MATH 502 Schedule of Complex Analysis Lectures

Lecture Date Topics
1 1/13/97 algebra and analytic geometry of complex numbers, the Riemann sphere
2 1/15/97 power series and their convergence
3 1/17/97 differentiability and analyticity, differentiability of power series
4 1/20/97 elementary functions in the complex domain, mapping properties
5 1/22/97 conformality
6 1/24/97 Cauchy-Riemann equations, partial z and partial z bar, harmonic functions, existence of harmonic conjugates
7 1/27/97 Mobius transformations
8 1/29/97 Path integrals, Cauchy's formula and theorem for a disc, analytic implies power series at each point,
9 1/31/97 Cauchy's estimate, Maximum Modulus Principle, Liouville's Theorem, Fundamental Theorem of Algebra, isolation and finite multiplicity of roots
10 2/3/97 homotopy of paths and Cauchy's theorem, motivation for winding numbers
11 2/5/97 winding numbers and Cauchy's integral formula, zero counting, Open Mapping Theorem
12 2/7/97 Morera's Theorem, theorem on existence of primitives, Goursat's Theorem, isolated singularities
13 2/10/97 poles, essential singularities, Casorati-Weierstrass Theorem, Laurent expansion
14 2/12/97 The Residue Theorem, residue integrals
15 2/14/97 Argument Principle, Rouche's Theorem, Weierstrass Theorem
16 2/17/97 Hurwitz's Theorem and Montel's Theorem
17 2/19/97 Schwarz's Lemma and applications
18 2/21/97 The Riemann Mapping Theorem
19 2/24/97 Complements on conformal mapping, mean value property and maximum priniciple harmonic functions on a disk
20 2/26/97 digression: statement of the little and great Picard theorems and integrability of functions with isolated singularities, Poisson kernel, Dirichlet problem on a disk
21 2/28/97 Harnack's inequality and principle, subharmonic and superharmonic functions
22 3/3/97 Perron's approach to the solution of the Dirichlet problem
23 3/5/97 Schwarz reflection principle, some loose ends, midsemester evaluation
24 3/7/97 Midterm Exam

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