
Fundamentals of Complex Analysis, 3/e
Edward Saff
Arthur D. Snider, both of the University of South Florida
Coming March, 2000 by Prentice Hall Engineering/Science/Mathematics
Copyright 2000, 495 pp.
Cloth
ISBN 0139078746

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For one/twosemester courses in Complex Analysis and Mathematical
Physics in departments of mathematics and engineering.
This book provides a comprehensive introduction to complex
variable theory and its applications to current engineering problems.
It is designed to make the fundamentals of the subject more easily
accessible to students who have little inclination to wade through
the rigors of the axiomatic approach. Modeled after standard calculus
books—both in level of exposition and layout—it incorporates
physical applications throughout the presentation, so that
the mathematical methodology appears less sterile to engineering students.
NEW—Downloadable MATLAB toolbox—A
stateoftheart computer aid.
 Enables visualization of complex arithmetic operations
and conformal maps using a graphicsbased computer program. Animations,
color, and accurate drawings enhance lecture presentations.
NEW—Section on Julia sets—The graphical
depiction of iterated complex functions leads to interesting fractal
patterns.
 Familiarizes students with a recent topic in complex
analysis research.
Physical interpretation of properties of analytic function
as equilibrium temperature profiles—Properties such as maximum
principles, boundary value determinations, and mean values are well
known for temperatures.
 Ex. Sections 2.4, 4.7, 7.6
Two alternative presentations of Cauchy's theorem are
given
(Ch. 4)—The first is based on the deformation of contours (homotopy).
The second interprets contour integrals in terms of line integrals
and invokes Green's theorem to complete the argument. These developments
are presented parallel to one another. Either one may be read, and
the other omitted, without disrupting the exposition.
 Provides an alternative approach that is understandable
and transparent to the novice because it is easy to visualize and
to apply in specific situations.
Modern exposition of the use of complex numbers in
linear analysis—AC circuits, kinematics, signal processing.
 Provides students with an alternative perspective on
these applications that is often more transparent than their ad hoc
introduction in engineering courses. Ex. Sections 1.7, 3.4, Ch. 8
Frequent use of analogies from elementary calculus or
algebra to introduce complex concepts.
 Helps students to anticipate results and to tie them
to familiar ideas.
New concepts illustrated with diagrams prior to rigorous
analyses—Whenever possible.
 Aids students in forming concrete interpretations.
Highly readable exposition on numerical conformal mapping—Draws
the distinction between direct methods, wherein a mapping must be
constructed to solve a specific problem, and indirect methods that
postulate a mapping and then investigate which problems it solves.
Dispels the impression, given in many older books, that all applications
of the technique fall in the latter category. An appendix describes
Trefethen's numerical implementation of the SchwarzChristoffel transformation.
Another appendix includes an easy touse table of known, closedform
conformal maps.
 Important for modern applications; relates to other areas
of mathematics.
Early introduction of Euler's formula.
 Obviates the clumsy “cis” notation.
A selfcontained approach—Includes all the proofs
that reflect the spirit of analytic function theory, and omits most
of those that involve deeper results from real analysis (e.g., does
not prove the convergence of Riemann sums for complex integrals, the
Cauchy criterion for convergence of a series, Goursat's generalization
of Cauchy's theorem, or the Riemann mapping theorem). Also shuns the
orderedpairs interpretations of complex numbers in favor of the more
intuitive approach (grounded in algebraic field extensions).
 Maintains the authors' philosophy of avoiding pedantics.
Applications to “real world” engineering problems.
Linear systems analysis used as a recurring application—The
basic ideas of frequency analysis are introduced in Ch. 3 following
the study of the transcendental functions; Smith charts, circuit synthesis,
and stability criteria are addressed at appropriate times; and the
development culminates in Ch. 8 with the exposition of Fourier, Mellin,
Laplace, Hilbert, and ztransforms. Proofs of the transform representations
are offered for those cases where analytic function theory applies.
Thorough coverage of the uses of residue theory in evaluating
integrals.
Optional sections (marked *).
 Enrich student understanding of either the mathematical
development or the physical interpretation.
About 150 fullyworked examples.
Summary and Suggested Reading sections—At the
end of each chapter.
Applications of complex algebra in celestial mechanics
and gear kinematics.
1. Complex Numbers.
2. Analytic Functions.
3. Elementary Functions.
4. Complex Integration.
5. Series Representations for Analytic Functions.
6. Residue Theory.
7. Conformal Mapping.
8. The Transforms of Applied Mathematics.
Appendix I. MATLAB ToolBox for Visualization of Conformal
Maps.
Appendix II. Numerical Construction of Conformal Maps.
Appendix III. Table of Conformal Mappings.
Answers.
Index.
