# §9.13 Generalized Airy Functions

## §9.13(i) Generalizations from the Differential Equation

Equations of the form

9.13.1

are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). The general solution of (9.13.1) is given by

where

9.13.3
,
,

and is any linear combination of the modified Bessel functions and 10.25(ii)).

Swanson and Headley (1967) define independent solutions and of (9.13.1) by

when is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of .) When and become and , respectively.

The distribution in and asymptotic properties of the zeros of , , , and are investigated in Swanson and Headley (1967) and Headley and Barwell (1975).

In Olver (1977a, 1978) a different normalization is used. In place of (9.13.1) we have

9.13.13

where For real variables the solutions of (9.13.13) are denoted by , when is even, and by , when is odd. (The overbar has nothing to do with complex conjugates.) Their relations to the functions and are given by

Properties and graphs of , , are included in Olver (1977a) together with properties and graphs of real solutions of the equation

9.13.17 even,

which are denoted by , .

In , the solutions of (9.13.13) used in Olver (1978) are

The function on the right-hand side is recessive in the sector , and is therefore an essential member of any numerically satisfactory pair of solutions in this region.

Another normalization of (9.13.17) is used in Smirnov (1960), given by

9.13.19

where and . Solutions are , , where

and denotes the Bessel function (§10.2(ii)).

When is a positive integer the relation of these functions to , is as follows:

9.13.22
,
,

For properties of the zeros of the functions defined in this subsection see Laforgia and Muldoon (1988) and references given therein.

## §9.13(ii) Generalizations from Integral Representations

Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow:

with in all cases. The integration paths , , , are depicted in Figure 9.13.1. , , are depicted in Figure 9.13.2. When is not an integer the branch of in (9.13.25) is usually chosen to be with .

 Figure 9.13.1: -plane. Paths , , , . Symbols: : integration path Referenced by: §9.13(ii) Permalink: http://dlmf.nist.gov/9.13.F1 Encodings: pdf, png Figure 9.13.2: -plane. Paths , , . Symbols: : integration path Referenced by: §9.13(ii) Permalink: http://dlmf.nist.gov/9.13.F2 Encodings: pdf, png

Each of the functions and satisfies the differential equation

9.13.31

and the difference equation

9.13.32

Further properties of these functions, and also of similar contour integrals containing an additional factor , in the integrand, are derived in Reid (1972), Drazin and Reid (1981, Appendix), and Baldwin (1985). These properties include Wronskians, asymptotic expansions, and information on zeros.

For further generalizations via integral representations see Chin and Hedstrom (1978), Janson et al. (1993, §10), and Kamimoto (1998).