Scorer's function

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


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

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} $$