§9.2 Differential Equation

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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
See also:: Annotations for Ch.9

§9.2(i) Airy’s Equation

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Defines:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function and $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function
Keywords:: Airy functions, Airy’s equation, analytic properties, definitions, differential equation
Notes:: See Miller (1946) and Olver (1997b, pp. 392–393, 413–414).
Referenced by:: §13.18(iii), §13.21(iii), §13.6(iii), §36.10(i)
Permalink:: http://dlmf.nist.gov/9.2.i
See also:: Annotations for §9.2 and Ch.9

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

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Defines:: : ODE solution (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to and : complex variable
Source:: Olver (1997b, (1.01), p. 392)
A&S Ref:: 10.4.1 (in slightly different form)
Referenced by:: §36.8, (9.10.10), (9.10.20), (9.10.21), (9.10.8), (9.10.9), §9.10(iii), (9.11.2), §9.11(i), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), (9.2.16), §9.2(iii), §9.2(vi), (9.8.14), (9.8.16), (9.8.18), (9.8.19)
Permalink:: http://dlmf.nist.gov/9.2.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(i), §9.2 and Ch.9

All solutions are entire functions of .

Standard solutions are:

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : 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 : ODE solution
Source:: Olver (1997b, pp. 392–393, 413–414)
Permalink:: http://dlmf.nist.gov/9.2.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(i), §9.2 and Ch.9

§9.2(ii) Initial Values

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Keywords:: Airy functions, differential equation, initial values
Notes:: See Miller (1946, p. B9) and Olver (1997b, pp. 392–393).
Referenced by:: (9.12.11), (9.12.12), (9.12.13), (9.12.14), §9.12(v), §9.17(ii)
Permalink:: http://dlmf.nist.gov/9.2.ii
See also:: Annotations for §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.03), p. 392)
A&S Ref:: 10.4.4 (with more digits)
Permalink:: http://dlmf.nist.gov/9.2.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

9.2.4 $\operatorname{Ai}'\left(0\right)=-\frac{1}{3^{1/3}\Gamma\left(\tfrac{1}{3}% \right)}=-0.25881\;94037\ldots,$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.03), p. 392)
A&S Ref:: 10.4.5 (with more digits)
Permalink:: http://dlmf.nist.gov/9.2.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.11), p. 393)
A&S Ref:: 10.4.4 (in different form)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.2.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

9.2.6 $\operatorname{Bi}'\left(0\right)=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right% )}=0.44828\;83573\ldots.$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Source:: Olver (1997b, (1.11), p. 393)
A&S Ref:: 10.4.5 (in different form)
Referenced by:: (9.12.15)
Permalink:: http://dlmf.nist.gov/9.2.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(ii), §9.2 and Ch.9

§9.2(iii) Numerically Satisfactory Pairs of Solutions

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Keywords:: Airy functions, differential equation, numerically satisfactory solutions
Notes:: See Olver (1997b, pp. 392–393, 413–414).
Referenced by:: Erratum (V1.0.1) for Clarifications
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.
See also:: Annotations for §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, : real variable and : complex variable
Source:: Olver (1997b, pp. 413–414)
Notes:: See also Olver (1997b, p. 154–155).
Referenced by:: §9.2(iii)
Permalink:: http://dlmf.nist.gov/9.2.T1
See also:: Annotations for §9.2(iii), §9.2 and Ch.9

§9.2(iv) Wronskians

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Keywords:: Airy functions, Wronskians
Notes:: See Olver (1997b, pp. 393, 416).
Permalink:: http://dlmf.nist.gov/9.2.iv
See also:: Annotations for §9.2 and Ch.9

9.2.7 $\mathscr{W}\left\{\operatorname{Ai}\left(z\right),\operatorname{Bi}\left(z% \right)\right\}=\frac{1}{\pi},$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter and : complex variable
Source:: Olver (1997b, (1.19), p. 393)
A&S Ref:: 10.4.10
Referenced by:: (9.10.2), (9.10.3), (9.11.2), (9.8.13), (9.8.17), (9.9.3), (9.9.4)
Permalink:: http://dlmf.nist.gov/9.2.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(iv), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : complex variable
Proof sketch:: Derivable from Olver (1997b, p. 416).
A&S Ref:: 10.4.11 10.4.12
Permalink:: http://dlmf.nist.gov/9.2.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(iv), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : complex variable
Proof sketch:: Derivable from Olver (1997b, p. 416).
A&S Ref:: 10.4.13
Permalink:: http://dlmf.nist.gov/9.2.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(iv), §9.2 and Ch.9

§9.2(v) Connection Formulas

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Keywords:: Airy functions, connection formulas
Notes:: See Miller (1946, p. B9) and Olver (1997b, p. 414).
Referenced by:: §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.2.v
See also:: Annotations for §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : 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
Proof sketch:: Derivable from Olver (1997b, p. 414).
A&S Ref:: 10.4.6
Referenced by:: (9.10.19), (9.5.5)
Permalink:: http://dlmf.nist.gov/9.2.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(v), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : 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
Source:: Olver (1997b, (8.04), p. 414)
A&S Ref:: 10.4.9
Permalink:: http://dlmf.nist.gov/9.2.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(v), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : 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
Source:: Olver (1997b, (8.03), p. 414)
A&S Ref:: 10.4.7
Permalink:: http://dlmf.nist.gov/9.2.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(v), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : 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
Proof sketch:: Derivable from Olver (1997b, p. 414).
A&S Ref:: 10.4.8
Permalink:: http://dlmf.nist.gov/9.2.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(v), §9.2 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : 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
Source:: Olver (1997b, p. 414)
Permalink:: http://dlmf.nist.gov/9.2.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §9.2(v), §9.2 and Ch.9

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

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