36 Integrals with Coalescing Saddles Properties 36.1 Special Notation 36.3 Visualizations of Canonical Integrals

§36.2 Catastrophes and Canonical Integrals

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Referenced by:: §36.11, §36.15(iii)
Permalink:: http://dlmf.nist.gov/36.2
See also:: Annotations for Ch.36

§36.2(i) Definitions
§36.2(ii) Special Cases
§36.2(iii) Symmetries
§36.2(iv) Addendum to 36.2(ii) Special Cases

§36.2(i) Definitions

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Keywords:: canonical integrals, definitions
Notes:: The convergence of the oscillatory integrals (36.2.4)–(36.2.11) can be confirmed by rotating the integration paths in the complex plane. For (36.2.6) see Berry et al. (1979). For (36.2.7) shift the $s$ variable in (36.2.5) (with (36.2.2)) to remove the quadratic term, integrate, and then deform the contour of the remaining $t$ integration. For (36.2.8) see Berry and Howls (1990). For (36.2.9) integrate (36.2.5) (with (36.2.3)) with respect to $t$ .
Referenced by:: §36.2(iii)
Permalink:: http://dlmf.nist.gov/36.2.i
See also:: Annotations for §36.2 and Ch.36

Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension $K$

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Keywords:: cusp catastophe, cuspoids, fold catastrophe, normal forms, swallowtail catastrophe
See also:: Annotations for §36.2(i), §36.2 and Ch.36

36.2.1		$\Phi_{K}\left(t;\mathbf{x}\right)=t^{K+2}+\sum_{m=1}^{K}x_{m}t^{m}.$
ⓘ Defines: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ Symbols: $n$ : integer, $t$ : variable, $K$ : codimension and $x_{i}$ : real parameter Referenced by: §36.10(i), §36.12(i), §36.5(ii), §36.7(ii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E1 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

Special cases: $K=1$ , fold catastrophe; $K=2$ , cusp catastrophe; $K=3$ , swallowtail catastrophe.

Normal Forms for Umbilic Catastrophes with Codimension $K=3$

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Defines:: $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$
Keywords:: elliptic umbilic catastrophe, hyperbolic umbilic catastrophe, normal forms, umbilics
See also:: Annotations for §36.2(i), §36.2 and Ch.36

36.2.2	$\displaystyle\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$	$\displaystyle=s^{3}-3st^{2}+z(s^{2}+t^{2})+yt+xs,$
$\mathbf{x}=\{x,y,z\}$ ,
ⓘ Defines: $\Phi^{(\mathrm{E})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe Symbols: $y$ : real parameter, $z$ : real parameter, $t$ : variable, $s$ : variable and $x$ : real parameter Referenced by: §36.10(iii), §36.2(i), §36.7(iii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E2 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36
(elliptic umbilic).
36.2.3	$\displaystyle\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$	$\displaystyle=s^{3}+t^{3}+zst+yt+xs,$
$\mathbf{x}=\{x,y,z\}$ ,
ⓘ Defines: $\Phi^{(\mathrm{H})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic catastrophe Symbols: $y$ : real parameter, $z$ : real parameter, $t$ : variable, $s$ : variable and $x$ : real parameter Referenced by: §36.10(iii), §36.2(i), §36.8 Permalink: http://dlmf.nist.gov/36.2.E3 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36
(hyperbolic umbilic).

