§9.11 Products

Permalink:: http://dlmf.nist.gov/9.11

§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

Notes:: Use §1.13(v).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.11.i

9.11.1 $\frac{{d}^{3}w}{{dz}^{3}}-4z\frac{dw}{dz}-2w=0,$ $w=w_{1}w_{2}$ ,

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $w$ : function, $w_{1}$ : function and $w_{2}$ : function
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where $w_{1}$ and $w_{2}$ are any solutions of (9.2.1). For example, $w={\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(z\right)$ , $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(ze^{{\mp 2\pi i/3}}\right)$ , ${\mathop{M\/}\nolimits^{{2}}}\!\left(z\right)$ . Numerically satisfactory triads of solutions can be constructed where needed on $\Real$ or $\Complex$ by inspection of the asymptotic expansions supplied in §9.7.

§9.11(ii) Wronskian

Notes:: Use (9.2.1) and (9.2.7).
Keywords:: Airy functions, Wronskian
Permalink:: http://dlmf.nist.gov/9.11.ii

9.11.2 $\mathop{\mathscr{W}\/}\nolimits\left\{{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(z\right),\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right),{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(z\right)\right\}=2\pi^{{-3}}.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.11.E2
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§9.11(iii) Integral Representations

Notes:: See Lebedev (1965, p. 142) and Muldoon (1977).
Keywords:: Airy functions, Bessel functions
Permalink:: http://dlmf.nist.gov/9.11.iii

9.11.3 ${\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(x\right)=\frac{1}{4\pi\sqrt{3}}\int _{0}^{{\infty}}\mathop{J_{{0}}\/}\nolimits\!\left(\tfrac{1}{12}t^{3}+xt\right)tdt,$ $x\geq 0$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\int$ : integral and $x$ : real variable
Referenced by:: §9.5(i)
Permalink:: http://dlmf.nist.gov/9.11.E3
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where $\mathop{J_{{0}}\/}\nolimits$ is the Bessel function (§10.2(ii)).

9.11.4 ${\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(z\right)=\frac{1}{\pi^{{3/2}}}\int _{0}^{{\infty}}\mathop{\exp\/}\nolimits\!\left(zt-\tfrac{1}{12}t^{3}\right)t^{{-1/2}}dt.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral and $z$ : complex variable
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.11.E4
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For an integral representation of the Dirac delta involving a product of two $\mathop{\mathrm{Ai}\/}\nolimits$ functions see §1.17(ii).

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

§9.11(iv) Indefinite Integrals

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

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

9.11.5 $\int w_{1}w_{2}dz=-w^{{\prime}}_{1}w^{{\prime}}_{2}+zw_{1}w_{2},$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function
Permalink:: http://dlmf.nist.gov/9.11.E5
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9.11.6 $\int w_{1}w^{{\prime}}_{2}dz=\tfrac{1}{2}\left(w_{1}w_{2}+z\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1},w_{2}\right\}\right),$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function
Permalink:: http://dlmf.nist.gov/9.11.E6
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9.11.7 $\int w^{{\prime}}_{1}w^{{\prime}}_{2}dz=\tfrac{1}{3}(w_{1}w^{{\prime}}_{2}+w^{{\prime}}_{1}w_{2}+zw^{{\prime}}_{1}w^{{\prime}}_{2}-z^{2}w_{1}w_{2}),$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function
Permalink:: http://dlmf.nist.gov/9.11.E7
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9.11.8 $\int zw_{1}w_{2}dz=\tfrac{1}{6}(w_{1}w^{{\prime}}_{2}+w^{{\prime}}_{1}w_{2})-\tfrac{1}{3}(zw^{{\prime}}_{1}w^{{\prime}}_{2}-z^{2}w_{1}w_{2}),$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function
Permalink:: http://dlmf.nist.gov/9.11.E8
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9.11.9 $\int zw_{1}w^{{\prime}}_{2}dz=\tfrac{1}{2}w^{{\prime}}_{1}w^{{\prime}}_{2}+\tfrac{1}{4}z^{2}\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1},w_{2}\right\},$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function
Permalink:: http://dlmf.nist.gov/9.11.E9
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9.11.10 $\int zw^{{\prime}}_{1}w^{{\prime}}_{2}dz=\tfrac{3}{10}(-w_{1}w_{2}+zw_{1}w^{{\prime}}_{2}+zw^{{\prime}}_{1}w_{2})+\tfrac{1}{5}(z^{2}w^{{\prime}}_{1}w^{{\prime}}_{2}-z^{3}w_{1}w_{2}).$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable and $w$ : function
Permalink:: http://dlmf.nist.gov/9.11.E10
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For $\int z^{n}w_{1}w_{2}dz$ , $\int z^{n}w_{1}w^{{\prime}}_{2}dz$ , $\int z^{n}w^{{\prime}}_{1}w^{{\prime}}_{2}dz$ , 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 $\int\frac{1}{w_{1}^{2}}f^{{\prime}}\!\left(\frac{w_{2}}{w_{1}}\right)dz=\frac{1}{\mathop{\mathscr{W}\/}\nolimits\left\{ w_{1},w_{2}\right\}}f\!\left(\frac{w_{2}}{w_{1}}\right).$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $w$ : function and $f$ : function
Permalink:: http://dlmf.nist.gov/9.11.E11
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¶ Examples

