1). Using an infinite series and the Riemann zeta function
2). Using neither infinite series nor polylogarithms
The infinite geometric series holds for , which, except for the singularity at (x,y)=(1,1), satisfies for our region of integration; thus
we can exchange the order of integration and summation
and performing the integral , we see
we can separate this sum into sums over the odd and even integers
But the sum over the even integers is
, and so
and so .
2). Without series or polylogarithms.
If we try to integrate directly over one variable, say x, we see that ,
so we obtain single integral
and since , we have
which requires polylogarithms to integrate.
So instead, we need to make a change of variables. The key is to define new variables u and v, with and . To require a one-to-one mapping to positive x and y, we need , . Further, we see x≤1 gives us
and similarly, the upper limit on y gives
Thus, we see the region of integration R in the uv plane is the isoceles right triangle bounded by u=0, v=0, and .
Now, we need to find the Jacobian determinant, since
Thus, we have
and our integral is just equal to the area of the triangle R, which is
Monday Math 142