Register to reply 
Change of variable in integral of product of exponential and gaussian functions 
Share this thread: 
#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: 2,264

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: 761

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!



Register to reply 
Related Discussions  
Rate of Change of product of three functions  Calculus & Beyond Homework  1  
The PDF of the exponential of a Gaussian random variable  Set Theory, Logic, Probability, Statistics  4  
Product of gaussian random variable with itself  Set Theory, Logic, Probability, Statistics  3  
Challenging integral with exponential functions  Calculus & Beyond Homework  2  
2D Integral, Gaussian and 2 Sinc Functions  Calculus  2 