TITLE: the ratio of the derivative of a holomorphic function over itself
QUESTION [0 upvotes]: Let $h(x)$ be a holomorphic complex function. Is it true that the ratio :$h'(x)/h(x)$ is holomorphic as well? (please explain why)

REPLY [1 votes]: It won't necessarily be a holomorphic function. Actually, anywhere $h(x)=0$ it wont be holomorphic.
For example for $h(x)=x$, 
$h^{'}(x)/h(x) = 1/x$ which is not holomorphic at $x=0$.
It will be a meromorphic function tho
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