Canonical Integrals

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Keywords:: canonical integrals, cusp canonical integral, elliptic umbilic canonical integral, fold canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral
See also:: Annotations for §36.2(i), §36.2 and Ch.36

36.2.4		$\Psi_{K}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\exp\left(i\Phi_{K}% \left(t;\mathbf{x}\right)\right)\,\mathrm{d}t.$
ⓘ Defines: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $t$ : variable and $K$ : codimension Referenced by: §36.12(i), §36.2(i), §36.7(ii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E4 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

36.2.5		$\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)=\int_{-\infty}^{\infty}\int_{-% \infty}^{\infty}\exp\left(i\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)% \right)\,\mathrm{d}s\,\mathrm{d}t,$
$\mathrm{U}=\mathrm{E},\mathrm{H}$ .
ⓘ Defines: $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function and $\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)$ : umbilic canonical integral function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ , $t$ : variable and $s$ : variable Referenced by: §36.10(iii), §36.10(iv), §36.2(i), §36.7(iii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E5 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

36.2.6		$\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\sqrt{\ifrac{\pi}{3}}\,\exp\left(i% \left(\tfrac{4}{27}z^{3}+\tfrac{1}{3}xz-\tfrac{1}{4}\pi\right)\right)\int_{% \infty\exp\left(-7\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp\left(% i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2}}{12u^{2}}\right)\right)\,% \mathrm{d}u,$
ⓘ Symbols: $\pi$ : 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, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.2(i), §36.2(ii), §36.5(iii) Permalink: http://dlmf.nist.gov/36.2.E6 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

with the contour passing to the lower right of $u=0$ .

36.2.7	$\displaystyle\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$	$\displaystyle=\dfrac{4\pi}{3^{1/3}}\exp\left(i\left(\tfrac{2}{27}z^{3}-\tfrac{% 1}{3}xz\right)\right)\left(\exp\left(-i\dfrac{\pi}{6}\right)\mathrm{F}_{+}(% \mathbf{x})+\exp\left(i\dfrac{\pi}{6}\right)\mathrm{F}_{-}(\mathbf{x})\right),$
36.2.7	$\displaystyle\mathrm{F}_{\pm}(\mathbf{x})$	$\displaystyle=\int_{0}^{\infty}\cos\left(ry\exp\left(\pm i\dfrac{\pi}{6}\right% )\right)\exp\left(2ir^{2}z\exp\left(\pm i\dfrac{\pi}{3}\right)\right)% \operatorname{Ai}\left(3^{2/3}r^{2}+3^{-1/3}\exp\left(\mp i\dfrac{\pi}{3}% \right)\left(\tfrac{1}{3}z^{2}-x\right)\right)\,\mathrm{d}r.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.2(i) Permalink: http://dlmf.nist.gov/36.2.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

36.2.8		$\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=4\sqrt{\ifrac{\pi}{6}}\,\exp\left(i% \left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac{1}{4}\pi\right)\right)\*% \int_{\infty\exp\left(5\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp% \left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+x+y\right)u^{2}-\frac{(y-x% )^{2}}{24u^{2}}\right)\right)\,\mathrm{d}u,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Referenced by: §36.2(i), §36.2(ii), §36.5(iii) Permalink: http://dlmf.nist.gov/36.2.E8 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

with the contour passing to the upper right of $u=0$ .

36.2.9		$\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=\frac{2\pi}{3^{1/3}}\int_{\infty% \exp\left(5\pi i/6\right)}^{\infty\exp\left(\pi i/6\right)}\exp\left(i(s^{3}+% xs)\right)\operatorname{Ai}\left(\frac{zs+y}{3^{1/3}}\right)\,\mathrm{d}s.$
ⓘ 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 $x$ , $\exp\NVar{z}$ : exponential function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $y$ : real parameter, $z$ : real parameter, $s$ : variable and $x$ : real parameter Referenced by: §36.2(i), §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E9 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

Diffraction Catastrophes

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Keywords:: diffraction catastrophes
See also:: Annotations for §36.2(i), §36.2 and Ch.36

36.2.10		$\Psi_{K}(\mathbf{x};k)=\sqrt{k}\int_{-\infty}^{\infty}\exp\left(ik\Phi_{K}% \left(t;\mathbf{x}\right)\right)\,\mathrm{d}t,$
$k>0$ .
ⓘ Defines: $\Psi_{\NVar{K}}(\NVar{\mathbf{x}};k)$ : diffraction catastrophe Symbols: $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $k$ : variable, $t$ : variable and $K$ : codimension Referenced by: §36.12(i) Permalink: http://dlmf.nist.gov/36.2.E10 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

