Math 520a - Topics
The text will be Complex Analysis by Stein and Shakarchi.
The following is what I hope to cover in the first semester.
This is following the table of contents of the text.
1 Preliminaries
1.1 Complex numbers and the complex plane: convergence, point set topology
1.2 Functions on the complex plane: continuity, analyticity, power series
1.3 Integration along curves
2 Cauchy's theorem and applications
2.1 Goursat's theorem
2.2 Primitives and Cauchy's theorem in a disc
2.3 Evaluation of some integrals
2.4 Cauchy's integral formula
2.5 Further apps: Morera's thm, Schwarz reflection principle, Runge's approximation theorom
3 Meromorphic functions and the logarithm
3.1 Zeroes and poles
3.2 Residue formula
3.3 Singularities and meromorphic functions
3.4 Argument principle and applications
3.5 Simply connected domains
3.6 The complex logarithm
3.7 Fourier series and harmonic functions
5 Entire functions
5.1 Jensen's formula
5.2 Functions of finite order
5.3 Infinite products
5.4 Weierstrass infinite products
5.5 Hadamard's factorization theorem
6 Gamma and zeta functions
6.1 Gamma function: analytic continuation and properties
6.2 Zeta function: functional equation and analytic continuation
8 Conformal mappings
8.1 Definition and examples
8.2 Schwarz lemma
8.3 Riemann mapping theorem
If time permits some of
9 Introduction to elliptic functions