# §9.13 Generalized Airy Functions

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

Equations of the form

 9.13.1 $\frac{{d}^{2}w}{{dz}^{2}}=z^{n}w,$ $n=1,2,3,\ldots,$ Defines: $n$: parameter (locally) Symbols: $\frac{d\NVar{f}}{d\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $w$: function Referenced by: §9.13(i), §9.13(i), §9.13(i) Permalink: http://dlmf.nist.gov/9.13.E1 Encodings: TeX, pMML, png

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

 9.13.2 $w=z^{1/2}\mathop{\mathscr{Z}_{p}\/}\nolimits\!\left(\zeta\right),$ Defines: $w$: function (locally) Symbols: $\mathop{\mathscr{Z}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified cylinder function, $z$: complex variable, $p$: variable and $\zeta$: variable Permalink: http://dlmf.nist.gov/9.13.E2 Encodings: TeX, pMML, png

where

 9.13.3 $\displaystyle p$ $\displaystyle=\frac{1}{n+2}$, $\displaystyle\zeta$ $\displaystyle=\frac{2}{n+2}z^{(n+2)/2}=2pz^{1/(2p)}$, Defines: $p$: variable (locally) and $\zeta$: variable (locally) Symbols: $z$: complex variable and $n$: parameter Permalink: http://dlmf.nist.gov/9.13.E3 Encodings: TeX, TeX, pMML, pMML, png, png

and $\mathop{\mathscr{Z}_{p}\/}\nolimits$ is any linear combination of the modified Bessel functions $\mathop{I_{p}\/}\nolimits$ and $e^{p\pi i}\mathop{K_{p}\/}\nolimits$10.25(ii)).

Swanson and Headley (1967) define independent solutions $\mathop{A_{n}\/}\nolimits\!\left(z\right)$ and $\mathop{B_{n}\/}\nolimits\!\left(z\right)$ of (9.13.1) by

 9.13.4 $\displaystyle\mathop{A_{n}\/}\nolimits\!\left(z\right)$ $\displaystyle=(2p/\pi)\mathop{\sin\/}\nolimits\!\left(p\pi\right)z^{1/2}% \mathop{K_{p}\/}\nolimits\!\left(\zeta\right)$, $\displaystyle\mathop{B_{n}\/}\nolimits\!\left(z\right)$ $\displaystyle=(pz)^{1/2}\left(\mathop{I_{-p}\/}\nolimits\!\left(\zeta\right)+% \mathop{I_{p}\/}\nolimits\!\left(\zeta\right)\right)$,

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

Properties of $\mathop{A_{n}\/}\nolimits\!\left(z\right)$ and $\mathop{B_{n}\/}\nolimits\!\left(z\right)$ follow from the corresponding properties of the modified Bessel functions. They include:

 9.13.5 $\displaystyle\mathop{A_{n}\/}\nolimits\!\left(0\right)$ $\displaystyle=p^{1/2}\mathop{B_{n}\/}\nolimits\!\left(0\right)=\frac{p^{1-p}}{% \mathop{\Gamma\/}\nolimits\!\left(1-p\right)},$ $\displaystyle-\mathop{A_{n}\/}\nolimits'\!\left(0\right)$ $\displaystyle=p^{1/2}\mathop{B_{n}\/}\nolimits'\!\left(0\right)=\frac{p^{p}}{% \mathop{\Gamma\/}\nolimits\!\left(p\right)}.$
 9.13.6 $\displaystyle\mathop{A_{n}\/}\nolimits\!\left(-z\right)$ $\displaystyle=\begin{cases}pz^{1/2}\left(\mathop{J_{-p}\/}\nolimits\!\left(% \zeta\right)+\mathop{J_{p}\/}\nolimits\!\left(\zeta\right)\right),&n\text{ odd% },\\ p^{1/2}\mathop{B_{n}\/}\nolimits\!\left(z\right),&n\text{ even},\end{cases}$ 9.13.7 $\displaystyle\mathop{B_{n}\/}\nolimits\!\left(-z\right)$ $\displaystyle=\begin{cases}(pz)^{1/2}\left(\mathop{J_{-p}\/}\nolimits\!\left(% \zeta\right)-\mathop{J_{p}\/}\nolimits\!\left(\zeta\right)\right),&n\text{ odd% },\\ p^{-1/2}\mathop{A_{n}\/}\nolimits\!\left(z\right),&n\text{ even}.\end{cases}$
 9.13.8 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{A_{n}\/}\nolimits\!\left(z\right% ),\mathop{B_{n}\/}\nolimits\!\left(z\right)\right\}=\frac{2}{\pi}p^{1/2}% \mathop{\sin\/}\nolimits\!\left(p\pi\right).$

