# §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 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 See also: Annotations for 9.2(i)

All solutions are entire functions of $z$.

Standard solutions are:

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

## §9.2(ii) Initial Values

 9.2.3 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(0\right)$ $\displaystyle=\frac{1}{3^{2/3}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}% \right)}=0.35502\;80538\ldots,$ Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function and $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function A&S Ref: 10.4.4 (with more digits) Permalink: http://dlmf.nist.gov/9.2.E3 Encodings: TeX, pMML, png See also: Annotations for 9.2(ii) 9.2.4 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(0\right)$ $\displaystyle=-\frac{1}{3^{1/3}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}% \right)}=-0.25881\;94037\ldots,$ Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function and $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function A&S Ref: 10.4.5 (with more digits) Permalink: http://dlmf.nist.gov/9.2.E4 Encodings: TeX, pMML, png See also: Annotations for 9.2(ii) 9.2.5 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)$ $\displaystyle=\frac{1}{3^{1/6}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}% \right)}=0.61492\;66274\ldots,$ Symbols: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function and $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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 See also: Annotations for 9.2(ii) 9.2.6 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(0\right)$ $\displaystyle=\frac{3^{1/6}}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}% \right)}=0.44828\;83573\ldots.$ Symbols: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function and $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: 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 See also: Annotations for 9.2(ii)

## §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 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z% \right),\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)\right\}=\frac{1}{\pi},$
 9.2.8 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z% \right),\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{\mp 2\pi i/3}\right)\right% \}=\frac{e^{\pm\pi i/6}}{2\pi},$
 9.2.9 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathrm{Ai}\/}\nolimits\!\left(% ze^{-2\pi i/3}\right),\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{2\pi i/3}% \right)\right\}=\frac{1}{2\pi i}.$

## §9.2(v) Connection Formulas

 9.2.10 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)=e^{-\pi i/6}\mathop{\mathrm{Ai% }\/}\nolimits\!\left(ze^{-2\pi i/3}\right)+e^{\pi i/6}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(ze^{2\pi i/3}\right).$
 9.2.11 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{\mp 2\pi i/3}\right)=\tfrac{1}{2}e^% {\mp\pi i/3}\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\pm i\mathop{% \mathrm{Bi}\/}\nolimits\!\left(z\right)\right).$
 9.2.12 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)+e^{-2\pi i/3}\mathop{\mathrm{% Ai}\/}\nolimits\!\left(ze^{-2\pi i/3}\right)+e^{2\pi i/3}\mathop{\mathrm{Ai}\/% }\nolimits\!\left(ze^{2\pi i/3}\right)=0,$
 9.2.13 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)+e^{-2\pi i/3}\mathop{\mathrm{% Bi}\/}\nolimits\!\left(ze^{-2\pi i/3}\right)+e^{2\pi i/3}\mathop{\mathrm{Bi}\/% }\nolimits\!\left(ze^{2\pi i/3}\right)=0.$
 9.2.14 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(-z\right)$ $\displaystyle=e^{\pi i/3}\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{\pi i/3}% \right)+e^{-\pi i/3}\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{-\pi i/3}\right),$ 9.2.15 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)$ $\displaystyle=e^{-\pi i/6}\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{\pi i/3}% \right)+e^{\pi i/6}\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{-\pi i/3}\right).$

## §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 Permalink: http://dlmf.nist.gov/9.2.E16 Encodings: TeX, pMML, png See also: Annotations for 9.2(vi)

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