§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 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$, and the Scorer functions $\mathop{\mathrm{Gi}\/}\nolimits(z)$ and $\mathop{\mathrm{Hi}\/}\nolimits(z)$ (also known as inhomogeneous Airy functions).

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