Complex Analysis In the Spirit of Lipman Bers, Second Edition, Springer 2013, by Rodr ́ıguez, Kra, Gilman (ISBN 978-1-4419-7322-1), e-book free from Rutgers Library link
Familiarity with real analysis at the level, roughly, of W. Rudin Principles of Mathematical Analysis
The course will present a rigorous introduction to the basic ideas of Complex Analysis, focusing on the study of functions of one complex variable and explaining in detail how this theory is fundamentally different from that of functions of one or several real variables. Topics covered will be: the complex plane, complex differentiation, holomorphic functions, the Cauchy-Riemann equations and the Delta-bar operator, line integrals, Goursat's Theorem, homotopy of loops, simply connected domains, the Cauchy Integral Theorem and the Cauchy Integral Formula, homology of curves in the plane, the winding number, calculus of residues, Taylor and Laurent series, conformal mapping, the open mapping theorem and the maximum principle, the principle of analytic continuation, meromorphic functions, convergence of sequences of holomorphic functions, compact sets of holomorphic functions ("normal families"), harmonic functions of two real variables, and the Riemann Mapping Theorem.
REVIEW SESSIONS: are held every Wednesday during the Fall semester, 2nd period (10:20-11:40am) in Room 525.