Complex analysis / Elias M. Stein, Rami Shakarchi

Chapter 1 is devoted to preliminaries (complex numbers, power series) to complex analysis. Chapter 2 focusses attention on Cauchy's theorem and its applications. Cauchy integral formulae, Morera's theorem, Schwarz's reflection principle and Runge's approximation theorem are delineated. Chapter 3 deals with meromorphic functions and the applications are presented here. Chapter 4 gives an account of the Fourier transforms. Chapter 5 explores the notion of entire functions. Jensens's formulae and Hadamard's factorization theorem are formulated and proved. The gamma and zeta functions are studied in Chapter 6. Analytic continuation is also a topic in this chapter. Chapter 7 is on the zeta function and prime number theorem. Chapter 8 elucidates a detailed treatment of conformal mappings. The Schwarz lemma and the Riemann mapping theorem are included in this chapter. Chapter 9 is an introduction to elliptic functions. Applications of theta functions are given in Chapter 10. Appendix A contains asymptotics. Simple connectivity and the Jordan curve theorem form contents of Appendix B. Some exercises are quoted below in order to assess the standard of the text. (Zentralblatt)

Bibliogr. p. [369]-371. Notes bibliogr. Glossaire des symboles. Index