Ok I have an alternate way, other than double integrals. The moment I see my sinc functions, my signal processing harmones pester me to use fourier transforms.
Here is the fourier transform way.
Note:
From fourier transform theory, , where rect denotes the function defined by:
By the convolution-multiplication theorem, multiplication in one domain gives convlution in another domain.
Thus
Convolving the rect function is easy so i will skip it. It gives the triangle function:
Now by definition of fourier transform:
To compute the integral we want, put f=0.
Now substitute , to get:
Here are a couple more I don't have the answers to, so I would appreciate if someone with an advanced knowledge of integrals...or a very nice computer system ...could verify them?
Now we know that
So we may rewrite as follows
Now since the region of integration is rectangular by Fubini's Theorem we may rewrite as follows
Now by two iterations of Integration by Parts (just for a note to those who wish to duplicate this, for simplicities sake call ) the inner iterated integral is equivalent to
Now seeing that as due to the overpowering effect of , and evaluating at zero we get
Now for the calculation of
The substitution of
Gives
Now letting
Gives us the final answer of
So we then have that