§36.9 Integral Identities

Keywords:: Airy functions, Pearcey integral, canonical integrals, fold canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.9

36.9.1 $|\mathop{\Psi _{{1}}\/}\nolimits\!\left(x\right)|^{2}=2^{{5/3}}\int _{0}^{\infty}\mathop{\Psi _{{1}}\/}\nolimits\!\left(2^{{2/3}}(3u^{2}+x)\right)du;$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\int$ : integral and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E1
Encodings:: TeX, pMML, png

equivalently,

36.9.2 $(\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right))^{2}=\frac{2^{{2/3}}}{\pi}\int _{0}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(2^{{2/3}}(u^{2}+x)\right)du.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $dx$ : differential of $x$ , $\int$ : integral and $x$ : real parameter
Referenced by:: §9.5(i)
Permalink:: http://dlmf.nist.gov/36.9.E2
Encodings:: TeX, pMML, png

36.9.3 $|\mathop{\Psi _{{1}}\/}\nolimits\!\left(x\right)|^{2}=\sqrt{\frac{8\pi}{3}}\int _{0}^{\infty}u^{{-1/2}}\mathop{\cos\/}\nolimits\!\left(2u(x+u^{2})+\tfrac{1}{4}\pi\right)du.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E3
Encodings:: TeX, pMML, png

36.9.4 $|\mathop{\Psi _{{2}}\/}\nolimits\!\left(x,y\right)|^{2}=\int _{0}^{\infty}\left(\mathop{\Psi _{{1}}\/}\nolimits\!\left(\frac{4u^{3}+2uy+x}{u^{{1/3}}}\right)+\mathop{\Psi _{{1}}\/}\nolimits\!\left(\frac{4u^{3}+2uy-x}{u^{{1/3}}}\right)\right)\frac{du}{u^{{1/3}}}.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E4
Encodings:: TeX, pMML, png

36.9.5 $|\mathop{\Psi _{{2}}\/}\nolimits\!\left(x,y\right)|^{2}=2\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(2xu\right)\mathop{\Psi _{{1}}\/}\nolimits\!\left(2u^{{2/3}}(y+2u^{2})\right)\frac{du}{u^{{1/3}}}.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E5
Encodings:: TeX, pMML, png

36.9.6 $|\mathop{\Psi _{{3}}\/}\nolimits\!\left(x,y,z\right)|^{2}=2^{{4/5}}\int _{{-\infty}}^{\infty}\mathop{\Psi _{{3}}\/}\nolimits\!\left(2^{{4/5}}(x+2uy+3u^{2}z+5u^{4}),0,2^{{2/5}}(z+10u^{2})\right)du.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E6
Encodings:: TeX, pMML, png

36.9.7 $|\mathop{\Psi _{{3}}\/}\nolimits\!\left(x,y,z\right)|^{2}=\frac{2^{{7/4}}}{5^{{1/4}}}\int _{0}^{\infty}\realpart{\left(e^{{2iu(u^{4}+zu^{2}+x)}}\mathop{\Psi _{{2}}\/}\nolimits\!\left(\frac{2^{{7/4}}}{5^{{1/4}}}yu^{{3/4}},\sqrt{\frac{2u}{5}}(3z+10u^{2})\right)\right)}\frac{du}{u^{{1/4}}}.$

Symbols:: $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E7
Encodings:: TeX, pMML, png

36.9.8 $\left|\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(x,y,z\right)\right|^{2}=8\pi^{2}\left(\frac{2}{9}\right)^{{1/3}}\int _{{-\infty}}^{\infty}\int _{{-\infty}}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(\left(\frac{4}{3}\right)^{{1/3}}(x+zv+3u^{2})\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(\left(\frac{4}{3}\right)^{{1/3}}(y+zu+3v^{2})\right)dudv.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E8
Encodings:: TeX, pMML, png

36.9.9 $\left|\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(x,y,z\right)\right|^{2}=\frac{8\pi^{2}}{3^{{2/3}}}\int _{0}^{\infty}\int _{0}^{{2\pi}}\realpart{}\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(\frac{1}{3^{{1/3}}}\left(x+iy+2zu\mathop{\exp\/}\nolimits\!\left(i\theta\right)+3u^{2}\mathop{\exp\/}\nolimits\!\left(-2i\theta\right)\right)\right)\*\mathop{\mathrm{Bi}\/}\nolimits\!\left(\frac{1}{3^{{1/3}}}\left(x-iy+2zu\mathop{\exp\/}\nolimits\!\left(-i\theta\right)+3u^{2}\mathop{\exp\/}\nolimits\!\left(2i\theta\right)\right)\right)\right)udud\theta.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\realpart{}$ : real part, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter
Keywords:: elliptic umbilic canonical integral
Permalink:: http://dlmf.nist.gov/36.9.E9
Encodings:: TeX, pMML, png

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.