Definition 1

Suppose ${\displaystyle f}$ is an analytic function defined on some non-empty open subset of ${\displaystyle \mathbb {C} }$, ${\displaystyle \mathbb {A} }$. Suppose ${\displaystyle \mathbb {B} }$ is also an open subset of ${\displaystyle \mathbb {C} }$ with cardinality greater than that of ${\displaystyle \mathbb {A} }$, containing all elements of ${\displaystyle \mathbb {A} }$. If ${\displaystyle g}$ is also an analytic function, defined on ${\displaystyle \mathbb {B} }$, such that,

${\displaystyle f(s)=g(s)\forall s\in \mathbb {A} }$

then ${\displaystyle g}$ may be called an analytical continuation of ${\displaystyle f}$.