 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017/5/b/ii/solution

e^{ia z} sin (b z) = (1 + ia z + O (z^{2})) (b z + O (z^{3})) = b z + iab z^{2} + O (z^{3}),

and hence

f (z) = b / z + iab + O (z)

. Since

b > 0

, zero is a simple pole with residue $b$ . It is not a removable singularity, despite the vanishing numerator, because the denominator has a zero of order two.

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