Meromorphic Functions
Why
The entire functions “extend” the polynomial functions. For polynomial in $\C $, we can extend the class to the (complex) rational functions in $\C $. Can we similarly extend the class entire functions?1
Definition
A meromorphic function
(or fractional function)
is a function $f: \C \to \C $ for which there
exists entire functions $g: \C \to \C $ and
$h: \C \to \C $ so that
\[
f(z) = \frac{g(z)}{h(z)}
\]