MAT 527 -- Complex Variables I 
Fall 1999 Course Description

E-mail: bradley@gauss.umemat.maine.edu
Website: http://www.umemat.maine.edu/faculty/bradley/index.html

Text: John B. Conway, "Functions of One Complex Variable I," (2nd ed.) Springer-Verlag Graduate Texts in Mathematics #11, New York, 1978.

References:

• Serge Lang, "Complex Analysis," (2nd ed.) Springer-Verlag Graduate Texts in Mathematics #103, New York, 1985.
• Joseph Bak and Donald J. Newman, "Complex Analysis," (2nd ed.) Springer-Verlag, New York, 1997.
• 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.
• Einar Hille, "Analytic Function Theory," (2 Volumes), Blaisdell Publishing Company, New York, 1963.

Background: We assume familiarity with epsilon-delta arguments and a smattering of topology (metric spaces) such as might be had in an upper level undergraduate Real Analysis course. See Chapters I and II of Conway for a whirlwind tour of the necessary concepts.

Syllabus: Chapters III, VI, V, VI.1-2, VII.5-8, VIII, and XII of Conway. Other topics as time permits. In detail:

• Power series, Analytic functions, Möbius transformations
• Line Integrals, Winding Number of a Curve, Cauchy's Theorem and Integral Formula
• Classification of Singularities: The Casorati-Weierstrass Theorem
• Residues, The Argument Principle, The Maximum Modulus Theorem, Schwarz's Lemma
• Phragmén-Lindelöf Theorem, Borel-Caratheodory Theorem, Weierstrass Factorization Theorem
• The Gamma and Riemann zeta functions, Applications to Number Theory
• Runge's Theorem, Mittag-Leffler's Theorem, The Little and Great Picard Theorems

• The Riemann Mapping Theorem, Analytic Continuation, Schwarz Reflection Principle
• Entire Functions, Hadamard Factorization Theorem, Elliptic Functions
• Laplace and Mellin Transforms, Harmonic Functions, The Dirichlet Problem
• De Branges' Proof of the Bieberbach Conjecture