= Cubic Laplace transition integral
{title2=$I(x)=\pi\operatorname{Hi}(x)$}
The integral $I(x)=\int_0^\infty e^{xt-t^3/3}\,dt$ is a rescaled <Scorer Hi function>. It has the three useful limits
$$
I(-M)\sim M^{-1},\qquad I(0)=3^{-2/3}\Gamma(1/3),\qquad I(M)\sim\sqrt\pi M^{-1/4}e^{2M^{3/2}/3},\quad M\to\infty.
$$
For the negative-argument limit, scale $t=s/M$ and use the <dominated convergence theorem>. At zero, substitute $s=t^3/3$ and use the <Gamma integral>. For the positive-argument limit, the exponent has its maximum at $t=\sqrt M$, with second <derivative> $-2\sqrt M$; <Laplace's method> gives the displayed factor. These estimates describe a <cubic endpoint-to-saddle transition>.
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