How to evaluate this double integeration of a gaussian function?

[peter308]

how to integrate this equation?


∫∫e^-x^2 e^-x'^2 / |x-x'| dx^3 dx'^3 lower limit is 0 and upper limit is inf for x and x'

the result is an error function. But I would like to know the details of the process of integration.

some one suggest me to change the variable to x+x'=u x-x'=v but I got stucked. Can some one gave me any further suggestions or hints. Much appreciated!



With Best Regards
Tsung-Wen Yen

 

[kdbnlin78]

Perhaps try polar coordinates.
x=r sin /theta
x' = r cos /theta

 

[Bacle]

To extend on Kdblin78's comment:

Let I:= Int e^(-x^2)

Consider e^(-x^2) , and e^(-y^2)

Then consider I^2 as the product of the two integrals, and use polar coordinates

like Kdbnlin78 suggested, to evaluate. Notice that e^(-x^2) is a constant when

integrating with respect to y, and viceversa for e^(-y^2) . Then I^2 is the

integral of e^(-x^2- y^2 ), and polar kicks-in nicely.