need not be covered in the

lectures and can be left for self-study or used for reference. If mapping by elementary

functions is desired earlier in the course, one can skip to Chap. 8 immediately after

Chap. 3 on elementary functions.

In order to accommodate as wide a range of readers as possible, there are foot

notes referring to other texts that give proofs and discussions of the more delicate

results from calculus and advanced calculus that are occasionally needed. A bibli

ography of other books on complex variables, many of which are more advanced,

is provided in Appendix 1. A table of conformal transformations that are useful in

applications appears in Appendix 2.

The main changes in this edition appear in the first nine chapters. Many of

those changes have been suggested by users of the last edition. Some readers have

urged that sections which can be skipped or postponed without disruption be more

clearly identified. The statements of Taylor's theorem and Laurent's theorem, for

example, now appear in sections that are separate from the sections containing

their proofs. Another significant change involves the extended form of the Cauchy

integral formula for derivatives. The treatment of that extension has been completely

rewritten, and its immediate consequences are now more focused and appear together

in a single section.

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