Carrier G.F., Krook M., Pearson C.E. Functions of a Complex Variable: Theory and Technique

SIAM, 2005. — 453 p.
Chapters 1, 2, and 5 of this book provide concisely but honestly the classical aspects of the theory of functions of a complex variable; the rest of the book is devoted to a detailed account of various techniques and the ideas from which they evolve. Many of the illustrative examples are phrased in terms of the physical contexts in which they might arise; however, we have tried to be consistent in including a mathematical statement of each problem to be discussed.
For the acquisition of skill in the use of these techniques, practice is even more important than instruction. Accordingly, we have inserted many exercises, including some which are essential parts of the text. The reader who fails to carry out a substantial number of these exercises will have missed much of the value of this book.
The individual chapters segregate specific topical items, but many readers will find it profitable to study selected parts of Chaps. 3 to 7 as they encounter the related underlying theory in Chap. 2.Complex numbers and their elementary properties
Analytic functions
Contour integration
Conformal mapping
Special functions
Asymptotic methods
Transform methods
Special techniques