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Homemeromorphic extension

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# 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 is unique; i.e., 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.

Related:

AnalyticContinuationOfRiemannZeta, RestrictionOfAFunction

Synonym:

meromorphic continuation

Major Section:

Reference

Type of Math Object:

Definition

Parent:

## Mathematics Subject Classification

30D30*no label found*

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