§9.11 Products

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Permalink:: http://dlmf.nist.gov/9.11
See also:: Annotations for Ch.9

§9.11(i) Differential Equation
§9.11(ii) Wronskian
§9.11(iii) Integral Representations
§9.11(iv) Indefinite Integrals
§9.11(v) Definite Integrals

§9.11(i) Differential Equation

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Keywords:: Airy functions, differential equation, for products, products
Notes:: Use §1.13(v).
Permalink:: http://dlmf.nist.gov/9.11.i
See also:: Annotations for §9.11 and Ch.9

9.11.1		$\frac{{\mathrm{d}}^{3}w}{{\mathrm{d}z}^{3}}-4z\frac{\mathrm{d}w}{\mathrm{d}z}-% 2w=0,$
$w=w_{1}w_{2}$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $w$ : function, $w_{1}$ : function and $w_{2}$ : function Permalink: http://dlmf.nist.gov/9.11.E1 Encodings: TeX, pMML, png See also: Annotations for §9.11(i), §9.11 and Ch.9

where $w_{1}$ and $w_{2}$ are any solutions of (9.2.1). For example, $w={\operatorname{Ai}}^{2}\left(z\right)$ , $\operatorname{Ai}\left(z\right)\operatorname{Bi}\left(z\right)$ , $\operatorname{Ai}\left(z\right)\operatorname{Ai}\left(ze^{\mp 2\pi i/3}\right)$ , ${M}^{2}\left(z\right)$ . Numerically satisfactory triads of solutions can be constructed where needed on $\mathbb{R}$ or $\mathbb{C}$ by inspection of the asymptotic expansions supplied in §9.7.

§9.11(ii) Wronskian

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Keywords:: Airy functions, Wronskian, products
Permalink:: http://dlmf.nist.gov/9.11.ii
See also:: Annotations for §9.11 and Ch.9

9.11.2		$\mathscr{W}\left\{{\operatorname{Ai}}^{2}\left(z\right),\operatorname{Ai}\left% (z\right)\operatorname{Bi}\left(z\right),{\operatorname{Bi}}^{2}\left(z\right)% \right\}=2\pi^{-3}.$
ⓘ 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 Proof sketch: Use (9.2.1) and (9.2.7). Permalink: http://dlmf.nist.gov/9.11.E2 Encodings: TeX, pMML, png See also: Annotations for §9.11(ii), §9.11 and Ch.9

§9.11(iii) Integral Representations

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Keywords:: Airy functions, Bessel functions, definite integrals, integral representations, products
Notes:: See Lebedev (1965, p. 142) and Muldoon (1977).
Permalink:: http://dlmf.nist.gov/9.11.iii
See also:: Annotations for §9.11 and Ch.9

9.11.3		${\operatorname{Ai}}^{2}\left(x\right)=\frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}J% _{0}\left(\tfrac{1}{12}t^{3}+xt\right)t\,\mathrm{d}t,$
$x\geq 0$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real variable Source: Lebedev (1965, Problem 22, p. 142) Referenced by: §9.5(i) Permalink: http://dlmf.nist.gov/9.11.E3 Encodings: TeX, pMML, png See also: Annotations for §9.11(iii), §9.11 and Ch.9

where $J_{0}$ is the Bessel function (§10.2(ii)).

9.11.4		${\operatorname{Ai}}^{2}\left(z\right)+{\operatorname{Bi}}^{2}\left(z\right)=% \frac{1}{\pi^{3/2}}\int_{0}^{\infty}\exp\left(zt-\tfrac{1}{12}t^{3}\right)t^{-% 1/2}\,\mathrm{d}t.$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral and $z$ : complex variable Source: Muldoon (1977, p. 32, extended to complex $z$ by analytic continuation) Referenced by: §9.5(ii) Permalink: http://dlmf.nist.gov/9.11.E4 Encodings: TeX, pMML, png See also: Annotations for §9.11(iii), §9.11 and Ch.9

For an integral representation of the Dirac delta involving a product of two $\operatorname{Ai}$ functions see §1.17(ii).

For further integral representations see Reid (1995, 1997a, 1997b).

