Semester
Summer
Year offered
2018
- Course Description: This course, for graduate students in mathematics, addresses the theory of functions of one complex variable with a focus on the qualifying examination in complex analysis. The course covers the representation of holomorphic functions by power series and by integrals; complex line integrals, Cauchy's integral formula, and some applications; singularities of holomorphic functions, Laurent series, and computation of definite integrals by residues; the maximum principle and Schwarz's lemma, and conformal mapping, infinite products, the Weierstrass factorization theorem, Mittag-Leffler’s theorem, normal families, proof of the Riemann mapping theorem, analytic continuation, Runge’s approximation theorem, harmonic functions, and Picard’s theorems.
- References:
1. Functions of One Complex Variable I, by John B. Conway
2. Complex Made Simple, by David C. Ullrich
3. Complex Analysis, by Lars V. Ahlfors
4. The problems on past complex analysis qualifying exams.