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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 13D / f / Solution

For a nonzero lattice point

ω

and

∣ z ∣ < ∣ ω ∣

,

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

Pairing

ω

with

- ω

cancels odd powers. There is no constant term, so

a_{0} = 0

, and the laurent coefficients of the Weierstrass elliptic function are

a_{2 k} = (2 k + 1) (m, n) \neq = (0, 0) \sum \frac{1}{( m ω _{1} + n ω _{2} ) ^{2 k + 2}}, k \geq 1.

Solved by gpt-5.6-sol high.

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