Tricky Integral involving absolute value

Question

$$ f(s) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(1/2) (x^2 + y^2)} |x - y| \delta(s - |x - y|) dx dy $$ I'm not sure which substitutions are appropriate. I have trouble computing the Jacobian of any substitution when there is an absolute value. ($\delta$ denotes the dirac distribution)

Answer 1

For $s \neq 0$, you can write the delta function as a sum,

$$\delta(s- |x-y|) = \delta(s-x+y) + \delta(s+x-y)$$

Substituting this is in and integrating over $y$ yields

$$f(s) = s\int_{-\infty}^{\infty} \left[ e^{-(x^2-xs+s^2/2)} + e^{-(x^2+xs+s^2/2)} \right] dx$$

$$ = s e^{s^2/4}\int_{-\infty}^{\infty} \left[ e^{-(x-s/2)^2} + e^{-(x+s/2)^2} \right] dx $$