§9.13 Generalized Airy Functions

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Keywords:: from differential equation, generalized Airy functions
Permalink:: http://dlmf.nist.gov/9.13
See also:: Annotations for Ch.9

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

§9.13(i) Generalizations from the Differential Equation

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Defines:: $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function and $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function
Keywords:: Bessel functions, asymptotic approximations, definitions, from differential equation, generalized Airy functions, modified Bessel functions, relation to Bessel functions, relation to modified Bessel functions, relations to other functions
Permalink:: http://dlmf.nist.gov/9.13.i
See also:: Annotations for §9.13 and Ch.9

Equations of the form

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

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Defines:: : parameter (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : complex variable and : function
Source:: Swanson and Headley (1967, (1))
Referenced by:: §9.13(i), §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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 $w=z^{1/2}\mathscr{Z}_{p}\left(\zeta\right),$

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Defines:: : function (locally)
Symbols:: $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified cylinder function, : complex variable, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, p. 1401)
Permalink:: http://dlmf.nist.gov/9.13.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

where

9.13.3

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

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

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Defines:: : variable (locally) and $\zeta$ : variable (locally)
Symbols:: : complex variable and : parameter
Source:: Swanson and Headley (1967, (2))
Permalink:: http://dlmf.nist.gov/9.13.E3
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

and $\mathscr{Z}_{p}$ is any linear combination of the modified Bessel functions $I_{p}$ and $e^{p\pi\mathrm{i}}K_{p}$ (§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

9.13.4

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

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

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Symbols:: $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, : complex variable, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (3),(3’))
Permalink:: http://dlmf.nist.gov/9.13.E4
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

when is real and positive, and by analytic continuation elsewhere. (All solutions of (9.13.1) are entire functions of .) When , $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ become $\operatorname{Ai}\left(z\right)$ and $\operatorname{Bi}\left(z\right)$ , 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

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

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, : parameter and : variable
Source:: Swanson and Headley (1967, (4))
Permalink:: http://dlmf.nist.gov/9.13.E5
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, : complex variable, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (5a)–(5b))
Permalink:: http://dlmf.nist.gov/9.13.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, : complex variable, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (6a)–(6b))
Permalink:: http://dlmf.nist.gov/9.13.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, : complex variable, : parameter and : variable
Source:: Swanson and Headley (1967, p. 1403)
Permalink:: http://dlmf.nist.gov/9.13.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

As $z\to\infty$

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (13))
Permalink:: http://dlmf.nist.gov/9.13.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (14)–(15))
Permalink:: http://dlmf.nist.gov/9.13.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (16))
Permalink:: http://dlmf.nist.gov/9.13.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, : complex variable, $\delta$ : small positive constant, : parameter, : variable and $\zeta$ : variable
Source:: Swanson and Headley (1967, (17)–(18))
Permalink:: http://dlmf.nist.gov/9.13.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

The distribution in $\mathbb{C}$ and asymptotic properties of the zeros of $A_{n}\left(z\right)$ , $A_{n}'\left(z\right)$ , $B_{n}\left(z\right)$ , and $B_{n}'\left(z\right)$ 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}t}^{2}}=\tfrac{1}{4}m^{2}t^{m-2}w,$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : function and : index
Source:: Olver (1977a, (2.01))
Referenced by:: §9.13(i), §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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 $A_{n}\left(z\right)$ and $B_{n}\left(z\right)$ are given by

9.13.14

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

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Symbols:: : complex variable, : parameter, : variable, $\zeta$ : variable and : index
Permalink:: http://dlmf.nist.gov/9.13.E14
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $A_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : 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
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $B_{\NVar{n}}\left(\NVar{z}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : 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
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

even,

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Defines:: $W_{m}$ : function (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : function and : index
Source:: Olver (1977a, (2.11), p. 131)
Referenced by:: §9.13(i)
Permalink:: http://dlmf.nist.gov/9.13.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

In $\mathbb{C}$ , the solutions of (9.13.13) used in Olver (1978) are

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $U_{i}$ : Smirnov’s generalized Airy function, : function, : index and : index
Notes:: See Olver (1978).
Permalink:: http://dlmf.nist.gov/9.13.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

The function on the right-hand side is recessive in the sector $-(2j-1)\pi/m\leq\operatorname{ph}z\leq(2j+1)\pi/m$ , 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 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+x^{\alpha}w=0,$

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Defines:: $\alpha$ : parameter (locally) and : parameter (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to and : function
Source:: Smirnov (1960, (1.2), p. 1)
Permalink:: http://dlmf.nist.gov/9.13.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\alpha$ : parameter, : parameter and $U_{i}$ : Smirnov’s generalized Airy function
Source:: Smirnov (1960, (2.10), p. 10)
Referenced by:: 1st item
Permalink:: http://dlmf.nist.gov/9.13.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\alpha$ : parameter, : parameter and $U_{i}$ : Smirnov’s generalized Airy function
Source:: Smirnov (1960, (2.11), p. 11)
Referenced by:: 1st item
Permalink:: http://dlmf.nist.gov/9.13.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

and 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

$\alpha=m-2$ ,

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

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Symbols:: $\alpha$ : parameter, : parameter and : index
Permalink:: http://dlmf.nist.gov/9.13.E22
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\alpha$ : 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
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\alpha$ : 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
See also:: Annotations for §9.13(i), §9.13 and Ch.9

