# §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 Ch.9

All solutions are entire functions of $z$.

Standard solutions are:

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

## §9.2(ii) Initial Values

 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

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

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

## §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 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).