MAT 542 Syllabus

Rudiments of Complex Analysis

• Definition of a holomorphic function, Cauchy-Riemann equations
• Complex integration, Cauchy's Theorem, Morera's Theorem
• Power series representation of holomorphic functions
• Zeros, poles, meromorphic functions
• Local normal forms, Open Mapping Theorem
• Maximum Principle and its applications
• Residue Theorem, Argument Principle, Rouche's Theorem

Convergence and Approximation Theorems

• Compact convergence, Arzela-Ascoli Theorem
• Theorems of Weierstrass and Hurwitz
• Precompactness and Montel's Theorem
• Mittag-Leffler's Theorem
• Runge's Rational Approximation Theorem

Mobius Maps and Hyperbolic Geometry

• Classical Schwarz Lemma
• Automorphism groups of the disk, plane, and sphere
• Elementary properties of the Mobius group
• Geometry in the hyperbolic plane, connections to complex analysis

Conformal Mappings

• Elementary properties of conformal mappings
• The Riemann Mapping Theorem for simply-connected domains
• Schlicht functions, Area Theorem, Koebe 1/4-Theorem
• Boundary behavior of conformal mappings, Caratheodory's Theorem

Analytic Continuation

• Real Analytic functions, Schwarz Reflection Principle
• Basic removability results
• Natural boundary, Ostrowski's Overconvergence and Hadamard's Gap Theorem
• Analytic continuation along curves, function elements, homotopy invariance
• The Monodromy Theorem

Harmonic Functions

• Basic definitions
• The Poisson integral
• Harnack's Theorem, Mean-Value Property and Maximum Principle for harmonic functions
• Boundary behavior of Poisson integrals

Zeros of Holomorphic Functions

• Infinite products
• Weierstrass Product and Factorization Theorems
• Jensen's Formula
• Blaschke products and their zeros
• Order of an entire function, Hadamard's Factorization Theorem

Additional Topics (to choose from depending on time and interest)

• The elliptic modular function
• Picard's Little and Big Theorems
• The Uniformization Theorem for planar domains
• The Kobayashi and Caratheodory metrics for planar domains
• Hyperbolicity and its connection to curvature, Ahlfors's version of Schwarz Lemma
• Modulus and extremal length
• Proper holomorphic mappings, branched coverings, Riemann-Hurwitz Formula
• Harmonic measure and Green's function

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