§9.13 Generalized Airy Functions

Equations of the form

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

and ${𝒵}_p$ is any linear combination of the modified Bessel functions $I_p$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_p$ (§10.25(ii)).

Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by

when $z$ is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of $z$.) When $n=1$, $A_n\left(z\right)$ and $B_n\left(z\right)$ become $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, respectively.

Properties of $A_n\left(z\right)$ and $B_n\left(z\right)$ follow from the corresponding properties of the modified Bessel functions. They include:

As $z\to \mathrm{\infty }$

The distribution in $ℂ$ and asymptotic properties of the zeros of $A_n\left(z\right)$, $A_n^{\prime }\left(z\right)$, $B_n\left(z\right)$, and $B_n^{\prime }\left(z\right)$ 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

where $m=3,4,5,\mathrm{\dots }.$ For real variables the solutions of (9.13.13) are denoted by $U_m(t)$, $U_m(-t)$ when $m$ is even, and by $V_m(t)$, ${\overline{V}}_m(t)$ when $m$ is odd. (The overbar has nothing to do with complex conjugates.) Their relations to the functions $A_n\left(z\right)$ and $B_n\left(z\right)$ are given by

Properties and graphs of $U_m(t)$, $V_m(t)$, ${\overline{V}}_m(t)$ are included in Olver (1977a) together with properties and graphs of real solutions of the equation

which are denoted by $W_m(t)$, $W_m(-t)$.

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

The function on the right-hand side is recessive in the sector $-(2j-1)\pi /m\le \mathrm{ph}z\le (2j+1)\pi /m$, 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

where $\alpha >-2$ and $x>0$. Solutions are $w=U_1(x,\alpha )$, $U_2(x,\alpha )$, where

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

When $\alpha$ is a positive integer the relation of these functions to $W_m(t)$, $W_m(-t)$ is as follows:

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

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 $z\in ℂ$ in all cases. The integration paths ${ℒ}_0$, ${ℒ}_1$, ${ℒ}_2$, ${ℒ}_3$ are depicted in Figure 9.13.1. ${ℐ}_1$, ${ℐ}_2$, ${ℐ}_3$ are depicted in Figure 9.13.2. When $p$ is not an integer the branch of $t^{-p}$ in (9.13.25) is usually chosen to be $\exp \left(-p(\ln |t|+\mathrm{i}\mathrm{ph}t)\right)$ with .

When $p=0$

and

Each of the functions $A_k(z,p)$ and $B_k(z,p)$ satisfies the differential equation

and the difference equation

The $A_k(z,p)$ are related by

Connection formulas for the solutions of (9.13.31) include

Further properties of these functions, and also of similar contour integrals containing an additional factor ${(\ln t)}^q$, $q=1,2,\mathrm{\dots }$, 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).