Complex analysis / Serge Lang

Basic Theory -- Complex Numbers and Functions -- Power Series -- Cauchy's Theorem, First Part -- Winding Numbers and Cauchy's Theorem -- Applications of Cauchy's Integral Formula -- Calculus of Residues -- Conformal Mappings -- Harmonic Functions -- Geometric Function Theory -- Schwarz Reflection -- The Riemann Mapping Theorem -- Analytic Continuation Along Curves -- Applications of the Maximum Modulus Principle and Jensen's Formula -- Entire and Meromorphic Functions -- Elliptic Functions -- The Gamma and Zeta Functions -- The Prime Number Theorem

The present book is meant as a text for a course on complex analysis at the advanced undergraduate level, or first-year graduate level. The first half, more or less, can be used for a one-semester course addressed to undergraduates. The second half can be used for a second semester, at either level. Somewhat more material has been included than can be covered at leisure in one or two terms, to give opportunities for the instructor to exercise individual taste, and to lead the course in whatever directions strikes the instructor's fancy at the time as well as extra read­ ing material for students on their own. A large number of routine exercises are included for the more standard portions, and a few harder exercises of striking theoretical interest are also included, but may be omitted in courses addressed to less advanced students. In some sense, I think the classical German prewar texts were the best (Hurwitz-Courant, Knopp, Bieberbach, etc. ) and I would recommend to anyone to look through them. More recent texts have emphasized connections with real analysis, which is important, but at the cost of exhibiting succinctly and clearly what is peculiar about complex analysis: the power series expansion, the uniqueness of analytic continuation, and the calculus of residues.