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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/7/c/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 7 / c / ii / Solution by

Codex  0 2026-10-18

For

z \neq = 0

, singularities occur when

1/ z = nπ

, with

n \neq = 0

. Each is a simple pole, since the substitution has nonzero derivative there:

z_{n} = \frac{1}{nπ}, n \in Z ∖ {0} .

More explicitly,

1/ z - nπ \sim - (z - z_{n}) / z_{n}^{2}

, so

cot (1/ z) \sim - z_{n}^{2} / (z - z_{n})

and its residue is

- 1/ (n^{2} π^{2})

.

These poles accumulate at zero. Consequently

z = 0 is a non-isolated singularity

. The removable/pole/essential classification for isolated singularities cannot be applied at that accumulation point; it is not an isolated essential singularity.

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