Yes, we can do this with CA. I am skiddish about doing this. Hope I don't make a mistake somewhere.
Make the sub
Then, we can write it as
Try using a 'wedge' in the 1st quadrant of the complex plane with angle
By the residue theorem,
Because there are no singularities, the enclosed residue = 0.
Since the wedge is broken up into three different sections along the contour, we can break it up into three separate integrals. Call them a,b, and c for instance.
Take the radius of the wedge to be fixed for now at r=R and use the polar:
First, take a peek at curve a, which lies on the x-axis.
Along this curve,
Curve b, is circular in its path at radius R, so while varies, r is fixed. So, along b, .
Next, along c, is fixed. But this time
So, the integral from above becomes:
Now, let . On b, as ,
because is a lot faster than .
Remember that . Then we are left with
The first one is a familiar one which equals
By Ol' Euler:
Equate real and imaginary and note that
Therefore, from the top,