meromorphic extension

Let $A\subset B\subseteq\mathbb{C}$ and $f\colon A\to\mathbb{C}$ be analytic. A meromorphic extension of $f$ is a meromorphic function $g\colon B\to\mathbb{C}$ such that $g|_{A}=f$ .

The meromorphic extension of an analytic function to a larger domain (http://planetmath.org/Domain) is unique; i.e. (http://planetmath.org/Ie), using the above notation, if $h\colon B\to\mathbb{C}$ has the property that $h|_{A}=f$ , then $g=h$ on $B$ .

Occasionally, an analytic function and its meromorphic extension are denoted using the same notation. A common example of this phenomenon is the Riemann zeta function.

Title	meromorphic extension
Canonical name	MeromorphicExtension
Date of creation	2013-03-22 16:07:26
Last modified on	2013-03-22 16:07:26
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 30D30
Synonym	meromorphic continuation
Related topic	AnalyticContinuationOfRiemannZeta
Related topic	RestrictionOfAFunction