As $z\to\infty$

 9.13.9 $\displaystyle\mathop{A_{n}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sqrt{\ifrac{p}{\pi}}\mathop{\sin\/}\nolimits\!\left(p\pi\right)% z^{-n/4}e^{-\zeta}\left(1+\mathop{O\/}\nolimits\!\left(\zeta^{-1}\right)\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq 3p\pi-\delta$, 9.13.10 $\displaystyle\mathop{A_{n}\/}\nolimits\!\left(-z\right)$ $\displaystyle=\begin{cases}2\sqrt{p/\pi}\mathop{\cos\/}\nolimits\!\left(\tfrac% {1}{2}p\pi\right)z^{-n/4}\left(\mathop{\cos\/}\nolimits\!\left(\zeta-\tfrac{1}% {4}\pi\right)+e^{|\Im\zeta|}\mathop{O\/}\nolimits\!\left(\zeta^{-1}\right)% \right),&\text{|\mathop{\mathrm{ph}\/}\nolimits z|\leq 2p\pi-\delta, n odd% },\\ \sqrt{p/\pi}z^{-n/4}e^{\zeta}\left(1+\mathop{O\/}\nolimits\!\left(\zeta^{-1}% \right)\right),&\text{|\mathop{\mathrm{ph}\/}\nolimits z|\leq p\pi-\delta, % n even},\end{cases}$ 9.13.11 $\displaystyle\mathop{B_{n}\/}\nolimits\!\left(z\right)$ $\displaystyle={\pi}^{-1/2}z^{-n/4}e^{\zeta}\left(1+\mathop{O\/}\nolimits\!% \left(\zeta^{-1}\right)\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq p\pi-\delta$, 9.13.12 $\displaystyle\mathop{B_{n}\/}\nolimits\!\left(-z\right)$ $\displaystyle=\begin{cases}-(\ifrac{2}{\sqrt{\pi}})\mathop{\sin\/}\nolimits\!% \left(\tfrac{1}{2}p\pi\right)z^{-n/4}\left(\mathop{\sin\/}\nolimits\!\left(% \zeta-\tfrac{1}{4}\pi\right)+e^{\left|\imagpart{\zeta}\right|}\mathop{O\/}% \nolimits\!\left(\zeta^{-1}\right)\right),&\left|\mathop{\mathrm{ph}\/}% \nolimits z\right|\leq 2p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\mathop{\sin\/}\nolimits\!\left(p\pi\right)z^{-n/4}e^{-% \zeta}\left(1+\mathop{O\/}\nolimits\!\left(\zeta^{-1}\right)\right),&\left|% \mathop{\mathrm{ph}\/}\nolimits z\right|\leq 3p\pi-\delta,n\text{ even}.\end{cases}$

The distribution in $\Complex$ and asymptotic properties of the zeros of $\mathop{A_{n}\/}\nolimits\!\left(z\right)$, $\mathop{A_{n}\/}\nolimits'\!\left(z\right)$, $\mathop{B_{n}\/}\nolimits\!\left(z\right)$, and $\mathop{B_{n}\/}\nolimits'\!\left(z\right)$ 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

 9.13.13 $\frac{{d}^{2}w}{{dt}^{2}}=\tfrac{1}{4}m^{2}t^{m-2}w,$

where $m=3,4,5,\ldots.$ 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 $\mathop{A_{n}\/}\nolimits\!\left(z\right)$ and $\mathop{B_{n}\/}\nolimits\!\left(z\right)$ are given by

 9.13.14 $\displaystyle m$ $\displaystyle=n+2=1/p$, $\displaystyle t$ $\displaystyle=(\tfrac{1}{2}m)^{-2/m}z=\zeta^{2/m}$,
 9.13.15 $\sqrt{2\pi}\left(\tfrac{1}{2}m\right)^{(m-1)/m}\mathop{\csc\/}\nolimits\!\left% (\ifrac{\pi}{m}\right)\mathop{A_{n}\/}\nolimits\!\left(z\right)=\begin{cases}U% _{m}(t),&m\text{ even},\\ V_{m}(t),&m\text{ odd},\end{cases}$
 9.13.16 $\sqrt{\pi}\left(\tfrac{1}{2}m\right)^{(m-2)/(2m)}\mathop{\csc\/}\nolimits\!% \left(\ifrac{\pi}{m}\right)\mathop{B_{n}\/}\nolimits\!\left(z\right)=\begin{% cases}U_{m}(-t),&m\text{ even},\\ \overline{V}_{m}(t),&m\text{ odd}.\end{cases}$

