Branching theorem

Branching theorem In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.

Statement of the theorem Let {displaystyle X} and {displaystyle Y} be Riemann surfaces, and let {displaystyle f:Xto Y} be a non-constant holomorphic map. Fix a point {displaystyle ain X} and set {displaystyle b:=f(a)in Y} . Then there exist {displaystyle kin mathbb {N} } and charts {displaystyle psi _{1}:U_{1}to V_{1}} on {displaystyle X} and {displaystyle psi _{2}:U_{2}to V_{2}} on {displaystyle Y} such that {displaystyle psi _{1}(a)=psi _{2}(b)=0} ; and {displaystyle psi _{2}circ fcirc psi _{1}^{-1}:V_{1}to V_{2}} is {displaystyle zmapsto z^{k}.} This theorem gives rise to several definitions: We call {displaystyle k} the multiplicity of {displaystyle f} at {displaystyle a} . Some authors denote this {displaystyle nu (f,a)} . If {displaystyle k>1} , the point {displaystyle a} is called a branch point of {displaystyle f} . If {displaystyle f} has no branch points, it is called unbranched. See also unramified morphism. References Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1. This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

Categories: Theorems in complex analysisRiemann surfacesMathematical analysis stubs

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