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Error function help

Hey just trying to integrate |y-x|exp(-0.5(x^2+y^2)) with respect to x, over infinity to minus infinity. I need to have my answer in terms of the error function, tried using a few substitutions and getting kinda close but not really.

my main issue is finding a substitution that transforms the limits in to those of the form in the error function.

If anyone could give me tips on where to start that'd be cool.
Original post by Kim-Jong-Illest
Hey just trying to integrate |y-x|exp(-0.5(x^2+y^2)) with respect to x, over infinity to minus infinity. I need to have my answer in terms of the error function, tried using a few substitutions and getting kinda close but not really.


You want to find f(y)=∫−∞∞∣y−x∣e−x2+y22 dx\displaystyle f(y) = \int_{-\infty}^\infty |y-x| e^{-\frac{x^2+y^2}{2}} \ dx?
(edited 7 years ago)
Original post by atsruser
You want to find f(y)=∫−∞∞∣y−x∣e−x2+y22 dx\displaystyle f(y) = \int_{-\infty}^\infty |y-x| e^{-\frac{x^2+y^2}{2}} \ dx?


Yes
Well, integrating |x-y| is going to be horrible, but for a fixed y, you can rewrite this as

∫−∞y(y−x)e−(x2+y2)/2 dx+∫y∞(x−y)e−(x2+y2/2 dx\int_{-\infty}^y (y-x)e^{-(x^2+y^2)/2}\, dx + \int_y^\infty (x-y) e^{-(x^2+y^2/2}\, dx

Then, note that you can easily integrate xe−(x2+y2)/2 dxx e^{-(x^2+y^2)/2}\, dx, you'll be left with a couple of integrals that fairly directly translate into error functions.

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