36 Integrals with Coalescing Saddles Properties 36.8 Convergent Series Expansions 36.10 Differential Equations

§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, : differential of , $\int$ : integral and : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E1
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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, : differential of , $\int$ : integral and : real parameter
Referenced by:: §9.5(i)
Permalink:: http://dlmf.nist.gov/36.9.E2
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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, : differential of , $\int$ : integral and : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E3
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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, : differential of , $\int$ : integral, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E4
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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, : differential of , $\int$ : integral, : real parameter and : 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, : differential of , $\int$ : integral, : real parameter, : real parameter and : 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, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E7
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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, : differential of , $\int$ : integral, : real parameter, : real parameter and : real parameter
Permalink:: http://dlmf.nist.gov/36.9.E8
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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, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\realpart{}$ : real part, : real parameter, : real parameter and : 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.

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