Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.
Essential results: 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.
For more detailed information visit th Math 532 Wiki page.