§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		$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=zw.$
ⓘ Defines: $w$ : ODE solution (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : 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 $z$ .

Standard solutions are:

9.2.2		$w=\operatorname{Ai}\left(z\right),\;\operatorname{Bi}\left(z\right),\;% \operatorname{Ai}\left(ze^{\mp 2\pi\mathrm{i}/3}\right).$
ⓘ 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, $z$ : complex variable and $w$ : 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	$\displaystyle\operatorname{Ai}\left(0\right)$	$\displaystyle=\frac{1}{3^{2/3}\Gamma\left(\tfrac{2}{3}\right)}=0.35502\;80538\ldots,$
ⓘ 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	$\displaystyle\operatorname{Ai}'\left(0\right)$	$\displaystyle=-\frac{1}{3^{1/3}\Gamma\left(\tfrac{1}{3}\right)}=-0.25881\;9403% 7\ldots,$
ⓘ 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	$\displaystyle\operatorname{Bi}\left(0\right)$	$\displaystyle=\frac{1}{3^{1/6}\Gamma\left(\tfrac{2}{3}\right)}=0.61492\;66274\ldots,$
ⓘ 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	$\displaystyle\operatorname{Bi}'\left(0\right)$	$\displaystyle=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right)}=0.44828\;83573\ldots.$
ⓘ 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, $x$ : real variable and $z$ : 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		$\mathscr{W}\left\{\operatorname{Ai}\left(z\right),\operatorname{Bi}\left(z% \right)\right\}=\frac{1}{\pi},$
ⓘ 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 $z$ : 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		$\mathscr{W}\left\{\operatorname{Ai}\left(z\right),\operatorname{Ai}\left(ze^{% \mp 2\pi i/3}\right)\right\}=\frac{e^{\pm\pi i/6}}{2\pi},$
ⓘ 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 $z$ : 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		$\mathscr{W}\left\{\operatorname{Ai}\left(ze^{-2\pi i/3}\right),\operatorname{% Ai}\left(ze^{2\pi i/3}\right)\right\}=\frac{1}{2\pi i}.$
ⓘ 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 $z$ : 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		$\operatorname{Bi}\left(z\right)=e^{-\pi i/6}\operatorname{Ai}\left(ze^{-2\pi i% /3}\right)+e^{\pi i/6}\operatorname{Ai}\left(ze^{2\pi i/3}\right).$
ⓘ 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 $z$ : 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		$\operatorname{Ai}\left(ze^{\mp 2\pi i/3}\right)=\tfrac{1}{2}e^{\mp\pi i/3}% \left(\operatorname{Ai}\left(z\right)\pm i\operatorname{Bi}\left(z\right)% \right).$
ⓘ 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 $z$ : 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		$\operatorname{Ai}\left(z\right)+e^{-2\pi i/3}\operatorname{Ai}\left(ze^{-2\pi i% /3}\right)+e^{2\pi i/3}\operatorname{Ai}\left(ze^{2\pi i/3}\right)=0,$
ⓘ 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 $z$ : 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		$\operatorname{Bi}\left(z\right)+e^{-2\pi i/3}\operatorname{Bi}\left(ze^{-2\pi i% /3}\right)+e^{2\pi i/3}\operatorname{Bi}\left(ze^{2\pi i/3}\right)=0.$
ⓘ 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 $z$ : 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	$\displaystyle\operatorname{Ai}\left(-z\right)$	$\displaystyle=e^{\pi i/3}\operatorname{Ai}\left(ze^{\pi i/3}\right)+e^{-\pi i/% 3}\operatorname{Ai}\left(ze^{-\pi i/3}\right),$
ⓘ 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 $z$ : 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	$\displaystyle\operatorname{Bi}\left(-z\right)$	$\displaystyle=e^{-\pi i/6}\operatorname{Ai}\left(ze^{\pi i/3}\right)+e^{\pi i/% 6}\operatorname{Ai}\left(ze^{-\pi i/3}\right).$
ⓘ 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 $z$ : complex variable Source: Olver (1997b, p. 414) Permalink: http://dlmf.nist.gov/9.2.E15 Encodings: TeX, pMML, png See also: Annotations for §9.2(v), §9.2 and Ch.9

§9.2(vi) Riccati Form of Differential Equation

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Keywords:: Airy functions, Riccati form, differential equation
Permalink:: http://dlmf.nist.gov/9.2.vi
See also:: Annotations for §9.2 and Ch.9

9.2.16		$\frac{\mathrm{d}W}{\mathrm{d}z}+W^{2}=z,$
ⓘ Defines: $W$ : Riccati solution (locally) Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable Proof sketch: Derive properties using (9.2.1). Permalink: http://dlmf.nist.gov/9.2.E16 Encodings: TeX, pMML, png See also: Annotations for §9.2(vi), §9.2 and Ch.9

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

9.1 Special Notation 9.3 Graphics

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