Note: This schedule is subject to revision during the term.
Thursday, September 2
Real and complex linearity.
Elementary differentiation rules.
The Cauchy-Riemann equations.
Tuesday, September 7
Overview of the properties of holomorphic functions.
The square and the square root.
Multi-valuedness and the idea of Riemann surfaces.
Mean value property of harmonic functions.
Thursday, September 9
Schwarz reflection principle.
Uniform limit of harmonic functions.
Derivative bounds and analyticity.
Tuesday, September 14 - (Homework 1 due)
Locally uniform convergence.
Normal convergence of series.
Locally normal convergence.
Rearrangement and product series.
Compact and absolute convergences.
Thursday, September 16 - class time is now 10:05-11:25 in Burnside 1214
Termwise differentiation and integration.
Tuesday, September 21 - (Homework 2 due)
The exponential and logarithms.
Thursday, September 23
Rearrangement of power series.
The identity theorem.
Discreteness of the zero set.
Prelude to analytic continuation.
Tuesday, September 28 - (Homework 3 due)
Maximal property of convergence radius.
Multiplicative inversion of power series.
Fundamental theorem of algebra.
Open mapping theorem.
Principle of conservation of domain.
Thursday, September 30
Complex differential forms.
Fundamental theorem of calculus and Green's theorem.
Strengthened Goursat's theorem.
Tuesday, October 5 - (Homework 4 due)
Cauchy's theorem for homotopic loops.
Discussion on curved triangles and triangulation of domains.
Residue at infinity.
Thursday, October 7
Cauchy's theorem for homologous cycles.
Residue of a function and index of a cycle.
Cauchy integral formula.
Tuesday, October 12
Characterizations of holomorphy.
Consequences of simple connectedness.
Laurent series expansion.
Classification of isolated singularities.
Riemann's removable singularity theorem.
Characterizations of poles.
Thursday, October 14
The Riemann sphere.
Analyticity at infinity.
Tuesday, October 19
Index of a loop.
Thursday, October 21 - (Midterm due)
Automorphisms of the complex plane, the Riemann sphere, and the disk.
The Riemann mapping theorem.
Tuesday, October 26
Proof of the Riemann mapping theorem.
Koebe's square-root trick.
Weierstrass convergence theorem.
Hurwitz injection theorem.
The Caratheodory-Koebe expansion map.
Thursday, October 28
Generalized argument prinnciple.
Holomorphy of integral.
Holomorphic inverse function theorem.
Poincaré's uniqueness lemma.
Some history of the Riemann mapping theorem.
Proof by solving the Dirichlet problem.
Simply connected subsets of the Riemann sphere.
Mention of the uniformization theorem.
Tuesday, November 2 - (Homework 5 due)
Schwarz reflection principle.
Analyticity of Riemann maps up to boundary.
Accessible boundary points.
Continuity of Riemann maps up to boundary.
Thursday, November 4
Continuity of Riemann maps up to boundary (continued).
Tuesday, November 9 - (Homework 6 due)
Thursday, November 11
Proof of Picard's big theorem.
Infinite products of holomorphic functions.
Weiertsrass elementary factors.
Tuesday, November 16
Weiertsrass product and factorization theorems.
Hadamard factorization theorem.
Thursday, November 18
Hadamard factorization theorem (continued).
Tuesday, November 23 - (Homework 7 due)
Euler's integral represention of the Gamma function.
Wielandt's uniqueness theorem.
Riemann zeta function.
Euler product formula.
Von Mangoldt function.
Poisson summation formula.
Riemann xi function.
Thursday, November 25
Hadamard factorization of xi.
Mellin inversion formula.
Tuesday, November 30
Zeroes of the zeta function.
Prime number theorem.
Thursday, December 2