§ 9.13. 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

Permalink:: http://dlmf.nist.gov/9.13.SS1

Equations of the form

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

Defines:: $n$ : parameter
Symbols:: $z$ : complex variable and $w$ : function
Referenced by:: §9.13(i), §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E1
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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 $w=z^{{1/2}}\mathscr{Z}_{{p}}\!\left(\zeta\right),$

Defines:: $w$ : function
Symbols:: $z$ : complex variable, $p$ : variable and $\zeta$ : variable
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where

9.13.3

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

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

Defines:: $p$ : variable and $\zeta$ : variable
Symbols:: $z$ : complex variable and $n$ : parameter
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and $\mathscr{Z}_{{p}}$ is any linear combination of the modified Bessel functions $I_{{p}}$ and $e^{{p\pi i}}K_{{p}}$ (§Ch.10).

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

9.13.4

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

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

Defines:: $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function and $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function
Symbols:: $z$ : complex variable, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E4
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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:

9.13.5

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

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

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $n$ : parameter and $p$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E5
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9.13.6 $A_{{n}}\!\left(-z\right)=\begin{cases}pz^{{1/2}}\left(J_{{-p}}\!\left(\zeta\right)+J_{{p}}\!\left(\zeta\right)\right),&n\text{ odd},\\ p^{{1/2}}B_{{n}}\!\left(z\right),&n\text{ even},\end{cases}$

Symbols:: $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E6
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9.13.7 $B_{{n}}\!\left(-z\right)=\begin{cases}(pz)^{{1/2}}\left(J_{{-p}}\!\left(\zeta\right)-J_{{p}}\!\left(\zeta\right)\right),&n\text{ odd},\\ p^{{-1/2}}A_{{n}}\!\left(z\right),&n\text{ even}.\end{cases}$

Symbols:: $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E7
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9.13.8 $\mathscr{W}\left\{ A_{{n}}\!\left(z\right),B_{{n}}\!\left(z\right)\right\}=\frac{2}{\pi}p^{{1/2}}\sin\!\left(p\pi\right).$

Symbols:: $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $n$ : parameter and $p$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E8
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As $z\to\infty$

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

Symbols:: $O\!\left(x\right)$ : order symbol, $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E9
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9.13.10 $A_{{n}}\!\left(-z\right)=\begin{cases}2\sqrt{p/\pi}\cos\!\left(\tfrac{1}{2}p\pi\right)z^{{-n/4}}\left(\cos\!\left(\zeta-\tfrac{1}{4}\pi\right)+e^{{|\Im\zeta|}}O\!\left(\zeta^{{-1}}\right)\right),&\text{$|\mathrm{ph}z|\leq 2p\pi-\delta$, $n$ odd},\\ \sqrt{p/\pi}z^{{-n/4}}e^{{\zeta}}\left(1+O\!\left(\zeta^{{-1}}\right)\right),&\text{$|\mathrm{ph}z|\leq p\pi-\delta$, $n$ even},\end{cases}$

Symbols:: $O\!\left(x\right)$ : order symbol, $A_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E10
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9.13.11 $B_{{n}}\!\left(z\right)={\pi}^{{-1/2}}z^{{-n/4}}e^{{\zeta}}\left(1+O\!\left(\zeta^{{-1}}\right)\right),$ $|\mathrm{ph}z|\leq p\pi-\delta$ ,

Symbols:: $O\!\left(x\right)$ : order symbol, $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E11
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9.13.12 $B_{{n}}\!\left(-z\right)=\begin{cases}-(\ifrac{2}{\sqrt{\pi}})\sin\!\left(\tfrac{1}{2}p\pi\right)z^{{-n/4}}\left(\sin\!\left(\zeta-\tfrac{1}{4}\pi\right)+e^{{\left|\imagpart{\zeta}\right|}}O\!\left(\zeta^{{-1}}\right)\right),&\left|\mathrm{ph}z\right|\leq 2p\pi-\delta,n\text{ odd},\\ (\ifrac{1}{\sqrt{\pi}})\sin\!\left(p\pi\right)z^{{-n/4}}e^{{-\zeta}}\left(1+O\!\left(\zeta^{{-1}}\right)\right),&\left|\mathrm{ph}z\right|\leq 3p\pi-\delta,n\text{ even}.\end{cases}$