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

 9.13.17 $\frac{{d}^{2}w}{{dt}^{2}}=-\tfrac{1}{4}m^{2}t^{m-2}w,$ $m$ even, Defines: $W_{m}$: function (locally) Symbols: $\frac{d\NVar{f}}{d\NVar{x}}$: derivative of $f$ with respect to $x$, $w$: function and $m$: index Referenced by: §9.13(i) Permalink: http://dlmf.nist.gov/9.13.E17 Encodings: TeX, pMML, png

which are denoted by $W_{m}(t)$, $W_{m}(-t)$.

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

 9.13.18 $w=U_{m}(te^{-2j\pi i/m}),$ $j=0,\pm 1,\pm 2,\ldots.$

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

 9.13.19 $\frac{{d}^{2}w}{{dx}^{2}}+x^{\alpha}w=0,$ Defines: $\alpha$: parameter (locally) and $x$: parameter (locally) Symbols: $\frac{d\NVar{f}}{d\NVar{x}}$: derivative of $f$ with respect to $x$ and $w$: function Permalink: http://dlmf.nist.gov/9.13.E19 Encodings: TeX, pMML, png

where $\alpha>-2$ and $x>0$. Solutions are $w=U_{1}(x,\alpha)$, $U_{2}(x,\alpha)$, where

 9.13.20 $U_{1}(x,\alpha)=\frac{1}{(\alpha+2)^{1/(\alpha+2)}}\*\mathop{\Gamma\/}% \nolimits\!\left(\frac{\alpha+1}{\alpha+2}\right)x^{1/2}\mathop{J_{-1/(\alpha+% 2)}\/}\nolimits\!\left(\frac{2}{\alpha+2}x^{(\alpha+2)/2}\right),$
 9.13.21 $U_{2}(x,\alpha)=(\alpha+2)^{1/(\alpha+2)}\*\mathop{\Gamma\/}\nolimits\!\left(% \frac{\alpha+3}{\alpha+2}\right)x^{1/2}\mathop{J_{1/(\alpha+2)}\/}\nolimits\!% \left(\frac{2}{\alpha+2}x^{(\alpha+2)/2}\right),$

and $\mathop{J\/}\nolimits$ 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:

 9.13.22 $\displaystyle\alpha$ $\displaystyle=m-2$, $\displaystyle x$ $\displaystyle=(m/2)^{2/m}t$, Symbols: $\alpha$: parameter, $x$: parameter and $m$: index Permalink: http://dlmf.nist.gov/9.13.E22 Encodings: TeX, TeX, pMML, pMML, png, png
 9.13.23 $U_{1}(x,\alpha)=\frac{\pi^{1/2}}{2^{(m+2)/(2m)}\mathop{\Gamma\/}\nolimits\!% \left(1/m\right)}\left(W_{m}(t)+W_{m}(-t)\right),$
 9.13.24 $U_{2}(x,\alpha)=\frac{\pi^{1/2}m^{2/m}}{2^{(m+2)/(2m)}\mathop{\Gamma\/}% \nolimits\!\left(-1/m\right)}\left(W_{m}(t){-}W_{m}(-t)\right).$

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:

 9.13.25 $\mathop{A_{k}\/}\nolimits\!\left(z,p\right)=\frac{1}{2\pi i}\int_{\mathscr{L}_% {k}}t^{-p}\mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $k=1,2,3$, $p\in\Complex$, Defines: $k$: index (locally) and $p$: parameter (locally) Symbols: $\mathop{A_{\NVar{k}}\/}\nolimits\!\left(\NVar{z},\NVar{p}\right)$: generalized Airy function, $\Complex$: complex plane, $d\NVar{x}$: differential of $x$, $\in$: element of, $\mathop{\exp\/}\nolimits\NVar{z}$: exponential function, $\int$: integral, $z$: complex variable and $\mathscr{L}$: integration path Referenced by: §9.13(ii) Permalink: http://dlmf.nist.gov/9.13.E25 Encodings: TeX, pMML, png
 9.13.26 $\mathop{B_{0}\/}\nolimits\!\left(z,p\right)=\frac{1}{2\pi i}\int_{\mathscr{L}_% {0}}t^{-p}\mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $p=0,\pm 1,\pm 2,\ldots,$
 9.13.27 $\mathop{B_{k}\/}\nolimits\!\left(z,p\right)=\int_{\mathscr{I}_{k}}t^{-p}% \mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $k=1,2,3$, $p=0,\pm 1,\pm 2,\ldots,$

