Re: Sinc integrals

On Sep 1, 9:43 am, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:



Kaba <n...@xxxxxxxx> wrote:
Let sinc(x) = sin(pi * x) / (pi * x)

I'm looking for a solution to:

integrate[-r, r] sinc(x) * sinc(x / r) dx

(This is a generalization of the Lanczos filter, and I am looking for a
normalization constant such that the filter function integrates to 1)

How do you arrive to it?

According to Mathematica 5.2, your integral above equals

((1 + r)*Si(pi*(1 + r)) - (1 - r)*Si(pi*(1 - r)))/pi

where Si denotes the sine integral, as discussed at
<http://functions.wolfram.com/GammaBetaErf/SinIntegral/>.

In a later message, you said "The first integral is clearly too tricky to
evaluate in my program." But note that the link above gives information
(such as series) which might allow you to adequately approximate your first
integral in your program.

Also, various approximation formulas could be given. For example, if r is
large, then your integral is approximately

1 - 2 sin(pi r) / (pi^3 r^2)

More accurate approximations for large r could given, if you're interested.

You could also precompute some values, store them in a table, and
interpolate.

.