Math 532: Complex Analysis
Math 352 or instructor`s consent.
A second course in complex analysis including the theory of infinite products, gamma and zeta functions, elliptic functions and the Riemann mapping theorem.
Desired Learning Outcomes
A knowledge of complex analysis at the level of a first course such as Math 352.
Minimal learning outcomes
Students should be familiar with the following concepts. They should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving simple results.
- Essential results from a first course. Review of power series, integration along curves, Goursat theorem, Cauchy's theorem in a disc, Taylor series, Morera's theorem, singularities, residue calculus, Laurent series, argument principle, harmonic functions, maximum modulus principle.
- Entire functions. Jensen's formula, functions of finite order, Weierstrass infinite products, Hadamard factorization theorem.
- The gamma and zeta functions. Analytic continuation of gamma function, further properties of Γ, functional equation and analytic continuation of zeta function.
- Conformal mappings. Conformal equivalence, Schwarz lemma, Montel's theorem, Riemann mapping theorem.
- Elliptic Functions. Liouville's Theorems, Poles and zeros of elliptic functions, Weierstrass elliptic functions.
Possible textbooks for this course include (but are not limited to):
These are at the discretion of the instructor as time allows. Some possible additional topics include:
Modular Functions and Theta Functions. The modular group, Eisenstein series, product formula for Jacobi theta function, transformation laws of theta functions, application to sums of squares.