 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2020

Paper 3

For each nonzero <period lattice> point $\omega$, the <binomial series> gives
$$
\frac1{(z-\omega)^2}-\frac1{\omega^2}
=\sum_{k=1}^{\infty}(k+1)\frac{z^k}{\omega^{k+2}}.
$$
Pairing $\omega$ with $-\omega$ cancels every odd power. Hence the <Laurent series> of the <Weierstrass elliptic function> is
$$
\wp(z)=\frac1{z^2}
+3z^2\sum_{\omega\ne0}\omega^{-4}
+5z^4\sum_{\omega\ne0}\omega^{-6}+O(z^6)
=\frac1{z^2}+\frac{g_2}{20}z^2+\frac{g_3}{28}z^4+O(z^6),
$$
which is the stated <Laurent coefficients of the Weierstrass elliptic function>[coefficient formula].

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 7E / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 7E / i / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 7E / i / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)