University of Illinois at Urbana-Champaign

Math 542 (Complex Analysis) Section D1 Fall 2007

Course Description

This is the first graduate course in complex analysis, one of the most elegant and beautiful branches of modern mathematics. Much of the power of the subject stems from the close interplay between algebra, analysis and geometry which greatly informs its development. Topics which we will discuss include: the Cauchy--Riemann equations, analytic and meromorphic functions, conformal mappings, the local and global Cauchy theory, Liouville's theorem, the maximum modulus theorem and the Schwarz Lemma, the classification of isolated singularities, residue theory with applications, normal families, Montel's theorem, the Riemann mapping theorem, infinite products and entire functions, the Weierstrass factorization theorem, analytic continuation, Runge's approximation theorem, the Dirichlet problem for harmonic functions, and the Poisson integral formula.

Contents

Lecture Log and homework assignments






Tentative Syllabus
  1. Complex numbers, topology of the complex plane and the Riemann sphere, stereographic projection, basic conformal mappings
  2. Cauchy-Riemann equations, analytic functions, meromorphic functions
  3. Contour integration, local Cauchy theory, Cauchy integral theorem, uniform convergence, Taylor series
  4. Liouville's theorem, maximum modulus theorem, Schwarz Lemma, automorphisms of the disc, Mobius transformations, cross ratio
  5. Laurent series, classification of isolated singularities
  6. Residue theory and applications, argument principle, Rouche's theorem, winding number, general Cauchy theorem and integral formula, simple connectivity
  7. Arzela-Ascoli theorem, normal families, Montel and Hurwitz theorems, Riemann mapping theorem
  8. infinite products, entire functions, Weierstrass factorization theorem, analytic continuation, introduction to Riemann surfaces
  9. Runge's approximation theorem
  10. harmonic functions, Laplace equation, analytic completion, Dirichlet problem, Poisson integral formula