Complex Analysis

# Analytic functions

A function is **holomorphic** if it is complex differentiable in a neighborhood of every point in its domain.

A function is **analytic** if it can be written as a convergent power series in a neighborhood of every point in its domain.

It can be showen that holomorphic functions are complex analytic, and complex analytic functions are holomorphic.

# Liouville’s theorem

Let *f*(*z*) be an analytic function on the complex plane. If *f*(*z*) is bounded, then *f*(*z*) is constant.

# Möbius transformations

A **Möbius transformation** can be written in the form

\begin{align} f(z) = \frac{az + b}{cz + d} \end{align}

where *a*, *b*, *c*, and *d* are complex numbers.

# sums of power series

# exponential and logarithm functions

# spherical representation

# Cauchy’s theorem

If *U* is an open, simply-connected subset of ℂ, *f*(*z*): *U* → ℂ holomorphic, and *γ* a rectifiable curve, then

\begin{align} \oint_\gamma f(z) \; dz = 0 \end{align}

# Other

- Goursat’s proof
- consequences of Cauchy integral formula,
- isolated singularities
- Casorati-Weierstrass theorem
- open mapping theorem
- maximum principle
- Morera’s theorem,
- Schwarz reflection principle
- Cauchy’s theorem on multiply connected domains
- residue theorem
- the argument principle
- Rouchés theorem
- evaluation of definite integrals
- Harmonic functions
- conjugate functions
- maximum principle
- mean value property
- Poisson integrals
- Dirichlet problem for a disk
- Harnack’s principle
- Schwarz lemma and the hyperbolic metric
- Compact families of analytic and harmonic functions
- series and product developments
- Hurwitz theorem
- Mittag-Leffler theorem
- infinite products
- Weierstrass product theorem
- Poisson-Jensen formula
- Conformal mappings
- Elementary mappings
- Riemann mapping theorem
- mapping of polygons
- reflections across analytic boundaries
- mappings of finitely connected domains
- Subharmonic functions and the Dirichlet problem.
- monodromy theorem
- Picard’s theorem
- Elementary facts about elliptic functions.

page revision: 12, last edited: 11 Apr 2017 15:18