Scorer's function

Remarks

They solve the inhomogeneous Airy equations $\operatorname{Gi}'' - x \operatorname{Gi} = -\frac{1}{\pi}$ and $\operatorname{Hi}'' - x \operatorname{Hi} = \frac{1}{\pi}$. Check for $\operatorname{Hi}$: differentiating under the integral sign, $\operatorname{Hi}'' - x \operatorname{Hi} = \frac{1}{\pi} \int_0^\infty (t^2 - x)\, e^{-t^3/3 + xt}\, dt = -\frac{1}{\pi} \left[ e^{-t^3/3 + xt} \right]_0^\infty = \frac{1}{\pi}$. By comparing with the integral that defines the Airy function $\operatorname{Bi}$, one has $\operatorname{Gi} + \operatorname{Hi} = \operatorname{Bi}$. A non-example: $\operatorname{Bi}$ itself is not a Scorer function; it solves the homogeneous equation $y'' - x y = 0$, not one with right-hand side $\pm \frac{1}{\pi}$. Both functions extend to entire functions of $z \in \C$.