Complex Analysis

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Typical Scheduling: 
Every spring semester

Complex integration, including Goursat's theorem; classification of singularities, the argument principle, the maximum principle; Riemann Mapping theorem; analytic continuation and Riemann surfaces; range of an analytic function, including Picard's theorem.


MATH 4317 and MATH 4320 or equivalent

Course Text: 

At the level of Conway, Functions of One Complex Variable

Topic Outline: 
  • Analytic functions
  • Series and integration theorems and formulas; Goursat's theorem
  • Singularities, the argument principle, Rouche's theorem
  • Conformal mapping by elementary functions
  • Harmonic families and Poisson's formula
  • The maximum principle and Schwarz's lemma
  • Spaces of analytic functions and normal families
  • The Riemann mapping theorem and the Weierstrass factorization theorem
  • Analytic continuation, multi-valued analytic functions, and Riemann surfaces
  • Additional topics as time permits and interest dictates, e.g., the theorems of Runge, Picard, and Mittag-Leffler, Bergman's kernel, moment problems, elliptic functions, zeros of analytic functions, the Schwarz-Christoffel transformation