fundamental theorems in complex analysis

The following is a list of fundamental theorems in the subject of complex analysis (single complex variable). If a theorem does not yet appear in the encyclopedia, please consider adding it — Planet Math is a work in progress and some basic results have not yet been entered. Likewise, if some basic theorem has been overlooked in this list, please add it.

• Cauchy-Riemann equations

• Cauchy’s integral theorem

• second form of Cauchy integral theorem

• Morera’s theorem

• Cauchy’s integral formula

• Cauchy’s residue theorem

• Cauchy’s argument principle

• Rouché’s theorem

• identity theorem of power series

• rigidity theorem for analytic functions

• Riemann’s removable singularity theorem

• Casorati-Weierstrass theorem

• implicit function theorem for complex analytic functions (I gave proofs of this and the next theorem in a posting to a forum and must convert them to an encyclopaedia entry.)

• inverse function theorem for complex analytic functions

• maximal modulus principle

• Schwarz lemma

• Liouville’s theorem

• characterization of rational functions

• Weierstrass’ factorization theorem

• Weierstrass’ criterion of uniform convergence

• Mittag-Leffler’s theorem

• Möbius circle transformation theorem

• Riemann mapping theorem

• Gauss’ mean value theorem

• Schwarz’ reflection principle

• Harnack’s principle

• Bloch theorem

• Picard’s theorem (http://planetmath.org/PicardsTheorem)

• Little Picard theorem

• Monodromy theorem

• Runge’s theorem

• Mergelyan’s theorem

• Montel’s theorem

• Marty’s theorem

• Hurwitz’s theorem

• Bieberbach’s conjecture

• Koebe one-fourth theorem

• Factorization theorem for $H^{\mathrm{\infty }}$ functions (http://planetmath.org/FactorizationTheoremForHinftyFunctions)

• Plemelj formulas

• Harnack theorem

• Schwarz and Poisson formulas