Reciprocal of Holomorphic Function

Theorem
Let $f: \C \to \C$ be a complex function.

Let $U \subseteq \C$ be an open set such that $f$ has no zeros in $U$.

Suppose further that $f$ is holomorphic in $U$.

Then the complex function
 * $\displaystyle \frac 1 {f_{\restriction U}} : U \to \C$

is holomorphic.