Meromorphic Functions

A function \(f:\Omega\to{\mathbf{C}}\) is meromorphic iff there exists a sequence \(\left\{{z_n}\right\}\) such that

  • \(\left\{{z_n}\right\}\) has no limit points in \(\Omega\).
  • \(f\) is holomorphic in \(\Omega\setminus\left\{{z_n}\right\}\).
  • \(f\) has poles at the points \(\left\{{z_n}\right\}\).

Equivalently, \(f\) is holomorphic on \(\Omega\) with a discrete set of points delete which are all poles of \(f\).

Meromorphic functions on \({\mathbf{C}}\) are rational functions.

Consider \(f(z) - P(z)\), subtracting off the principal part at each pole \(z_0\), to get a bounded entire function and apply Liouville.

If \(f\) is analytic on a region \(\Omega\) containing \(z_0\), then \(f\) can be written as \begin{align*} f(z) =\left(\sum_{k=0}^{n-1} \frac{f^{(k)}\left(z_{0}\right)}{k !}\left(z-z_{0}\right)^{k}\right)+ R_{n}(z)\left(z-z_{0}\right)^{n} ,\end{align*} where \(R_n\) is analytic.