Title: Alte Karten / Funktionentheorie

sine cosine

Def: holomorphic

Prop: Differentiation properties of holomorphic functions

Def: Cauchy-Riemann-equations

Theorem: Holomorphicity and CR-equations

Def: Power Series

Theorem: Convergence of Power Series

Theorem: Power Series defines holomorphic function

Def: Path Integral

Prop: Path Integral properties

Q: Def: Primitive
A: 1/x does not have a primitive on C\{0}

Theorem: Integral of function with primitive

Korollar: derivative of holomorphic function is zero

Theorem: Goursat

Korollar: Goursat over Rectangles

Theorem: Goursat prime

Theorem: Existence of primitive on open disc

Theorem: Cauchy

Theorem: Cauchy prime

Theorem: Cauchy Integral formula

Theorem: local power series expansion

Theorem: Cauchy Integral for derivatives

Korollar: Cauchys estimate

Theorem: Liouville

Def: Limit point

Def: Order at point of hol. func.

Prop: properties of order of holomorphic function

Theorem: isolated zero

Theorem: identity theorem

Korollar: Principle of analytic continuation

Theorem: zero divisors

Theorem: Moreras Theorem

Def: uniform convergence of functions

Def: locally uniformly convergent

Prop: uniformly convergent sequence of continuous functions

Theorem: locally uniformly convergent sequence of holomorphic funcitons

Theorem: derivatives of locally uniformly convergence sequence of holomorphic functions

Theorem: Weierstrass M-Test

Theorem: Parameterintegral for holomorphic functions

Def: isolated singularity

Def: removable singularity

Def: pole of complex function

Theorem: Riemann continuation theorem

Theorem: Characterisation of pole of order n

Theorem: local Power Series expansion with poles

Theorem: Calculating residue of pole

Lemma: Calculating residues

Theorem: Residue theorem

Prop: Another characterisation of a pole of a complex function

Def: essential singularity

Theorem: Caseroti-Weierstrass

Theorem: Great Picard

Def: extended complex plane

Def: Meromorphic function

Prop: Properties of meromorphic functions

Def: valuation at a point

Prop: Properties of valuation at a point

Def: isolated singularity at infinity

Theorem: Meromorphic functions on the extended plane

Lemma: Logarithmic derivative

Theorem: Argument Principle

Korollar: Argument principle applied to polynimials

Theorem: Rouché

Theorem: Open mapping theorem

Theorem: Maximum modulus principle

Theorem: Homotopy theorem

Theorem: holomorphic function on simply connected domain has primitive

Def: branch of logarithm

Theorem: branch of logarithm on simply connected domain

Def: principal branch of logarithm

Theorem: logarithm of complex function

Theorem: General Residue theorem

Def: conformal map

Def: winding number

Prop: conformal map has holomorphic inverse

Korollar: Conformal equivalence defines linear transform

Beispiel: Conformal equivalence on upper half plane and unit disc

Beispiel: important non-example of conformal equivalence

Lemma: Schwarz's Lemma

Lemma: Automorphisms of the unit disc

Bemerkung: Cayley Map

Theorem: Montel

Theorem: Riemann Mapping theorem

Aufg: Generalized Liouville

Aufg: Integration identity involving conjugation

Aufg: Schwarz's reflection principle

Aufg: Dense image!

Aufg: Poles at infinity

Aufg: Symmetric Rouché

Aufg: fractional residues

Aufg: Schwarz-Pick's Lemma

complex log power series expansion