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

(1)
\begin{align} f(z) = \frac{az + b}{cz + d} \end{align}

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

# Cauchy’s theorem

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

(2)
\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