MAT 527 -- Complex Variables I 
Course Information

Text: Serge Lang, Complex Analysis, (4th ed.) Springer-Verlag Graduate Texts in Mathematics #103, New York, 1999.

References:

• John B. Conway, Functions of One Complex Variable I, (2nd ed.) Springer-Verlag Graduate Texts in Mathematics #11, New York, 1978.
• Joseph Bak and Donald J. Newman, Complex Analysis, (2nd ed.) Springer-Verlag, New York, 1997.
• John Stalker, Complex Analysis: Fundamentals of the Classical Theory of Functions, Birkhäuser, Boston, 1998.
• E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, London, 1969.
• Konrad Knopp, Theory of Functions, (5 Volumes), Dover, New York, 1975.
• Stanislaw Saks and Antoni Zygmund, Analytic Functions, (2nd ed.) Polska Akademia Nauk, Warsaw, 1965.
• C. Carathéodory, Theory of Functions of a Complex Variable, (F. Steinhardt, trans.) Vol. 1, Chelsea Publishing, New York, 1983.
• Lars V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966.
• Einar Hille, Analytic Function Theory, (2 Volumes), Blaisdell Publishing Company, New York, 1963.

Syllabus: Chapters I through VII of Lang. Other topics as time permits. In more detail:

• Complex differentiability, Cauchy-Riemann equations
• Power series, analytic functions
• Inverse function theorem, open mapping theorem, local maximum modulus principle
• Line integrals, winding number, Cauchy's theorem
• Laurent series, classification of singularities, the Casorati-Weierstrass theorem
• Residues, evaluation of definite integrals
• Conformal maps, Schwarz's lemma, Möbius transformations

• Analytic continuation
• Harmonic functions
• Riemann mapping theorem
• Weierstrass products, Mittag-Leffler expansions
• Special functions: elliptic functions, gamma function, Riemann zeta function
• The prime number theorem
• Laplace transforms, Mellin transforms
• The little and great Picard theorems