 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/7e/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

For each nonzero period lattice point

ω

\frac{1}{( z - ω ) ^{2}} - \frac{1}{ω ^{2}} = \sum_{k = 1}^{\infty} (k + 1) \frac{z ^{k}}{ω ^{k + 2}} .

ω

with

- ω

℘ (z) = \frac{1}{z ^{2}} + 3 z^{2} \sum_{ω \neq = 0} ω^{- 4} + 5 z^{4} \sum_{ω \neq = 0} ω^{- 6} + O (z^{6}) = \frac{1}{z ^{2}} + \frac{g _{2}}{20} z^{2} + \frac{g _{3}}{28} z^{4} + O (z^{6}),

which is the stated coefficient formula.

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