# §36.9 Integral Identities

 36.9.1 $\displaystyle|\mathop{\Psi_{1}\/}\nolimits\!\left(x\right)|^{2}$ $\displaystyle=2^{5/3}\int_{0}^{\infty}\mathop{\Psi_{1}\/}\nolimits\!\left(2^{2% /3}(3u^{2}+x)\right)du;$ equivalently, 36.9.2 $\displaystyle(\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right))^{2}$ $\displaystyle=\frac{2^{2/3}}{\pi}\int_{0}^{\infty}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(2^{2/3}(u^{2}+x)\right)du.$
 36.9.3 $\displaystyle|\mathop{\Psi_{1}\/}\nolimits\!\left(x\right)|^{2}$ $\displaystyle=\sqrt{\frac{8\pi}{3}}\int_{0}^{\infty}u^{-1/2}\mathop{\cos\/}% \nolimits\!\left(2u(x+u^{2})+\tfrac{1}{4}\pi\right)du.$ 36.9.4 $\displaystyle|\mathop{\Psi_{2}\/}\nolimits\!\left(x,y\right)|^{2}$ $\displaystyle=\int_{0}^{\infty}\left(\mathop{\Psi_{1}\/}\nolimits\!\left(\frac% {4u^{3}+2uy+x}{u^{1/3}}\right)+\mathop{\Psi_{1}\/}\nolimits\!\left(\frac{4u^{3% }+2uy-x}{u^{1/3}}\right)\right)\frac{du}{u^{1/3}}.$ 36.9.5 $\displaystyle|\mathop{\Psi_{2}\/}\nolimits\!\left(x,y\right)|^{2}$ $\displaystyle=2\int_{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(2xu\right)% \mathop{\Psi_{1}\/}\nolimits\!\left(2u^{2/3}(y+2u^{2})\right)\frac{du}{u^{1/3}}.$
 36.9.6 $\displaystyle|\mathop{\Psi_{3}\/}\nolimits\!\left(x,y,z\right)|^{2}$ $\displaystyle=2^{4/5}\int_{-\infty}^{\infty}\mathop{\Psi_{3}\/}\nolimits\!% \left(2^{4/5}(x+2uy+3u^{2}z+5u^{4}),0,2^{2/5}(z+10u^{2})\right)du.$ 36.9.7 $\displaystyle|\mathop{\Psi_{3}\/}\nolimits\!\left(x,y,z\right)|^{2}$ $\displaystyle=\frac{2^{7/4}}{5^{1/4}}\int_{0}^{\infty}\realpart{\left(e^{2iu(u% ^{4}+zu^{2}+x)}\mathop{\Psi_{2}\/}\nolimits\!\left(\frac{2^{7/4}}{5^{1/4}}yu^{% 3/4},\sqrt{\frac{2u}{5}}(3z+10u^{2})\right)\right)}\frac{du}{u^{1/4}}.$
 36.9.8 $\left|\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(x,y,z\right)\right|^{2}=8% \pi^{2}\left(\frac{2}{9}\right)^{1/3}\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(\left(\frac{4}{3}\right)^{1/3}(x% +zv+3u^{2})\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\left(\frac{4}{3}% \right)^{1/3}(y+zu+3v^{2})\right)dudv.$
 36.9.9 $\left|\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(x,y,z\right)\right|^{2}=% \frac{8\pi^{2}}{3^{2/3}}\int_{0}^{\infty}\int_{0}^{2\pi}\realpart{}\left(% \mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{1}{3^{1/3}}\left(x+iy+2zu\mathop{% \exp\/}\nolimits\!\left(i\theta\right)+3u^{2}\mathop{\exp\/}\nolimits\!\left(-% 2i\theta\right)\right)\right)\*\mathop{\mathrm{Bi}\/}\nolimits\!\left(\frac{1}% {3^{1/3}}\left(x-iy+2zu\mathop{\exp\/}\nolimits\!\left(-i\theta\right)+3u^{2}% \mathop{\exp\/}\nolimits\!\left(2i\theta\right)\right)\right)\right)udud\theta.$

For these results and also integrals over doubly-infinite intervals see Berry and Wright (1980). This reference also provides a physical interpretation in terms of Lagrangian manifolds and Wigner functions in phase space.