§9.2 Differential Equation

Referenced by:: ¶ ‣ §1.17(ii), §10.19(iii), §10.20(i), §15.12(iii), ¶ ‣ §18.15(iv), §18.32, §18.34(iii), §18.35(iii), §2.4(v), §2.8(i), §2.8(iii), ¶ ‣ §3.3(v), §32.10(ii), §32.11(ii), §32.14, §32.5, §33.12(i), §36.1, §36.12(ii), §36.2(ii), §36.7(i)
Permalink:: http://dlmf.nist.gov/9.2

§9.2(i) Airy’s Equation
§9.2(ii) Initial Values
§9.2(iii) Numerically Satisfactory Pairs of Solutions
§9.2(iv) Wronskians
§9.2(v) Connection Formulas
§9.2(vi) Riccati Form of Differential Equation

§9.2(i) Airy’s Equation

Notes:: See Miller (1946) and Olver (1997b, pp. 392–393, 413–414).
Defines:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function and $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function
Keywords:: Airy functions, Airy’s equation
Referenced by:: §13.18(iii), §13.21(iii), §13.6(iii), ¶ ‣ §36.10(i)
Permalink:: http://dlmf.nist.gov/9.2.i

9.2.1 $\frac{{d}^{2}w}{{dz}^{2}}=zw.$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable and : ODE solution
A&S Ref:: 10.4.1 (in slightly different form)
Referenced by:: §36.8, §9.10(iii), §9.10(iii), §9.10(viii), §9.11(i), §9.11(ii), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), §9.2(iii), §9.2(vi), §9.2(vi)
Permalink:: http://dlmf.nist.gov/9.2.E1
Encodings:: TeX, pMML, png

All solutions are entire functions of .

Standard solutions are:

9.2.2 $w=\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),\;\mathop{\mathrm{Bi}\/}% \nolimits\!\left(z\right),\;\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{{\mp 2% \pi i/3}}\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function, : complex variable and : ODE solution
Permalink:: http://dlmf.nist.gov/9.2.E2
Encodings:: TeX, pMML, png

§9.2(ii) Initial Values

Notes:: See Miller (1946, p. B9) and Olver (1997b, pp. 392–393).
Keywords:: Airy functions
Referenced by:: §9.12(v), §9.17(ii)
Permalink:: http://dlmf.nist.gov/9.2.ii

9.2.3 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(0\right)=\frac{1}{3^{{2/3}}\mathop{% \Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)}=0.35502\;80538\ldots,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function and $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 10.4.4 (with more digits)
Permalink:: http://dlmf.nist.gov/9.2.E3
Encodings:: TeX, pMML, png

9.2.4 ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(0\right)=-\frac{1}{3^{{1/3% }}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right)}=-0.25881\;94037\ldots,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function and $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 10.4.5 (with more digits)
Permalink:: http://dlmf.nist.gov/9.2.E4
Encodings:: TeX, pMML, png

9.2.5 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)=\frac{1}{3^{{1/6}}\mathop{% \Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)}=0.61492\;66274\ldots,$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function and $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 10.4.4 (in different form)
Referenced by:: §9.12(vi)
Permalink:: http://dlmf.nist.gov/9.2.E5
Encodings:: TeX, pMML, png

9.2.6 ${\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(0\right)=\frac{3^{{1/6}}}{% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right)}=0.44828\;83573\ldots.$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function and $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function
A&S Ref:: 10.4.5 (in different form)
Referenced by:: §9.12(vi)
Permalink:: http://dlmf.nist.gov/9.2.E6
Encodings:: TeX, pMML, png

§9.2(iii) Numerically Satisfactory Pairs of Solutions

Notes:: See Olver (1997b, pp. 392–393, 413–414).
Keywords:: Airy functions
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/9.2.iii
Clarification (effective with 1.0.1):: It was clarified that numerically satisfactory pairs of solutions are stated for intervals as well as regions.
Reported 2010-11-23

Table 9.2.1 lists numerically satisfactory pairs of solutions of (9.2.1) for the stated intervals or regions; compare §2.7(iv).

Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.

Pair	Interval or Region
$\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right),\mathop{\mathrm{Bi}\/}% \nolimits\!\left(x\right)$	$-\infty<x<\infty$
$\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),\mathop{\mathrm{Bi}\/}% \nolimits\!\left(z\right)$	$\left\{\begin{array}[]{l}\|\mathop{\mathrm{ph}\/}\nolimits z\|\leq\tfrac{1}{3}% \pi\\ -\infty<z\leq 0\end{array}\right.$
$\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),\mathop{\mathrm{Ai}\/}% \nolimits\!\left(ze^{{-2\pi i/3}}\right)$	$-\tfrac{1}{3}\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\pi$
$\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),\mathop{\mathrm{Ai}\/}% \nolimits\!\left(ze^{{2\pi i/3}}\right)$	$-\pi\leq\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{3}\pi$
$\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{{\mp 2\pi i/3}}\right)$	$\|\mathop{\mathrm{ph}\/}\nolimits(-z)\|\leq\tfrac{2}{3}\pi$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : real variable and : complex variable
Referenced by:: §9.2(iii)
Permalink:: http://dlmf.nist.gov/9.2.T1

