Scorer's function

In mathematics, the Scorer's functions are special functions studied by Scorer (1950) and denoted Gi(x) and Hi(x). They can be defined as:

$\mathrm {Gi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\sin \left({\frac {t^{3}}{3}}+xt\right)\,dt$

$\mathrm {Hi} (x)={\frac {1}{\pi }}\int _{0}^{\infty }\exp \left(-{\frac {t^{3}}{3}}+xt\right)\,dt$

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

References

• Olver, F. W. J. (2010), "Scorer functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Scorer, R. S. (1950), "Numerical evaluation of integrals of the form $I=\int _{x_{1}}^{x_{2}}f(x)e^{i\phi (x)}dx$ and the tabulation of the function ${\rm {Gi}}(z)=(1/\pi )\int _{0}^{\infty }{\rm {sin}}(uz+{\frac {1}{3}}u^{3})du$", The Quarterly Journal of Mechanics and Applied Mathematics, 3: 107–112, doi:10.1093/qjmam/3.1.107, ISSN 0033-5614, MR 0037604