# 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’s integral theorem

• Morera’s theorem

• Cauchy’s integral formula

• Cauchy’s residue theorem

• Cauchy’s argument principle

• Rouché’s theorem

• Riemann’s removable singularity 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

• Liouville’s theorem

• characterization of rational functions

• Weierstrass’ factorization theorem

• Weierstrass’ criterion of uniform convergence

• Mittag-Leffler’s theorem

• MÃÂ¶bius circle transformation theorem

• Gauss’ mean value theorem

• Harnack’s principle

• Bloch theorem

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

• Runge’s theorem

• Mergelyan’s theorem

• Montel’s theorem

• Marty’s theorem

• Hurwitz’s theorem

• Bieberbach’s conjecture

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

Title fundamental theorems in complex analysis FundamentalTheoremsInComplexAnalysis 2013-03-22 14:57:33 2013-03-22 14:57:33 rspuzio (6075) rspuzio (6075) 26 rspuzio (6075) Topic msc 30-00 TopicEntryOnComplexAnalysis