# §9.2 Differential Equation

## §9.2(i) Airy’s Equation

 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 9

All solutions are entire functions of $z$.

Standard solutions are:

 9.2.2 $w=\mathrm{Ai}\left(z\right),\;\mathrm{Bi}\left(z\right),\;\mathrm{Ai}\left(ze^% {\mp 2\pi\mathrm{i}/3}\right).$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $z$: 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 9

## §9.2(ii) Initial Values

 9.2.3 $\displaystyle\mathrm{Ai}\left(0\right)$ $\displaystyle=\frac{1}{3^{2/3}\Gamma\left(\tfrac{2}{3}\right)}=0.35502\;80538\ldots,$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: 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 9 9.2.4 $\displaystyle\mathrm{Ai}'\left(0\right)$ $\displaystyle=-\frac{1}{3^{1/3}\Gamma\left(\tfrac{1}{3}\right)}=-0.25881\;9403% 7\ldots,$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: 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 9 9.2.5 $\displaystyle\mathrm{Bi}\left(0\right)$ $\displaystyle=\frac{1}{3^{1/6}\Gamma\left(\tfrac{2}{3}\right)}=0.61492\;66274\ldots,$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: 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 9 9.2.6 $\displaystyle\mathrm{Bi}'\left(0\right)$ $\displaystyle=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right)}=0.44828\;83573\ldots.$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: 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 9

## §9.2(iii) Numerically Satisfactory Pairs of Solutions

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

## §9.2(iv) Wronskians

 9.2.7 $\mathscr{W}\left\{\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right)\right\}=% \frac{1}{\pi},$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: 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 9
 9.2.8 $\mathscr{W}\left\{\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(ze^{\mp 2\pi i/3}% \right)\right\}=\frac{e^{\pm\pi i/6}}{2\pi},$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathscr{W}$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $z$: complex variable Source: 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 9
 9.2.9 $\mathscr{W}\left\{\mathrm{Ai}\left(ze^{-2\pi i/3}\right),\mathrm{Ai}\left(ze^{% 2\pi i/3}\right)\right\}=\frac{1}{2\pi i}.$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathscr{W}$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $z$: complex variable Source: 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 9

## §9.2(v) Connection Formulas

 9.2.10 $\mathrm{Bi}\left(z\right)=e^{-\pi i/6}\mathrm{Ai}\left(ze^{-2\pi i/3}\right)+e% ^{\pi i/6}\mathrm{Ai}\left(ze^{2\pi i/3}\right).$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $z$: complex variable Source: 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 9
 9.2.11 $\mathrm{Ai}\left(ze^{\mp 2\pi i/3}\right)=\tfrac{1}{2}e^{\mp\pi i/3}\left(% \mathrm{Ai}\left(z\right)\pm i\mathrm{Bi}\left(z\right)\right).$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function 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 9
 9.2.12 $\mathrm{Ai}\left(z\right)+e^{-2\pi i/3}\mathrm{Ai}\left(ze^{-2\pi i/3}\right)+% e^{2\pi i/3}\mathrm{Ai}\left(ze^{2\pi i/3}\right)=0,$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function 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 9
 9.2.13 $\mathrm{Bi}\left(z\right)+e^{-2\pi i/3}\mathrm{Bi}\left(ze^{-2\pi i/3}\right)+% e^{2\pi i/3}\mathrm{Bi}\left(ze^{2\pi i/3}\right)=0.$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $z$: complex variable Source: 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 9
 9.2.14 $\displaystyle\mathrm{Ai}\left(-z\right)$ $\displaystyle=e^{\pi i/3}\mathrm{Ai}\left(ze^{\pi i/3}\right)+e^{-\pi i/3}% \mathrm{Ai}\left(ze^{-\pi i/3}\right),$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function 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 9 9.2.15 $\displaystyle\mathrm{Bi}\left(-z\right)$ $\displaystyle=e^{-\pi i/6}\mathrm{Ai}\left(ze^{\pi i/3}\right)+e^{\pi i/6}% \mathrm{Ai}\left(ze^{-\pi i/3}\right).$

## §9.2(vi) Riccati Form of Differential Equation

 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 Source: 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 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).