How to evaluate this double integeration of a gaussian function?

In summary, the conversation discusses how to integrate a given equation and the result being an error function. The suggestion is to change variables and use polar coordinates to evaluate the integral. Further suggestions and hints are requested.
  • #1
peter308
15
0
how to integrate this equation?


∫∫e^-x^2 e^-x'^2 / |x-x'| dx^3 dx'^3 lower limit is 0 and upper limit is inf for x and x'

the result is an error function. But I would like to know the details of the process of integration.

some one suggest me to change the variable to x+x'=u x-x'=v but I got stucked. Can some one gave me any further suggestions or hints. Much appreciated!



With Best Regards
Tsung-Wen Yen
 
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  • #2
Perhaps try polar coordinates.
x=r sin /theta
x' = r cos /theta
 
  • #3
To extend on Kdblin78's comment:

Let I:= Int e^(-x^2)

Consider e^(-x^2) , and e^(-y^2)

Then consider I^2 as the product of the two integrals, and use polar coordinates

like Kdbnlin78 suggested, to evaluate. Notice that e^(-x^2) is a constant when

integrating with respect to y, and viceversa for e^(-y^2) . Then I^2 is the

integral of e^(-x^2- y^2 ), and polar kicks-in nicely.
 

Related to How to evaluate this double integeration of a gaussian function?

1. What is a double integration of a Gaussian function?

A double integration of a Gaussian function is the process of finding the area under the curve of a two-dimensional Gaussian (normal) distribution. It involves integrating the function twice, once with respect to one variable and then with respect to the other variable.

2. Why is it important to evaluate a double integration of a Gaussian function?

Evaluating a double integration of a Gaussian function is important in many fields of science, including physics, statistics, and engineering. It allows us to calculate probabilities, find the mean and variance of a distribution, and solve many types of differential equations.

3. What are the steps for evaluating a double integration of a Gaussian function?

The steps for evaluating a double integration of a Gaussian function are as follows: 1) Determine the limits of integration for each variable, 2) Rewrite the function in terms of the new variables, 3) Integrate the function with respect to one variable, treating the other variable as a constant, 4) Integrate the result from the first integration with respect to the second variable.

4. What is the final result of a double integration of a Gaussian function?

The final result of a double integration of a Gaussian function is a numerical value that represents the area under the curve of the Gaussian distribution. This value can be used to calculate probabilities, find the mean and variance, and solve differential equations.

5. Are there any special techniques for evaluating a double integration of a Gaussian function?

Yes, there are several techniques that can be used to evaluate a double integration of a Gaussian function, including the use of symmetry, substitution, and integration by parts. It is important to choose the most appropriate technique based on the complexity of the function and the limits of integration.

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