Can anyone help me with the following integral: $$ \int_{-\infty}^\infty dt \frac{\exp\left\{-t^2/2\right\}}{\sqrt{2\pi}} \left(\frac{1}{2}\text{erfc}\left\{\frac{a-bt}{\sqrt{2}}\right\}\right)^n, $$ where erfc is the complementary error function, $n$ is any positive integer, $a$ & $b$ can be any real number (the special case where $b =1$ is trivial...)? Many thanks in advance
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The exact answer for $n=1$ and $n=2$, as well as an approximate solution for arbitrary $n$, is here – Sal Jun 20 '21 at 02:59
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Thank you for your reply. But I need to know all the exact results of the same expression with arbitrary integer $n>0$. In fact, what I really want to calculate is $$ \int_{-\infty}^\infty dt \frac{\exp\left{-t^2/2\right}}{\sqrt{2\pi}} \log\left{c + \frac{d}{2} \text{erfc}\left{\frac{a-bt}{\sqrt{2}}\right}\right},$$ where for now $c$ & $d$ could be approximately $0$ & $1$, but $b \not=1$ as this is a trivial case. – Jimmy Jun 21 '21 at 00:50