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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 312

For opposite contour branches the two vertex integrals factorize. Since
$$
\int_{-\infty(1-i\epsilon)}^0\eta^2e^{iE\eta}\,d\eta=\frac{2i}{E^3},
$$
their product contributes $4/(E_L^3E_R^3)$. The two assignments $LR$ and $RL$ then give
$$
\boxed{B_{4,s}^{(lr+rl)}
=\frac{\lambda^2H^8s}{4k_1k_2k_3k_4}
\frac1{(E_RE_L)^3}}.
$$
Consequently
$$
\boxed{C_4=\frac{\lambda^2H^8}{4},
\qquad \alpha_4=0,
\qquad \beta_4=3}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1 / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1 / iv / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 1 / iv / Solution

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