How Do You Evaluate This Integral with Variables?

[forumfann]

Could anyone help me evaluate the integral
$\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}|sx+ty|e^{-s^{2}/2}e^{-t^{2}/2}dsdt$, which should be a function of x and y?

By the way, this is not a homework problem.

Thanks

 

[arildno]

Well, make a shift to polar coordinates:
$$s=r\cos\theta,t=r\sin\theta$$
$$x=R\cos\phi,y=R\sin\phi$$

Thus, your integral becomes:
$$R\int_{0}^{\infty}\int_{0}^{2\pi}|\cos(\theta-\phi)|r^{2}e^{-\frac{r^{2}}{2}}d\theta{d}r$$

 

[forumfann]

Thanks a lot for arildno's help. So I am able to get the value of the integral with 2 variable now.

But then how about 3 variables, i.e.
$$\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}|rx+sy+tz|e^{-r^{2}/2}e^{-s^{2}/2}e^{-t^{2}/2}drdsdt ?$$

 

[arildno]

Spherical coordinates, perchance??

 

[Count Iblis]

Instead of polar or spherical coordinates, you can also rotate your axis in the (r,s,t,...) space so that one of your axis becomes aligned with the (x,y,z,...) vector. The expression in the exponential is invariant under ratations, so what happens is that the integration becomes:


Integral dt1 dt2...dtn |y t1| exp(-t1^2/2)exp(-t2^2/2)...
exp(-tn^2/2) =

2|y| (2pi)^[(n-1)/2]

where, of course, y = the length of your (x,y,z,...) vector