with $z\in\Complex$ in all cases. The integration paths $\mathscr{L}_{0}$, $\mathscr{L}_{1}$, $\mathscr{L}_{2}$, $\mathscr{L}_{3}$ are depicted in Figure 9.13.1. $\mathscr{I}_{1}$, $\mathscr{I}_{2}$, $\mathscr{I}_{3}$ 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 $\mathop{\exp\/}\nolimits\!\left(-p(\mathop{\ln\/}\nolimits|t|+i\mathop{\mathrm% {ph}\/}\nolimits t)\right)$ with $0\leq\mathop{\mathrm{ph}\/}\nolimits t<2\pi$.

When $p=0$

 9.13.28 $\mathop{A_{1}\/}\nolimits\!\left(z,0\right)=\mathop{\mathrm{Ai}\/}\nolimits\!% \left(z\right),$
 9.13.29 $\displaystyle\mathop{A_{2}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=e^{2\pi i/3}\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{2\pi i/3}\right)$, $\displaystyle\mathop{A_{3}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=e^{-2\pi i/3}\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{-2\pi i/% 3}\right)$,

and

 9.13.30 $\displaystyle\mathop{B_{0}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=0$, $\displaystyle\mathop{B_{1}\/}\nolimits\!\left(z,0\right)$ $\displaystyle=\pi\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$.

Each of the functions $\mathop{A_{k}\/}\nolimits\!\left(z,p\right)$ and $\mathop{B_{k}\/}\nolimits\!\left(z,p\right)$ satisfies the differential equation

 9.13.31 $\frac{{d}^{3}w}{{dz}^{3}}-z\frac{dw}{dz}+(p-1)w=0,$ Defines: $w$: function (locally) Symbols: $\frac{d\NVar{f}}{d\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $p$: parameter Referenced by: §9.13(ii) Permalink: http://dlmf.nist.gov/9.13.E31 Encodings: TeX, pMML, png

and the difference equation

 9.13.32 $f(p-3)-zf(p-1)+(p-1)f(p)=0.$ Defines: $f$: function (locally) Symbols: $z$: complex variable and $p$: parameter Permalink: http://dlmf.nist.gov/9.13.E32 Encodings: TeX, pMML, png

The $\mathop{A_{k}\/}\nolimits\!\left(z,p\right)$ are related by

 9.13.33 $\displaystyle\mathop{A_{2}\/}\nolimits\!\left(z,p\right)$ $\displaystyle=e^{-2(p-1)\pi i/3}\mathop{A_{1}\/}\nolimits\!\left(ze^{2\pi i/3}% ,p\right)$, $\displaystyle\mathop{A_{3}\/}\nolimits\!\left(z,p\right)$ $\displaystyle=e^{2(p-1)\pi i/3}\mathop{A_{1}\/}\nolimits\!\left(ze^{-2\pi i/3}% ,p\right)$.

Connection formulas for the solutions of (9.13.31) include

 9.13.34 $\mathop{A_{1}\/}\nolimits\!\left(z,p\right)+\mathop{A_{2}\/}\nolimits\!\left(z% ,p\right)+\mathop{A_{3}\/}\nolimits\!\left(z,p\right)+\mathop{B_{0}\/}% \nolimits\!\left(z,p\right)=0,$
 9.13.35 $\mathop{B_{2}\/}\nolimits\!\left(z,p\right)-\mathop{B_{3}\/}\nolimits\!\left(z% ,p\right)=2\pi i\mathop{A_{1}\/}\nolimits\!\left(z,p\right),$
 9.13.36 $\mathop{B_{3}\/}\nolimits\!\left(z,p\right)-\mathop{B_{1}\/}\nolimits\!\left(z% ,p\right)=2\pi i\mathop{A_{2}\/}\nolimits\!\left(z,p\right),$
 9.13.37 $\mathop{B_{1}\/}\nolimits\!\left(z,p\right)-\mathop{B_{2}\/}\nolimits\!\left(z% ,p\right)=2\pi i\mathop{A_{3}\/}\nolimits\!\left(z,p\right).$

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