Problem.Evaluate

*Proof. * Let denote the positively oriented boundary of the square whose edges lie along the lines

where is positive integer.

Define . Then at , has pole of order because

Note that

So

For , has simple pole at . So

In , has poles. So by Residue theorem, we have

Note that

Also note that on the vertical side of the square, and on the horizontal sides. So there is a constant such that for all .

So

So as ,

Therefore,

i.e.,