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.