Scorer's function explained

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

Hi(x) and -Gi(x) solve the equation

y''(x) - x y(x) = (1)/(\pi)

and are given by

Gi (x) = (1)/(\pi) \int ^{inf ty} _{0} \sin\left( (t ^{3})/(3) + xt\right) dt, Hi (x) = (1)/(\pi) \int ^{inf ty} _{0} \exp\left( - (t ^{3})/(3) + xt\right) dt.

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

\begin{align} Gi (x) &{} = Bi (x) \int ^{inf ty} _{x} Ai (t) dt + Ai (x) \int ^{x} _{0} Bi (t) dt, \\ Hi (x) &{} = Bi (x) \int ^{x} _{-inf ty} Ai (t) dt - Ai (x) \int ^{x} _{-inf ty} Bi (t) dt. \end{align}