Maximum Modulus principle, Schwarz’ Lemma, Taylor series and Laurent series. Singularities, Cauchy’s residue theorem. Calculus of residues, bilinear transformations. Zeros and poles of meromorphic functions, Rouche’s theorem, Argument Principle.
Definitions and examples of conformal mappings. Infinite products, Weierstrass theorem, Mittagleffer’s theorem, Canonical product, Analytic Continuation through power series (basic ideas), Natural boundary, the Gamma function and Riemann Zeta function.