Solving Difficult Integral: \int_{-\infty}^{+\infty}Exp(-x^2)*Erf(x^2 - a^2)dx

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In summary, the conversation revolves around a difficult integral that cannot be solved by Mathematica. The integral in question is \int_{-\infty}^{+\infty}Exp(-x^2)*Erf(x^2 - a^2)dx and the person asking for help believes it should be integrable due to its simplicity. Mathematica also does not provide an analytical solution, but it is suggested to try using the derivative of the error function to solve it.
  • #1
Heimdall
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Hi,

I have an integral that I find quite difficult, I can't obtain anything from mathematica (but I'm far from being an expert).

Would some of you have a hint ? is it analytical ? It seems to be a "simple" function, from the physicist I am it should be integrable...

this integral is :

[tex]\int_{-\infty}^{+\infty}Exp(-x^2)*Erf(x^2 - a^2)dx[/tex]Thanks !
 
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  • #2
Mathematica doesn't give me an analytical solution either.
But the integrand seems to be sharply peaked around x = 0, and we have the nice result
[tex]\frac{d}{dx} \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} e^{-x^2}[/tex]
so maybe you can do something like steepest descent (expand the integrand around 0)?
 

1. What is the purpose of solving this difficult integral?

The purpose of solving this difficult integral is to find the area under the curve of the function Exp(-x^2)*Erf(x^2 - a^2). This can be useful in many applications, such as probability and statistics, physics, and engineering.

2. What does Erf(x^2 - a^2) represent in the integral?

Erf(x^2 - a^2) represents the error function, which is a mathematical function used to calculate the probability of a normally distributed variable falling within a certain range of values. In this integral, it is multiplied by the exponential function to form a more complex function.

3. What makes this integral difficult to solve?

This integral is difficult to solve because it contains both an exponential function and an error function, which are both non-elementary functions. This means that they cannot be expressed as a finite combination of elementary functions (e.g. polynomials, trigonometric functions, exponentials).

4. Are there any specific techniques or methods for solving this integral?

Yes, there are several techniques that can be used to solve this integral, such as integration by parts, substitution, and using special functions. However, the specific technique used may vary depending on the specific form of the integral.

5. What are some real-life applications of this integral?

This integral has many applications in various fields, such as in physics for calculating the probability of particles in a quantum system, in statistics for calculating the probability of data falling within a certain range, and in engineering for modeling and analyzing complex systems.

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