Chapter 15.P, Problem 6P

Leonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved that

${\displaystyle \underset{n-1}{\overset{\infty }{\sum }}\frac{1}{n^2}=\frac{{\pi }^2}{6}}$

In this problem we ask you to prove this fact by evaluating the double integral in Problem 5. Start by making the change of variables

$x=\frac{u-v}{\sqrt{2}}y=\frac{u+v}{\sqrt{2}}$

This gives a rotation about the origin through the angle $\pi /4$. You will need to sketch the corresponding region in the uv-plane.

[Hint: If, in evaluation the integral, you encounter either of the expressions $\left(1-\sin \theta \right)/\cos \theta$ or $\left(\cos \theta \right)/\left(1+\sin \theta \right)$, you might like to use the identity $\cos \theta =\sin \left(\left(\pi /2\right)-\theta \right)$ and the corresponding identity for $\sin \theta$.]