§9.13 Generalized Airy Functions

Keywords:: generalized Airy functions
Permalink:: http://dlmf.nist.gov/9.13

§9.13(i) Generalizations from the Differential Equation
§9.13(ii) Generalizations from Integral Representations

§9.13(i) Generalizations from the Differential Equation

Defines:: $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function and $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function
Keywords:: Bessel functions, generalized Airy functions, modified Bessel functions
Permalink:: http://dlmf.nist.gov/9.13.i

Equations of the form

9.13.1 $\frac{{d}^{2}w}{{dz}^{2}}=z^{n}w,$ $n=1,2,3,\ldots,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable, : parameter and : 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 $w=z^{{1/2}}\mathop{\mathscr{Z}_{{p}}\/}\nolimits\!\left(\zeta\right),$

Symbols:: $\mathop{\mathscr{Z}_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified cylinder function, : complex variable, : function, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E2
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where

9.13.3

$p=\frac{1}{n+2}$ ,

$\zeta=\frac{2}{n+2}z^{{(n+2)/2}}=2pz^{{1/(2p)}}$ ,

Symbols:: : complex variable, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E3
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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

$\mathop{A_{{n}}\/}\nolimits\!\left(z\right)=(2p/\pi)\mathop{\sin\/}\nolimits\!% \left(p\pi\right)z^{{1/2}}\mathop{K_{{p}}\/}\nolimits\!\left(\zeta\right)$ ,

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

Symbols:: $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E4
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when is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of .) When $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ 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

$\mathop{A_{{n}}\/}\nolimits\!\left(0\right)=p^{{1/2}}\mathop{B_{{n}}\/}% \nolimits\!\left(0\right)=\frac{p^{{1-p}}}{\mathop{\Gamma\/}\nolimits\!\left(1% -p\right)},$

$-{\mathop{A_{{n}}\/}\nolimits^{{\prime}}}\!\left(0\right)=p^{{1/2}}{\mathop{B_% {{n}}\/}\nolimits^{{\prime}}}\!\left(0\right)=\frac{p^{p}}{\mathop{\Gamma\/}% \nolimits\!\left(p\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, : parameter and : variable
Permalink:: http://dlmf.nist.gov/9.13.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

9.13.6 $\mathop{A_{{n}}\/}\nolimits\!\left(-z\right)=\begin{cases}pz^{{1/2}}\left(% \mathop{J_{{-p}}\/}\nolimits\!\left(\zeta\right)+\mathop{J_{{p}}\/}\nolimits\!% \left(\zeta\right)\right),&n\text{ odd},\\ p^{{1/2}}\mathop{B_{{n}}\/}\nolimits\!\left(z\right),&n\text{ even},\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, : complex variable, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E6
Encodings:: TeX, pMML, png

9.13.7 $\mathop{B_{{n}}\/}\nolimits\!\left(-z\right)=\begin{cases}(pz)^{{1/2}}\left(% \mathop{J_{{-p}}\/}\nolimits\!\left(\zeta\right)-\mathop{J_{{p}}\/}\nolimits\!% \left(\zeta\right)\right),&n\text{ odd},\\ p^{{-1/2}}\mathop{A_{{n}}\/}\nolimits\!\left(z\right),&n\text{ even}.\end{cases}$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, : complex variable, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E7
Encodings:: TeX, pMML, png

9.13.8 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{A_{{n}}\/}\nolimits\!\left(z% \right),\mathop{B_{{n}}\/}\nolimits\!\left(z\right)\right\}=\frac{2}{\pi}p^{{1% /2}}\mathop{\sin\/}\nolimits\!\left(p\pi\right).$

Symbols:: $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : parameter and : variable
Permalink:: http://dlmf.nist.gov/9.13.E8
Encodings:: TeX, pMML, png

As $z\to\infty$

9.13.9 $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)=\sqrt{\ifrac{p}{\pi}}\mathop{\sin% \/}\nolimits\!\left(p\pi\right)z^{{-n/4}}e^{{-\zeta}}\left(1+\mathop{O\/}% \nolimits\!\left(\zeta^{{-1}}\right)\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq 3p\pi-\delta$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E9
Encodings:: TeX, pMML, png

9.13.10 $\mathop{A_{{n}}\/}\nolimits\!\left(-z\right)=\begin{cases}2\sqrt{p/\pi}\mathop% {\cos\/}\nolimits\!\left(\tfrac{1}{2}p\pi\right)z^{{-n/4}}\left(\mathop{\cos\/% }\nolimits\!\left(\zeta-\tfrac{1}{4}\pi\right)+e^{{|\Im\zeta|}}\mathop{O\/}% \nolimits\!\left(\zeta^{{-1}}\right)\right),&\text{$|\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}$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\imagpart{}$ : imaginary part, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E10
Encodings:: TeX, pMML, png