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

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Defines:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function and $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function
Keywords:: connection formulas, definitions, difference equation, differential equation, from integral representations, generalized Airy functions
Permalink:: http://dlmf.nist.gov/9.13.ii
See also:: Annotations for §9.13 and Ch.9

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

, $p\in\mathbb{C}$ ,

ⓘ

Defines:: : index (locally) and : parameter (locally)
Symbols:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : complex variable and $\mathscr{L}$ : integration path
Sources:: Reid (1972, (A4), p. 362); Drazin and Reid (1981, (A4), p. 466)
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

9.13.26 $B_{0}\left(z,p\right)=\frac{1}{2\pi i}\int_{\mathscr{L}_{0}}t^{-p}\exp\left(zt% -\tfrac{1}{3}t^{3}\right)\,\mathrm{d}t,$ $p=0,\pm 1,\pm 2,\ldots$ ,

ⓘ

Symbols:: $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : complex variable, : parameter and $\mathscr{L}$ : integration path
Source:: Reid (1972, (A16), p. 364)
Permalink:: http://dlmf.nist.gov/9.13.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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

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

ⓘ

Symbols:: $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral, : complex variable, : index, : parameter and $\mathscr{I}$ : integration path
Source:: Drazin and Reid (1981, (A20), p. 470)
Permalink:: http://dlmf.nist.gov/9.13.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

with $z\in\mathbb{C}$ 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 $\exp\left(-p(\ln|t|+i\operatorname{ph}t)\right)$ with $0\leq\operatorname{ph}t<2\pi$ .

Figure 9.13.1:

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

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Defines:: $\mathscr{L}$ : integration path (locally)
Source:: Drazin and Reid (1981, Fig. A2, p. 467)
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.F1
Encodings:: pdf, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

Figure 9.13.2:

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

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Defines:: $\mathscr{I}$ : integration path (locally)
Source:: Drazin and Reid (1981, Fig. A2, p. 467)
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.F2
Encodings:: pdf, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

When

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

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function and : complex variable
Sources:: Reid (1972, (A10), p. 364, with (A1)); Drazin and Reid (1981, (A10), p. 467, with (A1))
Permalink:: http://dlmf.nist.gov/9.13.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

9.13.29

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

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

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : complex variable
Sources:: Reid (1972, (A10), p. 364, with (A1)); Drazin and Reid (1981, (A10), p. 467, with (A1))
Permalink:: http://dlmf.nist.gov/9.13.E29
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

and

9.13.30

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

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

ⓘ

Symbols:: $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter and : complex variable
Source:: Drazin and Reid (1981, p. 470)
Permalink:: http://dlmf.nist.gov/9.13.E30
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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

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

ⓘ

Defines:: : function (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : complex variable and : parameter
Referenced by:: §9.13(ii)
Permalink:: http://dlmf.nist.gov/9.13.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

and the difference equation

9.13.32

ⓘ

Defines:: : function (locally)
Symbols:: : complex variable and : parameter
Source:: Reid (1972, (A9), p. 363)
Permalink:: http://dlmf.nist.gov/9.13.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : complex variable and : parameter
Source:: Reid (1972, (A7), p. 363)
Permalink:: http://dlmf.nist.gov/9.13.E33
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, : complex variable and : parameter
Source:: Reid (1972, (A19), p. 365)
Permalink:: http://dlmf.nist.gov/9.13.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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

ⓘ

Symbols:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, : complex variable and : parameter
Source:: Drazin and Reid (1981, (A23), p. 471)
Permalink:: http://dlmf.nist.gov/9.13.E35
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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

ⓘ

Symbols:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, : complex variable and : parameter
Source:: Drazin and Reid (1981, (A23), p. 471)
Permalink:: http://dlmf.nist.gov/9.13.E36
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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

ⓘ

Symbols:: $A_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $B_{\NVar{k}}\left(\NVar{z},\NVar{p}\right)$ : generalized Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, : complex variable and : parameter
Source:: Drazin and Reid (1981, (A23), p. 471)
Permalink:: http://dlmf.nist.gov/9.13.E37
Encodings:: TeX, pMML, png
See also:: Annotations for §9.13(ii), §9.13 and Ch.9

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).

9.12 Scorer Functions 9.14 Incomplete Airy Functions

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§9.13 Generalized Airy Functions

Contents

§9.13(i) Generalizations from the Differential Equation

§9.13(ii) Generalizations from Integral Representations