meromorphic extensionMeromorphicExtension2006-07-29 08:29:202008-02-23 04:34:01Definitionmeromorphic% this is the default PlanetMath preamble. as your knowledge
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Let $A \subset B \subseteq \mathbb{C}$ and $f \colon A \to \mathbb{C}$ be analytic. A {\em 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 \PMlinkname{domain}{Domain} is unique; \PMlinkname{i.e.}{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.