= Solution
{c}
The expression is $F(z)=e^{z^2+1/z}=e^{z^2}e^{1/z}$, which is <holomorphic> for every $z\ne0$. At zero,
$$
e^{1/z}=\sum_{n=0}^\infty\frac{z^{-n}}{n!}
$$
has infinitely many nonzero negative-power terms, so it has an <essential singularity>. Multiplication by the nonvanishing <entire function> $e^{z^2}$ preserves that classification: if $F$ had a <removable singularity> or a <pole>, multiplying by $e^{-z^2}$ would give the same type for $e^{1/z}$, a contradiction. Hence
$$
\boxed{0\text{ is the only finite singularity, and it is essential}.}
$$
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