Thursday, September 2Real and complex linearity. Complex differentiability. Elementary differentiation rules. The Cauchy-Riemann equations. Harmonic functions.
Tuesday, September 7Holomorphic functions. Overview of the properties of holomorphic functions. The argument. The square and the square root. Multi-valuedness and the idea of Riemann surfaces. Laplace equation. Mean value property of harmonic functions.
Thursday, September 9Maximum principles. Schwarz reflection principle. Uniform limit of harmonic functions. Derivative bounds and analyticity.
Tuesday, September 14 - (Homework 1 due)Uniform convergence. Weierstrass M-test. 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 1214Power series. Convergence radius. Cauchy-Hadamard formula. Ratio test. Abel's theorem. Termwise differentiation and integration.
Tuesday, September 21 - (Homework 2 due)The exponential and logarithms. Polar coordinates. Euler's identity.
Thursday, September 23Power functions. Analyticity. Rearrangement of power series. The identity theorem. Discreteness of the zero set. Prelude to analytic continuation. Cauchy estimates.
Tuesday, September 28 - (Homework 3 due)Maximal property of convergence radius. Liouville's theorem. Multiplicative inversion of power series. Fundamental theorem of algebra. Open mapping theorem. Maximum principle. Principle of conservation of domain. Contour integration.
Thursday, September 30Complex differential forms. Fundamental theorem of calculus and Green's theorem. Goursat's theorem. Strengthened Goursat's theorem. Integrability. Morera's theorem.
Tuesday, October 5 - (Homework 4 due)Cauchy's theorem for homotopic loops. Simple connectivity. Discussion on curved triangles and triangulation of domains. Residue formula. Residue at infinity.
Thursday, October 7Cauchy's theorem for homologous cycles. Residue of a function and index of a cycle. Cauchy integral formula. Cauchy-Taylor theorem.
Tuesday, October 12Characterizations of holomorphy. Consequences of simple connectedness. Laurent series expansion. Classification of isolated singularities. Riemann's removable singularity theorem. Characterizations of poles.
Thursday, October 14Casorati-Weierstrass theorem. Riemann surfaces. The Riemann sphere. Analyticity at infinity. Meromorphic functions. Residue calculus.
Tuesday, October 19Index of a loop. Residue theorem. Argument principle. Rouché's theorem.
Thursday, October 21 - (Midterm due)Biholomorphic mappings. Conformality. Local injections. Schwarz's lemma. Automorphisms of the complex plane, the Riemann sphere, and the disk. The Riemann mapping theorem.
Tuesday, October 26Proof of the Riemann mapping theorem. Koebe's square-root trick. Weierstrass convergence theorem. Hurwitz injection theorem. Montel's theorem. The Caratheodory-Koebe expansion map.
Thursday, October 28Generalized 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. Topological corollary. Simply connected subsets of the Riemann sphere. Mention of the uniformization theorem.
Tuesday, November 2 - (Homework 5 due)Schwarz reflection principle. Analytic curves. Analyticity of Riemann maps up to boundary. Accessible boundary points. Continuity of Riemann maps up to boundary.
Thursday, November 4Continuity of Riemann maps up to boundary (continued). Picard's theorems.
Tuesday, November 9 - (Homework 6 due)Normal families. Marty's theorem. Zalcman's lemma.
Thursday, November 11Montel's theorem. Proof of Picard's big theorem. Infinite products of holomorphic functions. Logarithmic differentiation. Weiertsrass elementary factors.
Tuesday, November 16Weiertsrass product and factorization theorems. Canonical products. Jensen's formula. Hadamard factorization theorem.
Thursday, November 18Hadamard factorization theorem (continued). Gamma function.
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. Functional equation.
Thursday, November 25Hadamard factorization of xi. Mellin transform. Mellin inversion formula.
Tuesday, November 30Explicit formula. Zeroes of the zeta function. Prime number theorem. Riemann hypothesis.
Thursday, December 2Presentations.