This book is a thorough revision of its earlier eight edition, which was published in 2009. That edition has served, just as the earlier ones did, as a textbook for a one term introductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier ones, the first two of which were written by the late Ruel V. Churchill alone.
The book has always had two main objectives. The first is to develop those parts of the theory that are prominent in applications of the subject. The second objective is to furnish an introduction to applications of residues and conformal mapping. The applications of residues include their use in evaluating real improper integrals, finding inverse Laplace transforms, and locating zeros of functions. Considerable attention is paid to the use of conformal mapping in solving boundary value problems that arise in studies of heat conduction and fluid flow. Hence the book may be considered as a companion volume to the authors' text "Fourier Series and boundary Value Problems", where another classical method for solving boundary value problems in partial differential equations is developed.
1. Complex Numbers
2. Analytic Functions
3. Elementary Functions
4. Integrals
5. Series
6. Residues and Poles
7. Applications of Residues
8. Mapping by Elementary Functions
9. Conformal Mapping
10. Applications of Conformal Mapping
11. The Schwarz-Christoffel Transformation
12. Integral Formulas of the Poisson Type