Here is a preliminary version of the course syllabus, subject to minor change:
- Definition of a holomorphic function, the Cauchy-Riemann equations
- Complex integration, local Cauchy's theorem, Liouville's theorem, Morera's theorem
- Power series representation of holomorphic functions
- Local normal forms, the open mapping theorem
- The maximum modulus principle
- Covering properties of exp, lifting criteria
- Winding numbers, cycles and homology
- Homology version of Cauchy's theorem
- Zeros, poles, and essential singularities
- The Riemann sphere, meromorphic functions, Laurent series
- Residue theorem, the argument principle, Rouche's theorem
- Compact convergence, the Arzela-Ascoli theorem
- Theorems of Weierstrass and Hurwitz
- Normal families, Montel's theorem
- Elementary properties of the Mobius group
- Classical version of the Schwarz lemma
- Automorphism groups of the disk, plane, and sphere
- Geometry in the hyperbolic plane, Pick's theorem
- Elementary properties of conformal mappings
- The Riemann mapping theorem
- Schlicht functions, Area theorem, Koebe's 1/4-theorem
- Harmonic functions, relations to holomorphic functions
- The mean-value property and maximum principle for harmonic functions
- The Poisson integral formula and its applications
- Harnack's inequalities
- Schwarz reflection principle
- Regular and singular points, natural boundary, Ostrowski's overconvergence and Hadamard's gap theorems
- Analytic continuation along curves, homotopy invariance, the monodromy theorem
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