 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2021/s01/1/a/solution

The apparent singularity at zero cancels: the Taylor series of the sine gives

\frac{s i n z}{z ^{2}} - \frac{1}{z} = - \frac{z}{6} + \frac{z ^{3}}{120} - \dots .

Thus zero is a removable singularity, with extension

f (0) = 1/2

. At

z = - 2

, the other terms are holomorphic, whereas

(z + 2)^{- 1}

z = 0 : removable, z = - 2 : simple pole with residue 1.

There are no other finite singularities. If “all” includes the point at infinity on the Riemann sphere, it is an essential singularity:

f (1/ w) = w / (1 + 2 w) + w^{2} sin (1/ w) - w

has infinitely many negative powers in its Laurent series at

w = 0

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