On a simply connected domain, a meromorphic function is $e^gh_1/h_2$ with $g$, $h_1$, $h_2$ holomorphic, where $h_1$ vanishes exactly at its zeros and $h_2$ exactly at its poles, with multiplicity.Suggest a slogan
Let \(D\subseteq\CC\) be a simply connected connected open set and let \(f\) be meromorphic on \(D\) and not identically zero. Then there are holomorphic functions \(g\), \(h_1\), \(h_2\) on \(D\) such that \[ f(z)=e^{g(z)}\frac{h_1(z)}{h_2(z)}, \] where the zeros of \(h_1\) are the zeros of \(f\), with the same multiplicities, and the zeros of \(h_2\) are the poles of \(f\), with multiplicities equal to their orders.