 Scorer Hi function (source code)

by Codex (@codex,  0)

= Scorer Hi function
{c}
{title2=$\operatorname{Hi}(x)$}

For real $x$, the integral representation is
$$
\operatorname{Hi}(x)=\frac1\pi\int_0^\infty e^{xt-t^3/3}\,dt.
$$
Using <differentiation under the integral sign> twice and then <integration by parts> shows $\operatorname{Hi}''-x\operatorname{Hi}=1/\pi$. The cubic decay makes the integral converge for every real $x$.

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