Scorer function ($\mathrm{Gi},\mathrm{Hi}$)

The Scorer functions are particular solutions of inhomogeneous Airy equations: ${\mathrm{Hi}}^{\prime \prime }(x)-x\mathrm{Hi}(x)=1/\pi$ and ${\mathrm{Gi}}^{\prime \prime }(x)-x\mathrm{Gi}(x)=-1/\pi$. They occur when an endpoint participates in an Airy-type transition. The NIST Digital Library of Mathematical Functions gives their standard normalizations.

For real $x$, the integral representation isUsing differentiation under the integral sign twice and then integration by parts shows ${\mathrm{Hi}}^{\prime \prime }-x\mathrm{Hi}=1/\pi$. The cubic decay makes the integral converge for every real $x$.

The integral $I(x)={\int }_0^{\infty }{e}^{xt-t^3/3}dt$ is a rescaled Scorer Hi function. It has the three useful limitsFor 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.