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I want to get a closed/ semi-closed form of the integral given below.

$$ \int_{-\infty}^{+\infty} \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\right) \exp{(i x \tau)} dx$$

Where $\text{erf}$ is the error function and $i = \sqrt{-1}$. As $\text{erf}$ itself is an integral, this is a double integral already.

I came across this integral because I wanted to derive an equation that is the Fourier Transform of a skewed Gaussian function.

$$ \exp{\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)} \left( 1 + \text{erf}\left(\alpha \frac{x-\mu}{\sqrt{2}\sigma}\right) \right) $$

the first term is a regular Gaussian function. The second term is a complicated one because it has the error function also. Is there a way to approximate the skewed Gaussian function so that it could have a Fourier Transform based on the parameters $\alpha, \mu$ and $\sigma$ ?

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    $\begingroup$ there is little hope for a "closed" form (or even some reduction to a special function), not even if $\mu=0$. $\endgroup$
    – Carlo Beenakker
    Commented Jan 20, 2023 at 14:25
  • $\begingroup$ I am adding a new purpose based on approximations. $\endgroup$
    – CfourPiO
    Commented Jan 23, 2023 at 9:23
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    $\begingroup$ That’s a slight improvement — but why do you want a Fourier transform of this, or a closed form for the transform? $\endgroup$
    – user44143
    Commented Jan 23, 2023 at 14:03
  • $\begingroup$ I want to construct a covariance function of a signal that has a skewed Gaussian like power spectrum. The covariance function is nothing but the inverse Fourier of the power spectrum. That is why I wanted this. A closed form would help because of computational issues. $\endgroup$
    – CfourPiO
    Commented Jan 23, 2023 at 15:14

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