Change of variable in integral of product of exponential and gaussian functions

[galuoises]

I have the integral

\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}

where \xi is a constant. I would like to transform by some change of variables in the form

\int_{-\infty}^{\infty}dx F(x) \int_{-\infty}^{\infty}dy G(y)

the problem is that due to absolute value in the integral one must take in account where x is greater or less than y,

can someone help me, please?

 

[lurflurf]

First observe that

e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}=e^{-\xi \vert x-y\vert}e^{-(x-y)^2/2}e^{-(x+y)^2/2}

Then you can either change variables such as
u=(x+y)/sqrt(2)
v=(x-y)/sqrt(2)
or break into two regions
x<y
x>y

 

[JJacquelin]

Hi !

the clolsed form of the integral involves a special function (erf).

 

[galuoises]

Nice trick! Thank you so much!