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

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SUMMARY

The integral \(\int_{-\infty}^{+\infty} e^{-x^2} \operatorname{Erf}(x^2 - a^2) dx\) presents significant challenges in obtaining an analytical solution, as confirmed by multiple users who attempted to solve it using Mathematica without success. The integrand is sharply peaked around \(x = 0\), suggesting that techniques such as the method of steepest descent may be applicable for approximation. Users recommend expanding the integrand around \(x = 0\) to facilitate further analysis.

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  • Understanding of integral calculus, particularly improper integrals.
  • Familiarity with the error function, \(\operatorname{Erf}\), and its properties.
  • Knowledge of asymptotic analysis and the method of steepest descent.
  • Basic proficiency in using Mathematica for symbolic computation.
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  • Research the method of steepest descent for evaluating integrals.
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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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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)?
 

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