9.13.11 $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)={\pi}^{{-1/2}}z^{{-n/4}}e^{{\zeta}% }\left(1+\mathop{O\/}\nolimits\!\left(\zeta^{{-1}}\right)\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq p\pi-\delta$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E11
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, : base of exponential function, $\imagpart{}$ : imaginary part, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E12
Encodings:: TeX, pMML, png

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

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

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : function and : index
Referenced by:: §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E13
Encodings:: TeX, pMML, png

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 is even, and by $V_{m}(t)$ , $\overline{V}_{m}(t)$ when 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

$t=(\tfrac{1}{2}m)^{{-2/m}}z=\zeta^{{2/m}}$ ,

Symbols:: : complex variable, : parameter, : variable, $\zeta$ : variable and : index
Permalink:: http://dlmf.nist.gov/9.13.E14
Encodings:: TeX, TeX, pMML, pMML, png, png

9.13.15 $\sqrt{2\pi}\left(\tfrac{1}{2}m\right)^{{(m-1)/m}}\mathop{\csc\/}\nolimits\!% \left(\ifrac{\pi}{m}\right)\mathop{A_{{n}}\/}\nolimits\!\left(z\right)=\begin{% cases}U_{m}(t),&m\text{ even},\\ V_{m}(t),&m\text{ odd},\end{cases}$

Symbols:: $\mathop{A_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{\csc\/}\nolimits z$ : cosecant function, : complex variable, : parameter, $U_{i}$ : Smirnov’s generalized Airy function, : index and $V_{m}$ : Olver’s generalized Airy function
Permalink:: http://dlmf.nist.gov/9.13.E15
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(z\right)$ : generalized Airy function, $\mathop{\csc\/}\nolimits z$ : cosecant function, : complex variable, : parameter, $U_{i}$ : Smirnov’s generalized Airy function, : index and $V_{m}$ : Olver’s generalized Airy function
Permalink:: http://dlmf.nist.gov/9.13.E16
Encodings:: TeX, pMML, png

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

even,

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : function, : index and $W_{m}$ : function
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 $w=U_{m}(te^{{-2j\pi i/m}}),$ $j=0,\pm 1,\pm 2,\ldots.$

Symbols:: : base of exponential function, $U_{i}$ : Smirnov’s generalized Airy function, : function, : index and : index
Permalink:: http://dlmf.nist.gov/9.13.E18
Encodings:: TeX, pMML, png

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 $\frac{{d}^{2}w}{{dx}^{2}}+x^{{\alpha}}w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\alpha$ : parameter, : parameter and : function
Permalink:: http://dlmf.nist.gov/9.13.E19
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\alpha$ : parameter, : parameter and $U_{i}$ : Smirnov’s generalized Airy function
Referenced by:: §9.18(vii)
Permalink:: http://dlmf.nist.gov/9.13.E20
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\alpha$ : parameter, : parameter and $U_{i}$ : Smirnov’s generalized Airy function
Referenced by:: §9.18(vii)
Permalink:: http://dlmf.nist.gov/9.13.E21
Encodings:: TeX, pMML, png

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

$\alpha=m-2$ ,

$x=(m/2)^{{2/m}}t$ ,

Symbols:: $\alpha$ : parameter, : parameter and : index
Permalink:: http://dlmf.nist.gov/9.13.E22
Encodings:: TeX, TeX, pMML, pMML, png, png

9.13.23 $U_{1}(x,\alpha)=\frac{\pi^{{1/2}}}{2^{{(m+2)/(2m)}}\mathop{\Gamma\/}\nolimits% \!\left(1/m\right)}\left(W_{m}(t)+W_{m}(-t)\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\alpha$ : parameter, : parameter, $U_{i}$ : Smirnov’s generalized Airy function, : index and $W_{m}$ : function
Permalink:: http://dlmf.nist.gov/9.13.E23
Encodings:: TeX, pMML, png

9.13.24 $U_{2}(x,\alpha)=\frac{\pi^{{1/2}}m^{{2/m}}}{2^{{(m+2)/(2m)}}\mathop{\Gamma\/}% \nolimits\!\left(-1/m\right)}\left(W_{m}(t){-}W_{m}(-t)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\alpha$ : parameter, : parameter, $U_{i}$ : Smirnov’s generalized Airy function, : index and $W_{m}$ : function
Permalink:: http://dlmf.nist.gov/9.13.E24
Encodings:: TeX, pMML, png

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

Defines:: $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function and $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function
Keywords:: generalized Airy functions
Permalink:: http://dlmf.nist.gov/9.13.ii

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 $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)=\frac{1}{2\pi i}\int_{{\mathscr{% L}_{k}}}t^{{-p}}\mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$

