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As is well known, there is no explicit formula for $\int_{-\infty}^\infty step(t−x)\cdot e^{−t^2/2}dt=\int_x^\infty e^{−t^2/2} dt$ for generic $x,$ where $step(z)$ is the step function, $step(z)=1$ for $z>0,$ $step(z)=0$ for $z<0.$

Is there a "nice" family of functions $s_\alpha:{\mathbb R}\to [0,1]$ which converge to $step$ as $\alpha\to 0$ and such that $\int_{-\infty}^\infty s_\alpha(t−x)⋅e^{−t^2/2} dt$ can be computed explicitly for generic $x$ and $\alpha$?

If not then perhaps a single function $s$ which is a "reasonable" approximation of $step$ for which the above integral can be computed for a generic $x$?

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You can explicitly compute the integral for polynomials $s_\alpha$. Take suitable linear combinations of Hermite polynomials approximating step in $L^2(\exp(-t^2/2))$.

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    $\begingroup$ I agree, but I was hoping for functions $\mathbb R\to [0,1]$... $\endgroup$
    – Adam
    Commented Jan 20, 2016 at 19:49
  • $\begingroup$ Also, I am looking for a "nice" family, which I hope can be explicitly written down. (Obviously, there are various brut force approaches possible, leading to some messy definitions of $s_\alpha$) $\endgroup$
    – Adam
    Commented Jan 20, 2016 at 19:58
  • $\begingroup$ Do the linear combinations have coefficients that are "not explicit formulas" in the sense of this question? $\endgroup$
    – Douglas Zare
    Commented Jan 21, 2016 at 17:27

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