Integral depending on coordinate differences

[Irid]

Homework Statement

Consider a function which depends only on a difference between two variables, and integrate it with respect to both:
<br /> \int_a^b \int_a^b f(x-y)\, dxdy<br />
Is there any way to simplify this expression, like reducing it into a 1-D integral?

 

[Dick]

Homework Statement

Consider a function which depends only on a difference between two variables, and integrate it with respect to both:
<br /> \int_a^b \int_a^b f(x-y)\, dxdy<br />
Is there any way to simplify this expression, like reducing it into a 1-D integral?

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.

 

[Irid]

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

 

[Dick]

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

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.

 

[Irid]

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