36.2.11		$\Psi^{(\mathrm{U})}(\mathbf{x};k)=k\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\exp\left(ik\Phi^{(\mathrm{U})}\left(s,t;\mathbf{x}\right)\right)\,% \mathrm{d}s\,\mathrm{d}t,$
$\mathrm{U=E,H}$ ; $k>0$ .
ⓘ Defines: $\Psi^{(\mathrm{E})}(\NVar{\mathbf{x}};\NVar{k})$ : elliptic umbilic canonical integral function, $\Psi^{(\mathrm{H})}(\NVar{\mathbf{x}};\NVar{k})$ : hyperbolic umbilic canonical integral function and $\Psi^{(\mathrm{U})}(\NVar{\mathbf{x}};\NVar{k})$ : umbilic canonical integral function Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$ : elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ , $k$ : variable, $t$ : variable and $s$ : variable Referenced by: §36.2(i) Permalink: http://dlmf.nist.gov/36.2.E11 Encodings: TeX, pMML, png See also: Annotations for §36.2(i), §36.2(i), §36.2 and Ch.36

For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975).

§36.2(ii) Special Cases

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Keywords:: Airy function, Airy functions, Pearcey integral, canonical integrals, definition, fold canonical integral, relation to Airy function, relation to umbilics, relations to other functions, special cases
Notes:: For (36.2.12) and (36.2.13) use (4.10.11) and (9.5.4), respectively. For (36.2.15) and (36.2.17) use (5.9.1). For (36.2.18) combine (36.2.6), (36.2.8), and (5.9.1) For (36.2.19) use (12.5.1) and (12.14.13). For (36.2.20) see Trinkaus and Drepper (1977). For (36.2.21) use (36.2.9).
Referenced by:: §36.2(iv)
Permalink:: http://dlmf.nist.gov/36.2.ii
See also:: Annotations for §36.2 and Ch.36

36.2.12		$\Psi_{0}=\sqrt{\pi}\exp\left(i\frac{\pi}{4}\right).$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function and $\mathrm{i}$ : imaginary unit Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E12 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

$\Psi_{1}$ is related to the Airy function (§9.2):

36.2.13		$\Psi_{1}\left(x\right)=\frac{2\pi}{3^{1/3}}\operatorname{Ai}\left(\frac{x}{3^{% 1/3}}\right).$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real parameter Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E13 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

$\Psi_{2}$ is the Pearcey integral (Pearcey (1946)):

36.2.14		$\Psi_{2}\left(\mathbf{x}\right)=P(x_{2},x_{1})=\int_{-\infty}^{\infty}\exp% \left(\mathrm{i}(t^{4}+x_{2}t^{2}+x_{1}t)\right)\,\mathrm{d}t.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $t$ : variable and $x_{i}$ : real parameter Referenced by: §36.12(i) Permalink: http://dlmf.nist.gov/36.2.E14 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

(Other notations also appear in the literature.)

36.2.15		$\Psi_{K}\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\Gamma\left(\frac{1}{K+2}% \right)\*\begin{cases}\exp\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\ \cos\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit and $K$ : codimension Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E15 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

36.2.16	$\displaystyle\Psi_{1}\left(\boldsymbol{{0}}\right)$	$\displaystyle=1.54669,$
	$\displaystyle\Psi_{2}\left(\boldsymbol{{0}}\right)$	$\displaystyle=1.67481+\mathrm{i}\,0.69373$
	$\displaystyle\Psi_{3}\left(\boldsymbol{{0}}\right)$	$\displaystyle=1.74646,$
	$\displaystyle\Psi_{4}\left(\boldsymbol{{0}}\right)$	$\displaystyle=1.79222+\mathrm{i}\,0.48022.$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function and $\mathrm{i}$ : imaginary unit Permalink: http://dlmf.nist.gov/36.2.E16 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