, $p\in\Complex$ ,

Symbols:: $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, $\Complex$ : complex plane (excluding infinity), : differential of , $\in$ : element of, $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : complex variable, : index, : parameter 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 $\mathop{B_{{0}}\/}\nolimits\!\left(z,p\right)=\frac{1}{2\pi i}\int_{{\mathscr{% L}_{0}}}t^{{-p}}\mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $p=0,\pm 1,\pm 2,\ldots,$

Symbols:: $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : complex variable, : parameter and $\mathscr{L}$ : integration path
Permalink:: http://dlmf.nist.gov/9.13.E26
Encodings:: TeX, pMML, png

9.13.27 $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)=\int_{{\mathscr{I}_{k}}}t^{{-p}}% \mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$

, $p=0,\pm 1,\pm 2,\ldots,$

Symbols:: $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, : complex variable, : index, : parameter and $\mathscr{I}$ : integration path
Permalink:: http://dlmf.nist.gov/9.13.E27
Encodings:: TeX, pMML, png

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 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 $0\leq\mathop{\mathrm{ph}\/}\nolimits t<2\pi$ .

Figure 9.13.1:

-plane. Paths $\mathscr{L}_{0}$ , $\mathscr{L}_{1}$ , $\mathscr{L}_{2}$ , $\mathscr{L}_{3}$ .

Symbols:: $\mathscr{L}$ : integration path
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.F1
Encodings:: pdf, png

Figure 9.13.2:

-plane. Paths $\mathscr{I}_{1}$ , $\mathscr{I}_{2}$ , $\mathscr{I}_{3}$ .

Symbols:: $\mathscr{I}$ : integration path
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.F2
Encodings:: pdf, png

When

9.13.28 $\mathop{A_{{1}}\/}\nolimits\!\left(z,0\right)=\mathop{\mathrm{Ai}\/}\nolimits% \!\left(z\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function and : complex variable
Permalink:: http://dlmf.nist.gov/9.13.E28
Encodings:: TeX, pMML, png

9.13.29

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

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

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : base of exponential function and : complex variable
Permalink:: http://dlmf.nist.gov/9.13.E29
Encodings:: TeX, TeX, pMML, pMML, png, png

and

9.13.30

$\mathop{B_{{0}}\/}\nolimits\!\left(z,0\right)=0$ ,

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

Symbols:: $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function) and : complex variable
Permalink:: http://dlmf.nist.gov/9.13.E30
Encodings:: TeX, TeX, pMML, pMML, png, png

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 $\frac{{d}^{3}w}{{dz}^{3}}-z\frac{dw}{dz}+(p-1)w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable, : parameter and : function
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.E31
Encodings:: TeX, pMML, png

and the difference equation

9.13.32

Symbols:: : complex variable, : parameter and : function
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

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

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

Symbols:: $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : base of exponential function, : complex variable and : parameter
Permalink:: http://dlmf.nist.gov/9.13.E33
Encodings:: TeX, TeX, pMML, pMML, png, png

Connection formulas for the solutions of (9.13.31) include

9.13.34 $\mathop{A_{{1}}\/}\nolimits\!\left(z,p\right)+\mathop{A_{{2}}\/}\nolimits\!% \left(z,p\right)+\mathop{A_{{3}}\/}\nolimits\!\left(z,p\right)+\mathop{B_{{0}}% \/}\nolimits\!\left(z,p\right)=0,$

Symbols:: $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : complex variable and : parameter
Permalink:: http://dlmf.nist.gov/9.13.E34
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9.13.35 $\mathop{B_{{2}}\/}\nolimits\!\left(z,p\right)-\mathop{B_{{3}}\/}\nolimits\!% \left(z,p\right)=2\pi i\mathop{A_{{1}}\/}\nolimits\!\left(z,p\right),$

Symbols:: $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : complex variable and : parameter
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9.13.36 $\mathop{B_{{3}}\/}\nolimits\!\left(z,p\right)-\mathop{B_{{1}}\/}\nolimits\!% \left(z,p\right)=2\pi i\mathop{A_{{2}}\/}\nolimits\!\left(z,p\right),$

Symbols:: $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : complex variable and : parameter
Permalink:: http://dlmf.nist.gov/9.13.E36
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9.13.37 $\mathop{B_{{1}}\/}\nolimits\!\left(z,p\right)-\mathop{B_{{2}}\/}\nolimits\!% \left(z,p\right)=2\pi i\mathop{A_{{3}}\/}\nolimits\!\left(z,p\right).$

Symbols:: $\mathop{A_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, $\mathop{B_{{k}}\/}\nolimits\!\left(z,p\right)$ : generalized Airy function, : complex variable and : parameter
Permalink:: http://dlmf.nist.gov/9.13.E37
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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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 9.12 Scorer Functions 9.14 Incomplete Airy Functions

§9.13 Generalized Airy Functions

Contents

§9.13(i) Generalizations from the Differential Equation

§9.13(ii) Generalizations from Integral Representations