§9.11(iv) Indefinite Integrals

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Keywords:: Airy functions, integrals, of products, products
Notes:: See Albright (1977) and Albright and Gavathas (1986).
Referenced by:: §9.10(iii), §9.10(ix)
Permalink:: http://dlmf.nist.gov/9.11.iv
See also:: Annotations for §9.11 and Ch.9

Let $w_{1},w_{2}$ be any solutions of (9.2.1), not necessarily distinct. Then

9.11.5		$\int w_{1}w_{2}\,\mathrm{d}z=-w^{\prime}_{1}w^{\prime}_{2}+zw_{1}w_{2},$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function Source: Albright (1977, (A.16)) Permalink: http://dlmf.nist.gov/9.11.E5 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

9.11.6		$\int w_{1}w^{\prime}_{2}\,\mathrm{d}z=\tfrac{1}{2}\left(w_{1}w_{2}+z\mathscr{W% }\left\{w_{1},w_{2}\right\}\right),$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function Source: Albright (1977, (A.17)) Permalink: http://dlmf.nist.gov/9.11.E6 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

9.11.7		$\int w^{\prime}_{1}w^{\prime}_{2}\,\mathrm{d}z=\tfrac{1}{3}(w_{1}w^{\prime}_{2% }+w^{\prime}_{1}w_{2}+zw^{\prime}_{1}w^{\prime}_{2}-z^{2}w_{1}w_{2}),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function Source: Albright (1977, (A.18)) Permalink: http://dlmf.nist.gov/9.11.E7 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

9.11.8		$\int zw_{1}w_{2}\,\mathrm{d}z=\tfrac{1}{6}(w_{1}w^{\prime}_{2}+w^{\prime}_{1}w% _{2})-\tfrac{1}{3}(zw^{\prime}_{1}w^{\prime}_{2}-z^{2}w_{1}w_{2}),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function Source: Albright (1977, (A.19)) Permalink: http://dlmf.nist.gov/9.11.E8 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

9.11.9		$\int zw_{1}w^{\prime}_{2}\,\mathrm{d}z=\tfrac{1}{2}w^{\prime}_{1}w^{\prime}_{2% }+\tfrac{1}{4}z^{2}\mathscr{W}\left\{w_{1},w_{2}\right\},$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function Source: Albright (1977, (A.20)) Permalink: http://dlmf.nist.gov/9.11.E9 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

9.11.10		$\int zw^{\prime}_{1}w^{\prime}_{2}\,\mathrm{d}z=\tfrac{3}{10}(-w_{1}w_{2}+zw_{% 1}w^{\prime}_{2}+zw^{\prime}_{1}w_{2})+\tfrac{1}{5}(z^{2}w^{\prime}_{1}w^{% \prime}_{2}-z^{3}w_{1}w_{2}).$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function Source: Albright (1977, (A.21)) Permalink: http://dlmf.nist.gov/9.11.E10 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

For $\int z^{n}w_{1}w_{2}\,\mathrm{d}z$ , $\int z^{n}w_{1}w^{\prime}_{2}\,\mathrm{d}z$ , $\int z^{n}w^{\prime}_{1}w^{\prime}_{2}\,\mathrm{d}z$ , where $n$ is any positive integer, see Albright (1977). For related integrals see Gordon (1969, Appendix B).

For any continuously-differentiable function $f$

9.11.11		$\int\frac{1}{w_{1}^{2}}f^{\prime}\!\left(\frac{w_{2}}{w_{1}}\right)\,\mathrm{d% }z=\frac{1}{\mathscr{W}\left\{w_{1},w_{2}\right\}}f\left(\frac{w_{2}}{w_{1}}% \right).$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $w$ : function and $f$ : function Source: Albright and Gavathas (1986, p. 2664) Permalink: http://dlmf.nist.gov/9.11.E11 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11 and Ch.9

Examples

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See also:: Annotations for §9.11(iv), §9.11 and Ch.9

9.11.12	$\displaystyle\int\frac{\,\mathrm{d}z}{{\operatorname{Ai}}^{2}\left(z\right)}$	$\displaystyle=\pi\frac{\operatorname{Bi}\left(z\right)}{\operatorname{Ai}\left% (z\right)},$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable Source: Albright and Gavathas (1986, (8)) Permalink: http://dlmf.nist.gov/9.11.E12 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11(iv), §9.11 and Ch.9
9.11.13	$\displaystyle\int\frac{\,\mathrm{d}z}{\operatorname{Ai}\left(z\right)% \operatorname{Bi}\left(z\right)}$	$\displaystyle=\pi\ln\left(\frac{\operatorname{Bi}\left(z\right)}{\operatorname% {Ai}\left(z\right)}\right),$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable Source: Albright and Gavathas (1986, (10)) Permalink: http://dlmf.nist.gov/9.11.E13 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11(iv), §9.11 and Ch.9
9.11.14	$\displaystyle\int\frac{\operatorname{Ai}\left(z\right)\operatorname{Bi}\left(z% \right)}{\left({\operatorname{Ai}}^{2}\left(z\right)+{\operatorname{Bi}}^{2}% \left(z\right)\right)^{2}}\,\mathrm{d}z$	$\displaystyle=\frac{\pi}{2}\frac{{\operatorname{Bi}}^{2}\left(z\right)}{{% \operatorname{Ai}}^{2}\left(z\right)+{\operatorname{Bi}}^{2}\left(z\right)}.$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $z$ : complex variable Source: Albright and Gavathas (1986, (5)) Permalink: http://dlmf.nist.gov/9.11.E14 Encodings: TeX, pMML, png See also: Annotations for §9.11(iv), §9.11(iv), §9.11 and Ch.9

