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.
1) Complex Analysis; Elias M.Stein & Rami Shakarchi. Princeton Lectures in Analysis, Book 2. Princeton University Press; First Edition (April 27, 2003); ISBN-13: 978-0691113852, ISBN-10: 0691113858.
2) Real and Complex Analysis; Walter Rudin (Higher Mathematics Series) McGraw-Hill Education; Third Edition (May 1, 1986); ISBN-10: 0070542341, ISBN-13: 978-0070542341.
Familiarity with real analysis at the level, roughly, of W. Rudin Principles of Mathematical Analysis