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DLMF: §36.9 Integral Identities ‣ Properties ‣ Chapter 36 Integrals with Coalescing Saddles

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

§36.9 Integral Identities

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Keywords:: Airy functions, Pearcey integral, canonical integrals, fold canonical integral, hyperbolic umbilic canonical integral, integral identities, integral identity, swallowtail canonical integral
Permalink:: http://dlmf.nist.gov/36.9
See also:: Annotations for Ch.36

36.9.1		$\displaystyle\|\Psi_{1}\left(x\right)\|^{2}$	$\displaystyle=2^{5/3}\int_{0}^{\infty}\Psi_{1}\left(2^{2/3}(3u^{2}+x)\right)\,% \mathrm{d}u;$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.9.E1 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36
equivalently,
36.9.2		$\displaystyle(\operatorname{Ai}\left(x\right))^{2}$	$\displaystyle=\frac{2^{2/3}}{\pi}\int_{0}^{\infty}\operatorname{Ai}\left(2^{2/% 3}(u^{2}+x)\right)\,\mathrm{d}u.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : 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 See also: Annotations for §36.9 and Ch.36

36.9.3		$\displaystyle\|\Psi_{1}\left(x\right)\|^{2}$	$\displaystyle=\sqrt{\frac{8\pi}{3}}\int_{0}^{\infty}u^{-1/2}\cos\left(2u(x+u^{% 2})+\tfrac{1}{4}\pi\right)\,\mathrm{d}u.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.9.E3 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36
36.9.4		$\displaystyle\|\Psi_{2}\left(x,y\right)\|^{2}$	$\displaystyle=\int_{0}^{\infty}\left(\Psi_{1}\left(\frac{4u^{3}+2uy+x}{u^{1/3}% }\right)+\Psi_{1}\left(\frac{4u^{3}+2uy-x}{u^{1/3}}\right)\right)\frac{\,% \mathrm{d}u}{u^{1/3}}.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.9.E4 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36
36.9.5		$\displaystyle\|\Psi_{2}\left(x,y\right)\|^{2}$	$\displaystyle=2\int_{0}^{\infty}\cos\left(2xu\right)\Psi_{1}\left(2u^{2/3}(y+2% u^{2})\right)\frac{\,\mathrm{d}u}{u^{1/3}}.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.9.E5 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36

36.9.6		$\displaystyle\|\Psi_{3}\left(x,y,z\right)\|^{2}$	$\displaystyle=2^{4/5}\int_{-\infty}^{\infty}\Psi_{3}\left(2^{4/5}(x+2uy+3u^{2}% z+5u^{4}),0,2^{2/5}(z+10u^{2})\right)\,\mathrm{d}u.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\,\mathrm{d}\NVar{x}$ : 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 See also: Annotations for §36.9 and Ch.36
36.9.7		$\displaystyle\|\Psi_{3}\left(x,y,z\right)\|^{2}$	$\displaystyle=\frac{2^{7/4}}{5^{1/4}}\int_{0}^{\infty}\Re\left({\mathrm{e}}^{2% iu(u^{4}+zu^{2}+x)}\Psi_{2}\left(\frac{2^{7/4}}{5^{1/4}}yu^{3/4},\sqrt{\frac{2% u}{5}}(3z+10u^{2})\right)\right)\frac{\,\mathrm{d}u}{u^{1/4}}.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.9.E7 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36

36.9.8		$\left\|\Psi^{(\mathrm{H})}\left(x,y,z\right)\right\|^{2}=8\pi^{2}\left(\frac{2}{% 9}\right)^{1/3}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\operatorname{Ai}% \left(\left(\frac{4}{3}\right)^{1/3}(x+zv+3u^{2})\right)\operatorname{Ai}\left% (\left(\frac{4}{3}\right)^{1/3}(y+zu+3v^{2})\right)\,\mathrm{d}u\,\mathrm{d}v.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.9.E8 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36

36.9.9		$\left\|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right\|^{2}=\frac{8\pi^{2}}{3^{2/3}% }\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left(\operatorname{Ai}\left(\frac{1}{3^{1% /3}}\left(x+iy+2zu\exp\left(i\theta\right)+3u^{2}\exp\left(-2i\theta\right)% \right)\right)\*\operatorname{Bi}\left(\frac{1}{3^{1/3}}\left(x-iy+2zu\exp% \left(-i\theta\right)+3u^{2}\exp\left(2i\theta\right)\right)\right)\right)u\,% \mathrm{d}u\,\mathrm{d}\theta.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Keywords: elliptic umbilic canonical integral, integral identity Permalink: http://dlmf.nist.gov/36.9.E9 Encodings: TeX, pMML, png See also: Annotations for §36.9 and Ch.36

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.

36.8 Convergent Series Expansions 36.10 Differential Equations

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