Math 633 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.


Measure theory (the equivalent of Math 631).


  1. Holomorphic (aka complex analytic) functions, including:  definition; branch of log; Cauchy-Riemann equations; as conformal maps; Mobius transformations.
  2. Integration, including: power series representations; Cauchy’s Estimate; zeros of analytic functions; Liouville’s Theorem; Maximum Modulus Theorem; Cauchy’s Theorem and Integral Formula; counting zeros; Open Mapping Theorem.
  3. Singularities, including: classification; Schwarz’s Lemma and classification of conformal automorphisms of sphere; Laurent Series; Casorati—Weierstrass Theorem; residues; Argument Principle; Rouche’s Theorem.
  4. Space of holomorphic functions, including: Hurwitz’s Theorem; Montel’s Theorem; Riemann Mapping Theorem; infinite products, the gamma function.
  5. Analytic continuation, including: germs and analytic continuation; Monodromy Theorem; Riemann surfaces (examples).
  6. Additional topics to be covered as time allows and interest directs may include:
    • Mittag-Leffler Theorem
    • Harmonic functions, including: Harnack's inequality; the Dirichlet Problem
    • Elliptic functions
    • The modular group; the J-function and Little Picard.


  • Conway.  Functions of One Complex Variable I
  • Ahlfors.  Complex Analysis