# Integrating exp(-i(px+qy)) dxdy over all space

1. Apr 12, 2013

### I<3NickTesla

In my diffraction notes, this integral comes up on the page about Babinet's principle:

$\int ^{y=\infty}_{y=-\infty} \int ^{x=\infty}_{x=-\infty} exp(-i(px+qy)) dx dy = \delta (p,q)$

I'm not sure how this integral is derived as carrying out the integration and putting in the limits seems to give infinity, is it something to do with the completeness relation?

A screenshot of the slide is attached

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2. Apr 12, 2013

### Staff: Mentor

I think that should be $\delta(p,0)\delta(q,0)$.
Quick analysis:
If you substitute x->x+1, the integral gets multiplied with e(-ip), but it has to keep its value at the same time. Therefore, it is 0 unless p=0. The same applies to q, of course.
If p=0 and q=0, you integrate over 1, this leads to the delta distributions.