Meromorphic Function with a Simple Pole at the Origin

Question

Let $z_0 \in \mathbb{C}$ and $f$ be a meromorphic function with a simple pole at the origin. Show that $\dfrac{-f^{'}(z)}{f(z)-z_0} = \dfrac{1}{z} + \sum\limits_{n=0}^{\infty} p_n(z_0).z^n$ where each $p_n$ is a polynomial of degree
$n+1$. I am getting nowhere with this problem. I tried to work this out using the Taylor series expansion about $0$ and since $z_0$ is arbitrary, this is making the problem even more difficult for me. Thanks for any help.

Answer 1

Note that the $m$-th derivative of $p_n$ can be expressed as $$p_n^{(m)}(z_0)=\frac{m!}{2 \pi \mathrm{i}}\int_\gamma\frac{-f’(z)}{z^{n+1}(f(z)-z_0)^{m+1}}\mathrm{d}z$$ for some small loop around $z=0$. Now if $m\geq n+2$ then the integrand is holomorphic at $z=0$ and therefore $p_n^{(m)}(z_0)=0$.