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9 Airy and Related FunctionsNotation

§9.1 Special Notation

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

k nonnegative integer, except in §9.9(iii).
x real variable.
z(=x+iy) complex variable.
δ arbitrary small positive constant.
primes derivatives with respect to argument.

The main functions treated in this chapter are the Airy functions Ai(z) and Bi(z), and the Scorer functions Gi(z) and Hi(z) (also known as inhomogeneous Airy functions).

Other notations that have been used are as follows: Ai(-x) and Bi(-x) for Ai(x) and Bi(x) (Jeffreys (1928), later changed to Ai(x) and Bi(x)); U(x)=πBi(x), V(x)=πAi(x) (Fock (1945)); A(x)=3-1/3πAi(-3-1/3x) (Szegö (1967, §1.81)); e0(x)=πHi(-x), e~0(x)=-πGi(-x) (Tumarkin (1959)).