Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics

by Saff, Edward B.; Snider, Arthur David

• ISBN13: 9780139078743
• ISBN10: 0139078746
• Edition: 3rd
• Format: Hardcover
• Copyright: 2002-12-31
• Publisher: Pearson

Summary

This book provides a comprehensive introduction to complex variable theory and its applications to current engineering problems and is designed to make the fundamentals of the subject more easily accessible to readers 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 applicationsthroughout,so that the mathematical methodology appears less sterile to engineers. It makes frequent use of analogies from elementary calculus or algebra to introduce complex concepts, includes fully worked examples, and provides a dual heuristic/analytic discussion of all topics. A downloadable MATLAB toolbox--a state-of-the-art computer aid--is available.Complex Numbers. Analytic Functions. Elementary Functions. Complex Integration. Series Representations for Analytic Functions. Residue Theory. Conformal Mapping. The Transforms of Applied Mathematics. MATLAB ToolBox for Visualization of Conformal Maps. Numerical Construction of Conformal Maps. Table of Conformal Mappings. Features coverage of Julia Sets; modern exposition of the use of complex numbers in linear analysis (e.g., AC circuits, kinematics, signal processing); applications of complex algebra in celestial mechanics and gear kinematics; and an introduction to Cauchy integrals and the Sokhotskyi-Plemeij formulas.For mathematicians and engineers interested in Complex Analysis and Mathematical Physics.

Table of Contents

Preface	ix
Complex Numbers	1	(52)
The Algebra of Complex Numbers	1	(6)
Point Representation of Complex Numbers	7	(7)
Vectors and Polar Forms	14	(12)
The Complex Exponential	26	(7)
Powers and Roots	33	(6)
Planar Sets	39	(5)
The Riemann Sphere and Stereographic Projection	44	(9)
Summary	51	(2)
Analytic Functions	53	(46)
Functions of a Complex Variable	53	(5)
Limits and Continuity	58	(7)
Analyticity	65	(8)
The Cauchy-Riemann Equations	73	(6)
Harmonic Functions	79	(8)
*Steady-State Temperature as a Harmonic Function	87	(4)
*Iterated Maps: Julia and Mandelbrot Sets	91	(8)
Summary	95	(4)
Elementary Functions	99	(50)
Polynomials and Rational Functions	99	(12)
The Exponential, Trigonometric, and Hyperbolic Functions	111	(7)
The Logarithmic Function	118	(7)
Washers, Wedges, and Walls	125	(6)
Complex Powers and Inverse Trigonometric Functions	131	(7)
*Application to Oscillating Systems	138	(11)
Summary	145	(4)
Complex Integration	149	(86)
Contours	149	(12)
Contour Integrals	161	(12)
Independence of Path	173	(7)
Cauchy's Integral Theorem	180	(24)
Deformation of Contours Approach	180	(11)
Vector Analysis Approach	191	(13)
Cauchy's Integral Formula and Its Consequences	204	(10)
Bounds for Analytic Functions	214	(7)
*Applications to Harmonic Functions	221	(14)
Summary	230	(5)
Series Representations for Analytic Functions	235	(72)
Sequences and Series	235	(7)
Taylor Series	242	(10)
Power Series	252	(10)
*Mathematical Theory of Convergence	262	(7)
Laurent Series	269	(8)
Zeros and Singularities	277	(10)
The Point at Infinity	287	(5)
*Analytic Continuation	292	(15)
Summary	304	(3)
Residue Theory	307	(62)
The Residue Theorem	307	(7)
Trigonometric Integrals over [0, 2π]	314	(4)
Improper Integrals of Certain Functions over (--∞, ∞)	318	(10)
Improper Integrals Involving Trigonometric Functions	328	(9)
Indented Contours	337	(8)
Integrals Involving Multiple-Valued Functions	345	(10)
The Argument Principle and Rouche's Theorem	355	(14)
Summary	367	(2)
Conformal Mapping	369	(76)
Invariance of Laplace's Equation	369	(8)
Geometric Considerations	377	(6)
Mobius Transformations	383	(12)
Mobius Transformations, Continued	395	(12)
The Schwarz-Christoffel Transformation	407	(12)
Applications in Electrostatics, Heat Flow, and Fluid Mechanics	419	(13)
Further Physical Applications of Conformal Mapping	432	(13)
Summary	443	(2)
The Transforms of Applied Mathematics	445
Fourier Series (The Finite Fourier Transform)	446	(18)
The Fourier Transform	464	(12)
The Laplace Transform	476	(10)
The z-Transform	486	(9)
Cauchy Integrals and the Hilbert Transform	495
Summary	509
A Numerical Construction of Conformal Maps	1	(18)
The Schwarz-Christoffel Parameter Problem	1	(4)
Examples	5	(5)
Numerical Integration	10	(3)
Conformal Mapping of Smooth Domains	13	(1)
Conformal Mapping Software	14	(5)
B Table of Conformal Mappings	19	(8)
Mobius Transformations	19	(2)
Other Transformations	21	(6)
Answers to Odd-Numbered Problems	27
Index	1