9.11.12 $\int\frac{dz}{{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(z\right)}=\pi\frac{\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)}{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.11.E12
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9.11.13 $\int\frac{dz}{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)}=\pi\mathop{\ln\/}\nolimits\!\left(\frac{\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)}{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)}\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.11.E13
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9.11.14 $\int\frac{\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)}{\left({\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(z\right)\right)^{2}}dz=\frac{\pi}{2}\frac{{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(z\right)}{{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(z\right)}.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\int$ : integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.11.E14
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§9.11(v) Definite Integrals

Notes:: For (9.11.15) see Reid (1995). For (9.11.16) and (9.11.17) see Reid (1997a). For (9.11.18) see Laurenzi (1993). For (9.11.19) extend the definitions of §9.8(i) to positive values of $x$ , obtain the indefinite integrals of $1/{\mathop{M\/}\nolimits^{{2}}}\!\left(x\right)$ and $x/{\mathop{N\/}\nolimits^{{2}}}\!\left(x\right)$ via the first two of (9.8.14), then combine the values of $\mathop{\theta\/}\nolimits\!\left(0\right)$ and $\mathop{\phi\/}\nolimits\!\left(0\right)$ given in §9.8(i) with $\mathop{\theta\/}\nolimits\!\left(+\infty\right)=\mathop{\phi\/}\nolimits\!\left(+\infty\right)=0$ obtained from (9.8.4), (9.8.8), and §9.7(ii). (Communicated by M.E. Muldoon.)
Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.11.v

9.11.15 $\int _{0}^{\infty}t^{{\alpha-1}}{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(t\right)dt=\frac{2\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}{\pi^{{1/2}}12^{{(2\alpha+5)/6}}\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{3}\alpha+\frac{5}{6}\right)},$ $\realpart{\alpha}>0$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral and $\realpart{}$ : real part
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.11.E15
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9.11.16 $\int _{{-\infty}}^{\infty}{\mathop{\mathrm{Ai}\/}\nolimits^{{3}}}\!\left(t\right)dt=\frac{{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(\frac{1}{3}\right)}{4\pi^{2}},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.11.E16
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9.11.17 $\int _{{-\infty}}^{\infty}{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(t\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt=\frac{{\mathop{\Gamma\/}\nolimits^{{2}}}\!\left(\frac{1}{3}\right)}{4\sqrt{3}\pi^{2}}.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.11.E17
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9.11.18 $\int _{0}^{\infty}{\mathop{\mathrm{Ai}\/}\nolimits^{{4}}}\!\left(t\right)dt=\frac{\mathop{\ln\/}\nolimits 3}{24\pi^{2}}.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.11.E18
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9.11.19 $\int _{0}^{\infty}\frac{dt}{{\mathop{\mathrm{Ai}\/}\nolimits^{{2}}}\!\left(t\right)+{\mathop{\mathrm{Bi}\/}\nolimits^{{2}}}\!\left(t\right)}=\int _{0}^{\infty}\frac{tdt}{{{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(t\right)+{{\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}^{{2}}}\!\left(t\right)}=\frac{\pi^{2}}{6}.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: §9.11(v)
Permalink:: http://dlmf.nist.gov/9.11.E19
Encodings:: TeX, pMML, png

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).