Math 722: Graduate complex analysis
2007-2008
Instructor: W. Craig
Meeting times: Mon & Thurs 9:30 - 11:00
Office hours: Tues 2:00 - 3:30, HH418
Textbook: Greene & Krantz, Function theory of one complex
variable
Syllabus:
0) Introduction
1) Algebra
i) the complex plane
ii) fundamental theorem of arithmetic
iii) complex values polynomials
iv) Cauchy - Riemann equations
v) real and holomorphic antiderivatives
2) Geometry
i) real and complex line integrals
ii) conformal maps
iii) holomorphic antiderivatives
iv) Cauchy integral theorem
v) Cauchy integral formula
3) Analysis
i) differentiability properties of holomorphic functions
ii) Taylor series expansions of holomorphic functions
iii) Cauchy estimates and Liouville's theorem
iv) uniform convergence of holomorphic functions
v) zeros of holomorphic functions
4) Meromorphic functions
i) isolated singularities
ii) Laurent expansions
iii) calculus of residues
iv) applications
5) Zeros and poles of meromorphic functions
i) principle of argument
ii) maximum modulus principle
iii) Schwarz' lemma
6) Holomorphic mappings
i) maps of C -> C and D -> D
ii) linear fractional transformations, PSL(2,C)
iii) Riemann mapping theorem
7) Riemann surfaces
i) analytic continuation along curves
ii) monodromy theorem
iii) examples of Riemann surfaces
iv) the modular surface and Picard's theorem
v) elliptic functions
8)* Bergman kernel and biholomorphic mappings
i) Hlbert spaces of holomorphic functions
ii) Bergman kernel
iii) boundary regularity of conformal maps