Excerpts

The raison d'existence forFundamentals of Complex Analysis with Applications to Engineering and Science, 3/eis our conviction that engineering, science, and mathematics undergraduates who have completed the calculus sequence are capable of understanding the basics of complex analysis and applying its methods to solve engineering problems. Accordingly, we address ourselves to this audience in our attempt to make the fundamentals of the subject more easily accessible to readers who have little inclination to wade through the rigors of the axiomatic approach. To accomplish this goal we have modeled the text after standard calculus books, both in level of exposition and layout, and have incorporated engineering applications throughout the text so that the mathematical methodology will appear less sterile to the reader.To be more specific about our mode of exposition, we begin by addressing the question most instructors ask first: To what extent is the book self contained, i.e., which results are proved and which are merely stated? Let us say that we have elected to include all the proofs that reflect the spirit of analytic function theory and to omit most of those that involve deeper results from real analysis (such as the convergence of Riemann sums for complex integrals, the Cauchy criterion for convergence, Goursat's generalization of Cauchy's theorem, or the Riemann mapping theorem). Moreover, in keeping with our philosophy of avoiding pedantics, we have shunned the ordered pairs interpretation of complex numbers and retained the more intuitive approach (grounded in algebraic field extensions).Cauchy's theorem is given two alternative presentations in Chapter 4. The first is based on the deformation of contours, or what is known to topologists as homotopy. We have taken some pains to make this approach understandable and transparent to the novice because it is easy to visualize and to apply in specific situations. The second treatment interprets contour integrals in terms of line integrals and invokes Green's theorem to complete the argument. These parallel developments constitute the two parts of Section 4 in Chapter 4; either one may be read, and the other omitted, without disrupting the exposition (although it should not be difficult to discern our preference, from this paragraph).Steady state temperature patterns in two dimensions are, in our opinion, the most familiar instances of harmonic functions, so we have principally chosen this interpretation for visualization of the theorems of analytic function theory. This application receives attention throughout the book, with special emphasis in Chapter 7 in the context of conformal mapping. There we draw 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. In doing so we hope to dispel the impression, given in many older books, that all applications of the technique fall in the latter category.In this third edition L. N. Trefethen and T. Driscoll have updated an appendix that reflects the progress made in recent years on the numerical construction of conformal mappings. A second appendix compiles a listing of some useful mappings having closed form expressions.Linear systems analysis is another application that recurs in the text. The basic, ideas of frequency analysis are introduced in Chapter 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 Chapter 8 with the exposition of the analytic-function aspects of Fourier, Mellin, Laplace, Hilbert, and z transforms, including new applications in signal processing and communications. We hope thereby that our book will continue to serve the reader as a reference resource for subsequent coursework in these areas.