# §9.1 Special Notation

(For other notation see Notation for the Special Functions.)

$k$ nonnegative integer, except in §9.9(iii). real variable. complex variable. arbitrary small positive constant. derivatives with respect to argument.

The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions).

Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi}\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi}\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-\ifrac{1}{3}}\pi\mathrm{Ai}\left(-3^{-\ifrac{1}{3}}x\right)$ (Szegő (1967, §1.81)); $e_{0}(x)=\pi\mathrm{Hi}(-x)$, $\widetilde{e}_{0}(x)=-\pi\mathrm{Gi}(-x)$ (Tumarkin (1959)).