Complex Analysis

Lecture 0: Survey, basic definitions (Conway Chapter 1)
Lecture 1: Cauchy-Riemann equations and Harmonic functions (Conway Chapter 3, Section 1-2)
Lecture 2: Radius of convergence, analytic functions and Logarithm (Conway Chapter 3, Section 1-2)
Lecture 3: Conformal map, Mobius tranformations (Conway Chapter 3, Section 3)
Lecture 4: Goursat's lemma, Existance of primitives (Ahlfors Chapter 4)
Lecture 5: Cauchy's integral formula (local form), Winding numbers
Lecture 6: Basics of Algebraic Topology, Primitives in simply connected domains
Lecture 7: Homotopy version of Cauchy's Theorem, Cauchy's Integral formula (Conway Cahpter 4, Section 6)
Lecture 8: Liouville's theorem, The identity principle, Open mapping Theorem (Conway Cahpter 4)
Lecture 9: Maximum modulus principle, Laurent series, Types of singularities (Conway Cahpter 5, Section 1)
Lecture 10: Characterization of singularities, Removable singularity theorem, Poles, Casaorati-Weierstrass (Conway Cahpter 5, Section 1)
Lecture 11: Cauchy's theorem for nullhomologous chains, Residue theorem (Conway Cahpter 5, Section 2)
Lecture 12: Cauchy's Integral formula (Homology version)
Lecture 13: Residues at infinity, Residue theorem on Riemann sphere
Lecture 14: The argument principle, Biholomorphisms, Rouche's theorem
Lecture 15: Convergence for holomorphic functions, Weierstrass theorem, Hurwitz's theorem

Fall 2024