| Description |
xiv, 450 pages : illustrations ; 25 cm |
| Contents |
1. Complex numbers -- 1.1. Introduction -- 1.2. The complex plane -- 1.3. Elementary complex functions -- 1.4. An application -- Exercises -- 2. Complex functions -- 2.1. Single-valued functions -- 2.2. Multiple-valued functions -- 2.3. Branch points and cuts -- Exercises -- 3. Differentiation and analyticity -- 3.1. Definition of the derivative -- 3.2. Analyticity -- 3.3. Some properties of analytic functions -- Exercises -- 4. Complex functions as mappings -- 4.1. Similarity mappings -- 4.2. Conformal mappings -- 4.3. Mobius transformations -- Exercises -- 5. Closed contours and homology -- 5.1. Topology of the complex plane -- 5.2. Winding number -- 5.3. Homology -- 5.4. Roots of f(z) = & alpha; -- 5.5. Rouche's theorem and two applications -- Exercises -- 6. Integration -- 6.1. The complex integral -- 6.2. Integral form of the winding number -- 6.3. Integral of a complex derivative -- 6.4. Cauchy-Goursat theorem -- 6.5. Deformation of integration contours -- Exercises -- 7. Cauchy's integral formula -- 7.1. The integral formula -- 7.2. Derivatives of analytic functions -- 7.3. Maximum and minimum theorems -- Exercises -- 8. Multiply connected domains -- 8.1. Analyticity and vector calculus in two dimensions -- 8.2. A theorem about harmonic functions -- Exercises -- 9. Power series -- 9.1. Taylor series -- 9.2. Laurent series -- 9.3. Roots and the argument principle -- 9.4. Analytic mappings -- Exercises -- 10. Sequences, series, and infinite products -- 10.1. Sequences of complex numbers -- 10.2. Convergence of series -- 10.3. Integration and differentiation of series -- 10.4. The Cauchy-Hadamard formula -- 10.5. Sequences of functions -- 10.6. Families of analytic functions -- 10.7. Analyticity of functions defined by integrals -- 10.8. Gamma function and infinite products -- Exercises -- 11. Isolated singularities -- 11.1. Classification of isolated singularities -- 11.2. Meromorphic functions -- 11.3. Partial fractions -- Exercises -- 12. The residue theorem -- 12.1. Residues and domains of analyticity -- 12.2. Computing residues -- 12.3. Residues at infinity -- 12.4. Computing the residue of a pole -- Exercises -- 13. Real integrals -- 13.1. Integrals of the form & fnof; & infin; -- & infin; Rm n (x) dx -- 13.2. Integrals of the form & fnof; & infin; e"iax dx -- 13.3. Integrals of the form & fnof;2 & pi; 0 f(sin & theta;, cos & theta;) d & theta; -- 13.4. Integrals of the form & fnof; & infin; 0 xp Rm n (x) dx -- 13.5. Related integrals -- 13.6. More general integrands -- Exercises -- 14. Infinite sums -- 14.1. Inverting a Taylor series -- 14.2. Partial fractions: examples and applications -- 14.3. Partial fraction expansions: general criteria -- 14.4. More real integrals -- Exercises -- 15. Factoring entire and meromorphic functions -- 15.1. Entire functions -- 15.2. Infinite products -- 15.3. The gamma function -- 15.4. Meromorphic functions defined by poles -- Exercises -- 16. Method of steepest descent -- 16.1. Asymptotic series for the gamma function -- 16.2. Steepest descent and asymptotic expansions -- 16.3. Steepest descent---some special cases -- Exercises -- 17. Integral representations of the gamma and zeta functions -- 17.1. Integral representation for the gamma function -- 17.2. The Riemann zeta function -- Exercises -- 18. Special functions and Fourier transforms -- 18.1. Legendre functions -- 18.2. The spherical functions -- 18.3. Bessel functions -- 18.4. The Fourier transform -- Exercises -- Appendix A. Glossary of topological terms -- Appendix B. Groups and matrices -- B.1. Groups -- B.2. Two-dimensional complex matrices -- Appendix C. Some extended proofs -- C.1. Expansion of a general closed contour -- C.2. Montel's criterion -- C.3. Theorems on the analyticity of integral representations -- Appendix D. Other functions -- D.1. Special functions and the wave equation -- Appendix E. Further reading |
| Summary |
This undergraduate textbook on the theory of functions of a complex variable explains the standard introductory material, clearly but in depth, with many examples and applications, and also introduces more advanced topics. Primarily an introductory text, it will be useful at a more advanced level and as a reference |
| Notes |
Formerly CIP. Uk |
| Bibliography |
Includes bibliographical references (pages 445-446) and index |
| Subject |
Functions of complex variables.
|
| Author |
Abers, Ernest S. (Ernest Stephen), 1936- editor
|
| LC no. |
2010930776 |
| ISBN |
0198509820 |
|
0198509839 |
|
9780198509820 |
|
9780198509837 |
|
