Scorer's function

\mathrm{Gi}(x)

and

\mathrm{Hi}(x)

In mathematics, the Scorer's functions are special functions studied by Scorer (1950) and denoted Gi(x) and Hi(x).

They solve the equation

y''(x) - x\ y(x) = \frac{1}{\pi}

and are given by

\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt,

\mathrm{Hi}(x) = \frac{1}{\pi} \int_0^\infty \exp\left(-\frac{t^3}{3} + xt\right)\, dt.

The Scorer's functions can also be defined in terms of Airy functions:

\begin{align} \mathrm{Gi}(x) &{}= \mathrm{Bi}(x) \int_x^\infty \mathrm{Ai}(t) \, dt + \mathrm{Ai}(x) \int_0^x \mathrm{Bi}(t) \, dt, \\ \mathrm{Hi}(x) &{}= \mathrm{Bi}(x) \int_{-\infty}^x \mathrm{Ai}(t) \, dt - \mathrm{Ai}(x) \int_{-\infty}^x \mathrm{Bi}(t) \, dt. \end{align}

References

Olver, F. W. J. (2010), "Scorer functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Scorer, R. S. (1950), "Numerical evaluation of integrals of the form $I=\int^{x_2}_{x_{1}}f(x)e^{i\phi(x)}dx$ and the tabulation of the function ${\rm Gi} (z)=\frac{1}{\pi}\int^\infty_0{\rm sin}\left(uz+\frac 13 u^3\right)du$ ", The Quarterly Journal of Mechanics and Applied Mathematics 3: 107–112, doi:10.1093/qjmam/3.1.107, ISSN 0033-5614, MR 0037604