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Complex Variables: Second Edition
by Stephen D. Fisher
Hundreds of solved examples, exercises, and applications help students gain a firm understanding of the most important topics in the theory and applications of complex variables. Topics include the complex plane, basic properties of analytic functions, analytic functions as mappings, analytic and harmonic functions in applications, and transform methods. Perfect for undergrads/grad students in science, mathematics, engineering. A three-semester course in calculus is sole prerequisite. 1990 edition. Appendices.
Table of Contents for Complex Variables: Second Edition
| 1. The complex plane | | 1.1 The complex numbers and the complex plane | | 1.1.1 A formal view of the complex numbers | | 1.2 Some geometry | | 1.3 Subsets of the plane | | 1.4 Functions and limits | | 1.5 The exponential, logarithm, and trigonometric functions | | 1.6 Line integrals and Green's theorem | | 2. Basic properties of analytic functions | | 2.1 Analytic and harmonic functions; the Cauchy-Riemann equations | | 2.1.1 Flows, fields, and analytic functions | | 2.2 Power series | | 2.3 Cauchy's theorem and Cauchy's formula | | 2.3.1 The Cauchy-Goursat theorem | | 2.4 Consequences of Cauchy's formula | | 2.5 Isolated singularities | | 2.6 The residue theorem and its application to the evaluation of definite integrals | | 3. Analytic functions as mappings | | 3.1 The zeros of an analytic function | | 3.1.1 The stability of solutions of a system of linear differential equations | | 3.2 Maximum modulus and mean value | | 3.3 Linear fractional transformations | | 3.4 Conformal mapping | | 3.4.1 Conformal mapping and flows | | 3.5 The Riemann mapping theorem and Schwarz-Christoffel transformations | | 4. Analytic and harmonic functions in applications | | 4.1 Harmonic functions | | 4.2 Harmonic functions as solutions to physical problems | | 4.3 Integral representations of harmonic functions | | 4.4 Boundary-value problems | | 4.5 Impulse functions and the Green's function of a domain | | 5. Transform methods | | 5.1 The Fourier transform: basic properties | | 5.2 Formulas Relating u and û | | 5.3 The Laplace transform | | 5.4 Applications of the Laplace transform to differential equations | | 5.5 The Z-Transform | | 5.5.1 The stability of a discrete linear system | | Appendix 1. The stability of a discrete linear system | | Appendix 2. A Table of Conformal Mappings | | Appendix 3. A Table of Laplace Transforms | | Solutions to Odd-Numbered Exercises | | Index |
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