Basic Complex Analysis

In the second half of 2015, the American Math Society will publish a five volume (total about 3000 pages) set of books that is a graduate analysis text with lots of additional bonus material. Included are hundreds of problems and copious notes which extend the text and provide historical background. Efforts have been made to find simple and elegant proofs and to keeping the writing style clear.

Cauchy Integral Theorem, Consequences of the Cauchy Integral Theorem (including holomorphic iff analytic, Local Behavior, Phragmén-Lindelöf, Reflection Principle, Calculation of Integrals), Montel, Vitali and Hurwitz’s Theorems, Fractional Linear Transformations, Conformal Maps, Zeros and Product Formulae, Elliptic Functions, Global Analytic Functions, Picard’s Theorem.

Selected topics include Goursat Argument, Ultimate and Ultra Cauchy Integral Formulas, Runge’s Theorem, complex interpolation, Marty’s Theorem, continued fraction analysis of real numbers, Riemann mapping theorem, Uniformization theorem (modulo results from Part 3), Mittag Leffler and Weirstrass product theorems, finite order and Hadamard product formula, Gamma function, Euler-Maclaurin Series and Stirling’s formula to all orders, Jensen’s formula and Blaschke products, Weierstrass and Jacobi elliptic functions, Jacobi theta functions, Paley-Wiener theorems, Hartog’s phenomenon, Poincaré’s theorem that in higher complex dimensions, the ball and polydisk are not conformally equivalent.