§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, $w={\mathop{\mathrm{Ai}\/}\nolimits^{2}}\!\left(z\right)$, $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Bi}\/}\nolimits% \!\left(z\right)$, $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Ai}\/}\nolimits% \!\left(ze^{\mp 2\pi i/3}\right)$, ${\mathop{M\/}\nolimits^{2}}\!\left(z\right)$. Numerically satisfactory triads of solutions can be constructed where needed on $\mathbb{R}$ or $\mathbb{C}$ by inspection of the asymptotic expansions supplied in §9.7.

§9.11(ii) Wronskian

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

§9.11(iii) Integral Representations

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

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

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

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

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

 9.11.5 $\int w_{1}w_{2}\mathrm{d}z=-w^{\prime}_{1}w^{\prime}_{2}+zw_{1}w_{2},$
 9.11.6 $\int w_{1}w^{\prime}_{2}\mathrm{d}z=\tfrac{1}{2}\left(w_{1}w_{2}+z\mathop{% \mathscr{W}\/}\nolimits\left\{w_{1},w_{2}\right\}\right),$
 9.11.7 $\int w^{\prime}_{1}w^{\prime}_{2}\mathrm{d}z=\tfrac{1}{3}(w_{1}w^{\prime}_{2}+% w^{\prime}_{1}w_{2}+zw^{\prime}_{1}w^{\prime}_{2}-z^{2}w_{1}w_{2}),$
 9.11.8 $\int zw_{1}w_{2}\mathrm{d}z=\tfrac{1}{6}(w_{1}w^{\prime}_{2}+w^{\prime}_{1}w_{% 2})-\tfrac{1}{3}(zw^{\prime}_{1}w^{\prime}_{2}-z^{2}w_{1}w_{2}),$
 9.11.9 $\int zw_{1}w^{\prime}_{2}\mathrm{d}z=\tfrac{1}{2}w^{\prime}_{1}w^{\prime}_{2}+% \tfrac{1}{4}z^{2}\mathop{\mathscr{W}\/}\nolimits\left\{w_{1},w_{2}\right\},$
 9.11.10 $\int zw^{\prime}_{1}w^{\prime}_{2}\mathrm{d}z=\tfrac{3}{10}(-w_{1}w_{2}+zw_{1}% w^{\prime}_{2}+zw^{\prime}_{1}w_{2})+\tfrac{1}{5}(z^{2}w^{\prime}_{1}w^{\prime% }_{2}-z^{3}w_{1}w_{2}).$

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 $\int\frac{1}{w_{1}^{2}}f^{\prime}\!\left(\frac{w_{2}}{w_{1}}\right)\mathrm{d}z% =\frac{1}{\mathop{\mathscr{W}\/}\nolimits\left\{w_{1},w_{2}\right\}}f\!\left(% \frac{w_{2}}{w_{1}}\right).$

Examples

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

§9.11(v) Definite Integrals

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

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).