 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2021/4/b/i/solution

The expression is

F (z) = e^{z^{2} + 1/ z} = e^{z^{2}} e^{1/ z}

, which is holomorphic for every

z \neq = 0

. 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

e^{- z^{2}}

would give the same type for

e^{1/ z}

0 is the only finite singularity, and it is essential .

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