§9.11 Products

§9.11(i) Differential Equation

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

where $w_{1}$ and $w_{2}$ are any solutions of (9.2.1). For example, $w={\mathrm{Ai}^{2}}\left(z\right)$, $\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right)$, $\mathrm{Ai}\left(z\right)\mathrm{Ai}\left(ze^{\mp 2\pi i/3}\right)$, ${M^{2}}\left(z\right)$. 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

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

§9.11(iii) Integral Representations

 9.11.3 ${\mathrm{Ai}^{2}}\left(x\right)=\frac{1}{4\pi\sqrt{3}}\int_{0}^{\infty}J_{0}% \left(\tfrac{1}{12}t^{3}+xt\right)t\mathrm{d}t,$ $x\geq 0$,

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

 9.11.4 ${\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)=\frac{1}{\pi^{% 3/2}}\int_{0}^{\infty}\exp\left(zt-\tfrac{1}{12}t^{3}\right)t^{-1/2}\mathrm{d}t.$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: 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 $\mathrm{Ai}$ functions see §1.17(ii).

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

§9.11(iv) Indefinite Integrals

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

 9.11.12 $\displaystyle\int\frac{\mathrm{d}z}{{\mathrm{Ai}^{2}}\left(z\right)}$ $\displaystyle=\pi\frac{\mathrm{Bi}\left(z\right)}{\mathrm{Ai}\left(z\right)},$ 9.11.13 $\displaystyle\int\frac{\mathrm{d}z}{\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(% z\right)}$ $\displaystyle=\pi\ln\left(\frac{\mathrm{Bi}\left(z\right)}{\mathrm{Ai}\left(z% \right)}\right),$ 9.11.14 $\displaystyle\int\frac{\mathrm{Ai}\left(z\right)\mathrm{Bi}\left(z\right)}{% \left({\mathrm{Ai}^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)\right)^{% 2}}\mathrm{d}z$ $\displaystyle=\frac{\pi}{2}\frac{{\mathrm{Bi}^{2}}\left(z\right)}{{\mathrm{Ai}% ^{2}}\left(z\right)+{\mathrm{Bi}^{2}}\left(z\right)}.$

§9.11(v) Definite Integrals

 9.11.15 $\int_{0}^{\infty}t^{\alpha-1}{\mathrm{Ai}^{2}}\left(t\right)\mathrm{d}t=\frac{% 2\Gamma\left(\alpha\right)}{\pi^{1/2}12^{(2\alpha+5)/6}\Gamma\left(\frac{1}{3}% \alpha+\frac{5}{6}\right)},$ $\Re\alpha>0$.
 9.11.16 $\displaystyle\int_{-\infty}^{\infty}{\mathrm{Ai}^{3}}\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{{\Gamma^{2}}\left(\frac{1}{3}\right)}{4\pi^{2}},$ 9.11.17 $\displaystyle\int_{-\infty}^{\infty}{\mathrm{Ai}^{2}}\left(t\right)\mathrm{Bi}% \left(t\right)\mathrm{d}t$ $\displaystyle=\frac{{\Gamma^{2}}\left(\frac{1}{3}\right)}{4\sqrt{3}\pi^{2}}.$ 9.11.18 $\displaystyle\int_{0}^{\infty}{\mathrm{Ai}^{4}}\left(t\right)\mathrm{d}t$ $\displaystyle=\frac{\ln 3}{24\pi^{2}}.$
 9.11.19 $\int_{0}^{\infty}\frac{\mathrm{d}t}{{\mathrm{Ai}^{2}}\left(t\right)+{\mathrm{% Bi}^{2}}\left(t\right)}=\int_{0}^{\infty}\frac{t\mathrm{d}t}{{\mathrm{Ai}'^{2}% }\left(t\right)+{\mathrm{Bi}'^{2}}\left(t\right)}=\frac{\pi^{2}}{6}.$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Bi}\left(\NVar{z}\right)$: 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 Source: 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).