§9.11(v) Definite Integrals

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Keywords:: Airy functions, integrals, of products, products
Notes:: See Reid (1995), Laurenzi (1993).
Permalink:: http://dlmf.nist.gov/9.11.v
See also:: Annotations for §9.11 and Ch.9

9.11.15		$\int_{0}^{\infty}t^{\alpha-1}{\operatorname{Ai}}^{2}\left(t\right)\,\mathrm{d}% t=\frac{2\Gamma\left(\alpha\right)}{\pi^{1/2}12^{(2\alpha+5)/6}\Gamma\left(% \frac{1}{3}\alpha+\frac{5}{6}\right)},$
$\Re\alpha>0$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : real part Source: Laurenzi (1993, p. 904) Permalink: http://dlmf.nist.gov/9.11.E15 Encodings: TeX, pMML, png See also: Annotations for §9.11(v), §9.11 and Ch.9

9.11.16	$\displaystyle\int_{-\infty}^{\infty}{\operatorname{Ai}}^{3}\left(t\right)\,% \mathrm{d}t$	$\displaystyle=\frac{{\Gamma}^{2}\left(\frac{1}{3}\right)}{4\pi^{2}},$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Source: Reid (1997a, (2.17)) Permalink: http://dlmf.nist.gov/9.11.E16 Encodings: TeX, pMML, png See also: Annotations for §9.11(v), §9.11 and Ch.9
9.11.17	$\displaystyle\int_{-\infty}^{\infty}{\operatorname{Ai}}^{2}\left(t\right)% \operatorname{Bi}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{{\Gamma}^{2}\left(\frac{1}{3}\right)}{4\sqrt{3}\pi^{2}}.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Source: Reid (1997a, (2.33)) Permalink: http://dlmf.nist.gov/9.11.E17 Encodings: TeX, pMML, png See also: Annotations for §9.11(v), §9.11 and Ch.9
9.11.18	$\displaystyle\int_{0}^{\infty}{\operatorname{Ai}}^{4}\left(t\right)\,\mathrm{d}t$	$\displaystyle=\frac{\ln 3}{24\pi^{2}}.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function Source: Laurenzi (1993, p. 905) Permalink: http://dlmf.nist.gov/9.11.E18 Encodings: TeX, pMML, png See also: Annotations for §9.11(v), §9.11 and Ch.9

9.11.19		$\int_{0}^{\infty}\frac{\,\mathrm{d}t}{{\operatorname{Ai}}^{2}\left(t\right)+{% \operatorname{Bi}}^{2}\left(t\right)}=\int_{0}^{\infty}\frac{t\,\mathrm{d}t}{{% \operatorname{Ai}'}^{2}\left(t\right)+{\operatorname{Bi}'}^{2}\left(t\right)}=% \frac{\pi^{2}}{6}.$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $M\left(\NVar{z}\right)$ : Airy modulus function, $N\left(\NVar{z}\right)$ : Airy modulus function, $\phi\left(\NVar{z}\right)$ : Airy phase function, $\theta\left(\NVar{z}\right)$ : Airy phase function and $x$ : real variable Proof sketch: Extend the definitions of §9.8(i) to positive values of $x$ , obtain the indefinite integrals of $1/{M}^{2}\left(x\right)$ and $x/{N}^{2}\left(x\right)$ via the first two of (9.8.14), then combine the values of $\theta\left(0\right)$ and $\phi\left(0\right)$ given in §9.8(i) with $\theta\left(+\infty\right)=\phi\left(+\infty\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.) Permalink: http://dlmf.nist.gov/9.11.E19 Encodings: TeX, pMML, png See also: Annotations for §9.11(v), §9.11 and Ch.9

For further definite integrals see Prudnikov et al. (1990, §1.8.2), Laurenzi (1993), Reid (1995, 1997a, 1997b), and Vallée and Soares (2010, Chapters 3, 4).

9.10 Integrals 9.12 Scorer Functions

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