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I am working on some Complex Analysis problems and am stuck on the following from Lang 3rd edition. I understand all the homework up to this point but don't even know where to begin here! I have attached a screenshot but the question is as follows:

Let f be analytic on an open disc centered at a point $z_o$, except at the point itself where f has a simple pole with residue equal to an integer n. Show that there is an analytic function g on the disc such that $f=\frac{g'}{g}$, and $g(z) = (z-z_o)^n h(z)$ where $h$ is analytic and $h(z_0)≠ 0$ We can assume that $z_0=0$ for simplicity.

Anything would be appreciated! Lang VI.1.22

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Note that $g \longmapsto \frac{g’}{g}$ maps products to sums, so you can split $f$ into a holomorphic part and $\frac{n}{z}$, so you can treat each part separately.

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