§9.2 Differential Equation

9.2.1		$$\frac{d^{2}w}{{dz}^2}=zw.$$
ⓘ Defines: $w$: ODE solution (locally) Symbols: $\frac{df}{dx}$: 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		$$w=\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right).$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: 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

9.2.3		$\mathrm{Ai}\left(0\right)$	$={\displaystyle \frac{1}{3^{2/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}}=\mathrm{0.35502\hspace{0.33em}80538}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: 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		${\mathrm{Ai}}^{\prime }\left(0\right)$	$=-{\displaystyle \frac{1}{3^{1/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}}=-\mathrm{0.25881\hspace{0.33em}94037}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: 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		$\mathrm{Bi}\left(0\right)$	$={\displaystyle \frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}}=\mathrm{0.61492\hspace{0.33em}66274}\mathrm{\dots },$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: 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		${\mathrm{Bi}}^{\prime }\left(0\right)$	$={\displaystyle \frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}}=\mathrm{0.44828\hspace{0.33em}83573}\mathrm{\dots }.$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: 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

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

9.2.7		$$𝒲\left\{\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right)\right\}=\frac{1}{\pi },$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $𝒲$: 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		$$𝒲\left\{\mathrm{Ai}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)\right\}=\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}/6}}{2\pi },$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $𝒲$: 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 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 Ch.9

9.2.9		$$𝒲\left\{\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right),\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)\right\}=\frac{1}{2\pi \mathrm{i}}.$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $𝒲$: 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 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 Ch.9

9.2.10		$$\mathrm{Bi}\left(z\right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right).$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: 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: 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		$$\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right)=\frac{1}{2}{\mathrm{e}}^{\mp \pi \mathrm{i}/3}\left(\mathrm{Ai}\left(z\right)\pm \mathrm{i}\mathrm{Bi}\left(z\right)\right).$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: 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		$$\mathrm{Ai}\left(z\right)+{\mathrm{e}}^{-2\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{2\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)=0,$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: 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		$$\mathrm{Bi}\left(z\right)+{\mathrm{e}}^{-2\pi \mathrm{i}/3}\mathrm{Bi}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{2\pi \mathrm{i}/3}\mathrm{Bi}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right)=0.$$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: 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: 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		$\mathrm{Ai}\left(-z\right)$	$={\mathrm{e}}^{\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}\mathrm{Ai}\left(z{\mathrm{e}}^{-\pi \mathrm{i}/3}\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: 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		$\mathrm{Bi}\left(-z\right)$	$={\mathrm{e}}^{-\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{\pi \mathrm{i}/3}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}\mathrm{Ai}\left(z{\mathrm{e}}^{-\pi \mathrm{i}/3}\right).$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: 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

9.2.16		$$\frac{dW}{dz}+W^2=z,$$
ⓘ Defines: $W$: Riccati solution (locally) Symbols: $\frac{df}{dx}$: 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 Ch.9

$W=(1/w)dw/dz$, where $w$ is any nontrivial solution of (9.2.1). See also Smith (1990).