Integral of a gaussian times ("shifted") erf squared

Question

I would like to know whether this integral could be solved analytically

$\int_{-\infty }^{\infty } e^{-x^2} \text{erf}(k-x)^2 \, \text{d}x$

where $k$ is a real constant.

P.S. I know that $\int e^{-x^2} \text{erf}(x)^2 \, \text{d}x = \frac{1}{6} \sqrt{\pi } \text{erf}(x)^3$ .

https://math.stackexchange.com/questions/2236490/closed-form-for-definite-integral-involving-erf-and-gaussian?rq=1 this may be useful — Siddhartha, Jun 24 '17 at 13:02

score 0 · Answer 1 · answered Jul 02 '17 at 11:25

0

For what it's worth, I can give you the approximate solution

$$ \int_{-\infty }^{\infty } e^{-x^2} \text{erf}(k-x)^2 \, \text{d}x\approx \sqrt{\pi }-\sqrt{\frac{8 \pi }{8+\pi ^2}}\text{exp}\left(-\frac{k^2 \pi ^2}{8+\pi ^2}\right) $$

answered Jul 02 '17 at 11:25

petru

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Integral of a gaussian times ("shifted") erf squared

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