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Change of variable in integral of product of exponential and gaussian functions 
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#1
Dec2412, 07:46 PM

P: 8

I have the integral
[itex]\int_{\infty}^{\infty}dx \int_{\infty}^{\infty}dy e^{\xi \vert xy\vert}e^{x^2}e^{y^2}[/itex] where [itex]\xi[/itex] is a constant. I would like to transform by some change of variables in the form [itex]\int_{\infty}^{\infty}dx F(x) \int_{\infty}^{\infty}dy G(y)[/itex] 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? 


#2
Dec2512, 04:18 AM

HW Helper
P: 2,263

First observe that
[itex]e^{\xi \vert xy\vert}e^{x^2}e^{y^2}=e^{\xi \vert xy\vert}e^{(xy)^2/2}e^{(x+y)^2/2}[/itex] Then you can either change variables such as u=(x+y)/sqrt(2) v=(xy)/sqrt(2) or break into two regions x<y x>y 


#3
Dec2612, 03:06 AM

P: 756

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


#4
Dec2612, 08:32 AM

P: 8

Change of variable in integral of product of exponential and gaussian functions
Nice trick! Thank you so much!



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