 Solution

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

The only possible finite singularities are

z_{\pm} = \pm iπ /2

, the zeros of

4 z^{2} + π^{2}

. They are simple zeros of the denominator, because its derivative

8 z_{\pm}

is nonzero. They are also simple zeros of the numerator:

cosh z_{\pm} = 0

and

sinh z_{\pm} = \pm i \neq = 0

The common simple factors cancel, so both points are removable singularities, with extension values

z \to z_{\pm} lim \frac{cosh z}{4 z ^{2} + π ^{2}} = \frac{sinh z _{\pm}}{8 z _{\pm}} = \frac{1}{4 π} .

There are no other finite singularities; after inserting these two values the function is entire.

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