§9.2(iv) Wronskians

Notes:: See Olver (1997b, pp. 393, 416).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.2.iv

9.2.7 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z% \right),\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)\right\}=\frac{1}{\pi},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian and : complex variable
A&S Ref:: 10.4.10
Referenced by:: §9.10(i), §9.11(ii), §9.8(ii)
Permalink:: http://dlmf.nist.gov/9.2.E7
Encodings:: TeX, pMML, png

9.2.8 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z% \right),\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{{\mp 2\pi i/3}}\right)% \right\}=\frac{e^{{\pm\pi i/6}}}{2\pi},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 10.4.11 10.4.12
Permalink:: http://dlmf.nist.gov/9.2.E8
Encodings:: TeX, pMML, png

9.2.9 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathrm{Ai}\/}\nolimits\!\left(% ze^{{-2\pi i/3}}\right),\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{{2\pi i/3}}% \right)\right\}=\frac{1}{2\pi i}.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, : base of exponential function and : complex variable
A&S Ref:: 10.4.13
Permalink:: http://dlmf.nist.gov/9.2.E9
Encodings:: TeX, pMML, png

§9.2(v) Connection Formulas

Notes:: See Miller (1946, p. B9) and Olver (1997b, p. 414).
Keywords:: Airy functions
Referenced by:: §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.2.v

9.2.10 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)=e^{{-\pi i/6}}\mathop{\mathrm{% Ai}\/}\nolimits\!\left(ze^{{-2\pi i/3}}\right)+e^{{\pi i/6}}\mathop{\mathrm{Ai% }\/}\nolimits\!\left(ze^{{2\pi i/3}}\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function and : complex variable
A&S Ref:: 10.4.6
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.2.E10
Encodings:: TeX, pMML, png

9.2.11 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{{\mp 2\pi i/3}}\right)=\tfrac{1}{2}% e^{{\mp\pi i/3}}\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\pm i% \mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function and : complex variable
A&S Ref:: 10.4.9
Permalink:: http://dlmf.nist.gov/9.2.E11
Encodings:: TeX, pMML, png

9.2.12 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)+e^{{-2\pi i/3}}\mathop{\mathrm% {Ai}\/}\nolimits\!\left(ze^{{-2\pi i/3}}\right)+e^{{2\pi i/3}}\mathop{\mathrm{% Ai}\/}\nolimits\!\left(ze^{{2\pi i/3}}\right)=0,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function and : complex variable
A&S Ref:: 10.4.7
Permalink:: http://dlmf.nist.gov/9.2.E12
Encodings:: TeX, pMML, png

9.2.13 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)+e^{{-2\pi i/3}}\mathop{\mathrm% {Bi}\/}\nolimits\!\left(ze^{{-2\pi i/3}}\right)+e^{{2\pi i/3}}\mathop{\mathrm{% Bi}\/}\nolimits\!\left(ze^{{2\pi i/3}}\right)=0.$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function and : complex variable
A&S Ref:: 10.4.8
Permalink:: http://dlmf.nist.gov/9.2.E13
Encodings:: TeX, pMML, png

9.2.14 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(-z\right)=e^{{\pi i/3}}\mathop{\mathrm{% Ai}\/}\nolimits\!\left(ze^{{\pi i/3}}\right)+e^{{-\pi i/3}}\mathop{\mathrm{Ai}% \/}\nolimits\!\left(ze^{{-\pi i/3}}\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function and : complex variable
Permalink:: http://dlmf.nist.gov/9.2.E14
Encodings:: TeX, pMML, png

9.2.15 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)=e^{{-\pi i/6}}\mathop{\mathrm% {Ai}\/}\nolimits\!\left(ze^{{\pi i/3}}\right)+e^{{\pi i/6}}\mathop{\mathrm{Ai}% \/}\nolimits\!\left(ze^{{-\pi i/3}}\right).$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : base of exponential function and : complex variable
Permalink:: http://dlmf.nist.gov/9.2.E15
Encodings:: TeX, pMML, png

§9.2(vi) Riccati Form of Differential Equation

Notes:: Use (9.2.1).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.2.vi

9.2.16 $\frac{dW}{dz}+W^{2}=z,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable and : Riccati solution
Permalink:: http://dlmf.nist.gov/9.2.E16
Encodings:: TeX, pMML, png

$W=(1/w)\ifrac{dw}{dz}$ , where is any nontrivial solution of (9.2.1). See also Smith (1990).