Topics

The UGA complex analysis prelim review topic list groups the material as follows; each topic links the page that treats it.

Preliminaries

• \(\CC\), \(\DD\), \(\HH\), \(\CP^1\)
• Complex arithmetic
• Arguments, branch cuts, and the complex logarithm
• Elementary geometry and conic sections
• Uniform continuity, uniform differentiability, uniform convergence, and the Heine–Cantor theorem
• Calculus — the implicit and inverse function theorems, Green's theorem, and Stokes' theorem

Differentiation and integration

• The Laplace equation and harmonic functions
• The Cauchy--Riemann equations
• Contours, parameterizations, and primitives
• Cauchy's integral formula
• Cauchy's inequality

Analytic functions and singularities

• Power series and standard series
• Laurent series
• Isolated singularities — poles, essential singularities, and Riemann’s removable singularity theorem
• Meromorphic functions

Integration

• Cauchy's theorem and Morera's theorem
• The maximum modulus principle
• The Schwarz reflection principle
• The mean value property
• Residues
• Contour integration

Entire functions

• Liouville's theorem
• The fundamental theorem of algebra

Zeros

• The argument principle
• Rouché's theorem
• Hurwitz's theorem

Conformal maps

• Conformal maps and the standard maps
• Möbius transformations
• Blaschke factors and automorphisms of the disc and the plane
• The Schwarz lemma
• The open mapping theorem

Omitted values

• The Casorati--Weierstrass theorem and the Picard theorems

Normal families

• The identity principle
• Normal families and Montel's theorem
• The Riemann mapping theorem and uniformization

Further topics

• \(\operatorname{SL}_2(\RR)\), modular forms, and elliptic curves
• The hyperbolic metric
• Riemann surfaces
• Special functions — the Gamma function, the zeta function, and elliptic functions
• Canonical products
• The Dirichlet problem