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Finite-dimensional distributions of a stochastic process: added link

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In [[mathematics]], the '''branching theorem''' is a [[theorem]] about [[Riemann surface]]s. Intuitively, it states that every non-constant [[holomorphic function]] is [[locally]] a [[polynomial]].

==Statement of the theorem==

Let <math>X</math> and <math>Y</math> be Riemann surfaces, and let <math>f : X \to Y</math> be a non-constant holomorphic map. Fix a point <math>a \in X</math> and set <math>b := f(a) \in Y</math>. Then there exist <math>k \in \N</math> and [[Chart_(topology)|chart]]s <math>\psi_{1} : U_{1} \to V_{1}</math> on <math>X</math> and <math>\psi_{2} : U_{2} \to V_{2}</math> on <math>Y</math> such that
* <math>\psi_{1} (a) = \psi_{2} (b) = 0</math>; and
* <math>\psi_{2} \circ f \circ \psi_{1}^{-1} : V_{1} \to V_{2}</math> is <math>z \mapsto z^{k}.</math>

This theorem gives rise to several definitions:
* We call <math>k</math> the ''[[Multiplicity (mathematics)|multiplicity
]]'' of <math>f</math> at <math>a</math>. Some authors denote this <math>\nu (f, a)</math>.
* If <math>k > 1</math>, the point <math>a</math> is called a ''branch point'' of <math>f</math>.
* If <math>f</math> has no branch points, it is called ''unbranched''. See also [[unramified morphism]].

== References ==
*{{citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|year=1953|publication-date=1979|edition=3rd|isbn=0-07-000657-1}}.

[[Category:Theorems in complex analysis]]
[[Category:Riemann surfaces]]
{{mathanalysis-stub}}

Finite-dimensional distribution - Revision history

en>Kanenas: /* Finite-dimensional distributions of a stochastic process */ added link