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

#### Prerequisites

Measure theory (the equivalent of Math 631).#### Syllabus

- Holomorphic (aka complex analytic) functions, including: definition; branch of log; Cauchy-Riemann equations; as conformal maps; Mobius transformations.
- 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.
- Singularities, including: classification; Schwarz’s Lemma and classification of conformal automorphisms of sphere; Laurent Series; Casorati—Weierstrass Theorem; residues; Argument Principle; Rouche’s Theorem.
- Space of holomorphic functions, including: Hurwitz’s Theorem; Montel’s Theorem; Riemann Mapping Theorem; infinite products, the gamma function.
- Analytic continuation, including: germs and analytic continuation; Monodromy Theorem; Riemann surfaces (examples).
- 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.

#### References

- Conway.
*Functions of One Complex Variable I* - Ahlfors.
*Complex Analysis*