Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S1 / 2 / 4 / Solution

At $n=3$ the bank difference is $-6\pi i{I}_2+12{\pi }^2{I}_1+8{\pi }^{3}i{I}_0$, while the residue result is $13i{\pi }^4/4$. Substitution of $I_0,I_1$ yields $I_2={\pi }^3/8$.

At $n=5$, the odd moments vanish and the remaining relation isThusAs an independent generating-function check, $F(s)={\int }_0^{\infty }{x}^s/(1+x^2)dx=\pi /[2\cos (\pi s/2)]$ for $|\mathrm{Re}s|<1$. Differentiating twice and four times at zero gives the same moments.