= Solution
{c}
The reciprocal map is holomorphic for $z\ne0$, and the <complex exponential> is entire. Its denominator vanishes exactly when
$$
e^{2\pi i/z}=1\quad\Longleftrightarrow\quad z^{-1}\in\mathbb Z.
$$
All these points are excluded from $U$. The set $U$ is open: it is the inverse image of $\mathbb C\setminus\mathbb Z$ under the reciprocal map on $\mathbb C\setminus\{0\}$. A reciprocal of a nonvanishing holomorphic function is holomorphic. Therefore \b[$f$ is holomorphic throughout $U$].
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