= 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 is
$$
-10\pi i I_4+80\pi^3 i I_2-32\pi^5 i I_0=-\frac{121}{16}i\pi^6.
$$
Thus
$$
\boxed{I_2=\frac{\pi^3}{8},\qquad I_4=\frac{5\pi^5}{32}.}
$$
As an independent generating-function check, $F(s)=\int_0^\infty x^s/(1+x^2)\,dx=\pi/[2\cos(\pi s/2)]$ for $|\operatorname{Re}s|<1$. Differentiating twice and four times at zero gives the same moments.
Back to article page
