# §36.9 Integral Identities

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

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.