Symbols:: $O\!\left(x\right)$ : order symbol, $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $\delta$ : small positive constant, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E12
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The distribution in $\Complex$ 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 (1977b, 1978) a different normalization is used. In place of (9.13.1) we have

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

Defines:: $m$ : index
Symbols:: $w$ : function
Referenced by:: §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E13
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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 $A_{{n}}\!\left(z\right)$ and $B_{{n}}\!\left(z\right)$ are given by

9.13.14

$m=n+2=1/p$ ,

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

Defines:: $m$ : index
Symbols:: $z$ : complex variable, $n$ : parameter, $p$ : variable and $\zeta$ : variable
Permalink:: http://dlmf.nist.gov/9.13.E14
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9.13.15 $\sqrt{2\pi}\left(\tfrac{1}{2}m\right)^{{(m-1)/m}}\csc\!\left(\ifrac{\pi}{m}\right)A_{{n}}\!\left(z\right)=\begin{cases}U_{m}(t),&m\text{ even},\\ V_{m}(t),&m\text{ odd},\end{cases}$

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

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

Defines:: $m$ : index and $V_{m}$ : Olver's generalized Airy function
Symbols:: $B_{{n}}\!\left(z\right)$ : generalized (ODE) Airy function, $z$ : complex variable, $n$ : parameter and $U_{i}$ : Smirnov's generalized Airy function
Permalink:: http://dlmf.nist.gov/9.13.E16
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Properties and graphs of $U_{m}(t)$ , $V_{m}(t)$ , $\overline{V}_{m}(t)$ are included in Olver (1977b) 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
Symbols:: $w$ : function and $m$ : index
Referenced by:: §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E17
Encodings:: TeX, pMathML, 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 $w=U_{m}(te^{{-2j\pi i/m}}),$ $j=0,\pm 1,\pm 2,\ldots.$

Defines:: $j$ : index
Symbols:: $U_{i}$ : Smirnov's generalized Airy function, $w$ : function and $m$ : index
Permalink:: http://dlmf.nist.gov/9.13.E18
Encodings:: TeX, pMathML, png

The function on the right-hand side is recessive in the sector $-(2j-1)\pi/m\leq\mathrm{ph}z\leq(2j+1)\pi/m$ , 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 and $x$ : parameter
Symbols:: $w$ : function
Permalink:: http://dlmf.nist.gov/9.13.E19
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where $\alpha>-2$ and $x>0$ . Solutions are $w=U_{1}(x,\alpha)$ , $U_{2}(x,\alpha)$ , where

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

Defines:: $U_{i}$ : Smirnov's generalized Airy function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\alpha$ : parameter and $x$ : parameter
Referenced by:: §9.18(vii)
Permalink:: http://dlmf.nist.gov/9.13.E20
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9.13.21 $U_{2}(x,\alpha)=(\alpha+2)^{{1/(\alpha+2)}}\*\Gamma\!\left(\frac{\alpha+3}{\alpha+2}\right)x^{{1/2}}J_{{1/(\alpha+2)}}\!\left(\frac{2}{\alpha+2}x^{{(\alpha+2)/2}}\right),$

Defines:: $U_{i}$ : Smirnov's generalized Airy function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\alpha$ : parameter and $x$ : parameter
Referenced by:: §9.18(vii)
Permalink:: http://dlmf.nist.gov/9.13.E21
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and $J$ denotes the Bessel function (§Ch.10).

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, $x$ : parameter and $m$ : index
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9.13.23 $U_{1}(x,\alpha)=\frac{\pi^{{1/2}}}{2^{{(m+2)/(2m)}}\Gamma\!\left(1/m\right)}\left(W_{m}(t)+W_{m}(-t)\right),$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\alpha$ : parameter, $x$ : parameter, $U_{i}$ : Smirnov's generalized Airy function, $m$ : index and $W_{m}$ : function
Permalink:: http://dlmf.nist.gov/9.13.E23
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9.13.24 $U_{2}(x,\alpha)=\frac{\pi^{{1/2}}m^{{2/m}}}{2^{{(m+2)/(2m)}}\Gamma\!\left(-1/m\right)}\left(W_{m}(t){-}W_{m}(-t)\right).$

Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\alpha$ : parameter, $x$ : parameter, $U_{i}$ : Smirnov's generalized Airy function, $m$ : index and $W_{m}$ : function
Permalink:: http://dlmf.nist.gov/9.13.E24
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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

Permalink:: http://dlmf.nist.gov/9.13.SS2

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 $A_{{k}}\!\left(z,p\right)=\frac{1}{2\pi i}\int _{{\mathscr{L}_{k}}}t^{{-p}}\exp\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $k=1,2,3$ , $p\in\Complex$ ,

Defines:: $A_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $k$ : index and $p$ : parameter
Symbols:: $z$ : complex variable and $\mathscr{L}$ : integration path
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.E25
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9.13.26 $B_{{0}}\!\left(z,p\right)=\frac{1}{2\pi i}\int _{{\mathscr{L}_{0}}}t^{{-p}}\exp\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $p=0,\pm 1,\pm 2,\ldots,$

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

9.13.27 $B_{{k}}\!\left(z,p\right)=\int _{{\mathscr{I}_{k}}}t^{{-p}}\exp\!\left(zt-\tfrac{1}{3}t^{3}\right)dt,$ $k=1,2,3$ , $p=0,\pm 1,\pm 2,\ldots,$

Defines:: $B_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function
Symbols:: $z$ : complex variable, $k$ : index, $p$ : parameter and $\mathscr{I}$ : integration path
Permalink:: http://dlmf.nist.gov/9.13.E27
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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 $\exp\!\left(-p(\ln|t|+i\mathrm{ph}t)\right)$ with $0\leq\mathrm{ph}t<2\pi$ .

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

Magnify

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

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

Magnify

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

When $p=0$

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

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $A_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.13.E28
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9.13.29

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

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

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $A_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.13.E29
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and

9.13.30

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

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

Symbols:: $B_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $\mathrm{Hi}\!\left(z\right)$ : Scorer function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.13.E30
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Each of the functions $A_{{k}}\!\left(z,p\right)$ and $B_{{k}}\!\left(z,p\right)$ satisfies the differential equation

9.13.31 $\frac{{d}^{3}w}{{dz}^{3}}-z\frac{dw}{dz}+(p-1)w=0,$

Defines:: $w$ : function
Symbols:: $z$ : complex variable and $p$ : parameter
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.E31
Encodings:: TeX, pMathML, png

and the difference equation

9.13.32 $f(p-3)-zf(p-1)+(p-1)f(p)=0.$

Defines:: $f$ : function
Symbols:: $z$ : complex variable and $p$ : parameter
Permalink:: http://dlmf.nist.gov/9.13.E32
Encodings:: TeX, pMathML, png

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

9.13.33

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

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

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

Connection formulas for the solutions of (9.13.31) include

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

Symbols:: $A_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $B_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $z$ : complex variable and $p$ : parameter
Permalink:: http://dlmf.nist.gov/9.13.E34
Encodings:: TeX, pMathML, png

9.13.35 $B_{{2}}\!\left(z,p\right)-B_{{3}}\!\left(z,p\right)=2\pi iA_{{1}}\!\left(z,p\right),$

Symbols:: $A_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $B_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $z$ : complex variable and $p$ : parameter
Permalink:: http://dlmf.nist.gov/9.13.E35
Encodings:: TeX, pMathML, png

9.13.36 $B_{{3}}\!\left(z,p\right)-B_{{1}}\!\left(z,p\right)=2\pi iA_{{2}}\!\left(z,p\right),$

Symbols:: $A_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $B_{{k}}\!\left(z,p\right)$ : generalized (integral) Airy function, $z$ : complex variable and $p$ : parameter
Permalink:: http://dlmf.nist.gov/9.13.E36
Encodings:: TeX, pMathML, png

9.13.37 $B_{{1}}\!\left(z,p\right)-B_{{2}}\!\left(z,p\right)=2\pi iA_{{3}}\!\left(z,p\right).$

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

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

§ 9.13. Generalized Airy Functions

Contents

§ 9.13(i). Generalizations from the Differential Equation

§ 9.13(ii). Generalizations from Integral Representations