Change of variable in integral of product of exponential and gaussian functions

by galuoises
Tags: exponential, functions, gaussian, integral, product, variable
galuoises is offline
Dec24-12, 07:46 PM
P: 8
I have the integral

[itex]\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy e^{-\xi \vert x-y\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?
Phys.Org News Partner Science news on
Internet co-creator Cerf debunks 'myth' that US runs it
Astronomical forensics uncover planetary disks in Hubble archive
Solar-powered two-seat Sunseeker airplane has progress report
lurflurf is online now
Dec25-12, 04:18 AM
HW Helper
P: 2,168
First observe that

[itex]e^{-\xi \vert x-y\vert}e^{-x^2}e^{-y^2}=e^{-\xi \vert x-y\vert}e^{-(x-y)^2/2}e^{-(x+y)^2/2}[/itex]

Then you can either change variables such as
or break into two regions
JJacquelin is offline
Dec26-12, 03:06 AM
P: 746
Hi !

the clolsed form of the integral involves a special function (erf).
Attached Thumbnails
Exp Erfc.JPG  

galuoises is offline
Dec26-12, 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