# 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 MeromorphicExtension 2013-03-22 16:07:26 2013-03-22 16:07:26 Wkbj79 (1863) Wkbj79 (1863) 10 Wkbj79 (1863) Definition msc 30D30 meromorphic continuation AnalyticContinuationOfRiemannZeta RestrictionOfAFunction