§9.1 Special Notation

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Keywords:: Airy functions, Scorer functions, generalized, inhomogeneous Airy functions, notation
Permalink:: http://dlmf.nist.gov/9.1
See also:: Annotations for Ch.9

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

$k$	nonnegative integer, except in §9.9(iii).
$x$	real variable.
$z(=x+\mathrm{i}y)$	complex variable.
$\delta$	arbitrary small positive constant.
primes	derivatives with respect to argument.

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

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

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