# Integral depending on coordinate differences

1. Nov 21, 2012

### Irid

1. The problem statement, all variables and given/known data
Consider a function which depends only on a difference between two variables, and integrate it with respect to both:
$$\int_a^b \int_a^b f(x-y)\, dxdy$$
Is there any way to simplify this expression, like reducing it into a 1-D integral?

2. Nov 21, 2012

### Dick

Use a change of variables like u=x-y, v=x+y. That will reduce it to a single integration over u after you do the dv integration.

3. Nov 21, 2012

### Irid

This gives me
$$dxdy = (dv^2-du^2)/4$$
and I don't see how this makes the integral any easier

4. Nov 21, 2012

### Dick

That's not how you do change of variables in double integration. dxdy is equal to dudv times a Jacobian factor, remember? http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant Then you have to change the limits.

5. Nov 21, 2012

### Irid

Oh, thanks Dick, I wasn't aware of this...

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