## 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

Cloth
ISBN 0-13-907874-6

mailings
on this subject.

Complex Analysis-Mathematics

Mathematical Physics-Physics/Astronomy

For one/two-semester 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.

• Enables visualization of complex arithmetic operations and conformal maps using a graphics-based 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 Schwarz-Christoffel transformation. Another appendix includes an easy- to-use table of known, closed-form conformal maps.
• Important for modern applications; relates to other areas of mathematics.
Early introduction of Euler's formula.
• Obviates the clumsy “cis” notation.
A self-contained 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 ordered-pairs 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 z-transforms. 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.
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.