36.2.17		$\displaystyle\frac{{\partial}^{p}}{{\partial x_{1}}^{p}}\Psi_{K}\left(% \boldsymbol{{0}}\right)$	$\displaystyle=\frac{2}{K+2}\Gamma\left(\frac{p+1}{K+2}\right)\cos\left(\frac{% \pi}{2}\left(\frac{p+1}{K+2}+p\right)\right),$
	$K$ odd,
		$\displaystyle\frac{{\partial}^{2q+1}}{{\partial x_{1}}^{2q+1}}\Psi_{K}\left(% \boldsymbol{{0}}\right)$	$\displaystyle=0,$
	$K$ even,
		$\displaystyle\frac{{\partial}^{2q}}{{\partial x_{1}}^{2q}}\Psi_{K}\left(% \boldsymbol{{0}}\right)$	$\displaystyle=\frac{2}{K+2}\Gamma\left(\frac{2q+1}{K+2}\right)\exp\left(i\frac% {\pi}{2}\left(\frac{2q+1}{K+2}+2q\right)\right),$
	$K$ even.
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $K$ : codimension and $x_{i}$ : real parameter Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

36.2.18	$\displaystyle\Psi^{(\mathrm{E})}\left(\boldsymbol{{0}}\right)$	$\displaystyle=\tfrac{1}{3}\sqrt{\pi}\Gamma\left(\tfrac{1}{6}\right)=3.28868,$
36.2.18	$\displaystyle\Psi^{(\mathrm{H})}\left(\boldsymbol{{0}}\right)$	$\displaystyle=\tfrac{1}{3}{\Gamma}^{2}\left(\tfrac{1}{3}\right)=2.39224.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function and $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function Referenced by: §36.2(ii), Erratum (V1.0.20) for Equation (36.2.18), Subsections §§36.12(i), 36.15(i), 36.15(ii) Permalink: http://dlmf.nist.gov/36.2.E18 Encodings: TeX, TeX, pMML, pMML, png, png Clarification (effective with 1.0.20): In the second equation for $\Psi^{(\mathrm{H})}$ , the argument, originally given as $0$ , has been clarified to read $\boldsymbol{{0}}$ . See also: Annotations for §36.2(ii), §36.2 and Ch.36

36.2.19		$\Psi_{2}\left(0,y\right)=\frac{\pi}{2}\sqrt{\frac{\|y\|}{2}}\exp\left(-i\frac{y^% {2}}{8}\right)\left(\exp\left(i\frac{\pi}{8}\right)J_{-\ifrac{1}{4}}\left(% \frac{y^{2}}{8}\right)-\operatorname{sign}\left(y\right)\exp\left(-i\frac{\pi}% {8}\right)J_{\ifrac{1}{4}}\left(\frac{y^{2}}{8}\right)\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\operatorname{sign}\NVar{x}$ : sign of and $y$ : real parameter Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E19 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

For the Bessel function $J$ see §10.2(ii).

36.2.20		$\Psi^{(\mathrm{E})}\left(x,y,0\right)=2\pi^{2}(\tfrac{2}{3})^{2/3}\Re\left(% \operatorname{Ai}\left(\frac{x+iy}{12^{1/3}}\right)\operatorname{Bi}\left(% \frac{x-iy}{12^{1/3}}\right)\right),$
ⓘ 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, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $y$ : real parameter and $x$ : real parameter Referenced by: §36.2(ii), §36.7(iii), §36.8 Permalink: http://dlmf.nist.gov/36.2.E20 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

36.2.21		$\Psi^{(\mathrm{H})}\left(x,y,0\right)=\frac{4\pi^{2}}{3^{2/3}}\operatorname{Ai% }\left(\frac{x}{3^{1/3}}\right)\operatorname{Ai}\left(\frac{y}{3^{1/3}}\right).$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $y$ : real parameter and $x$ : real parameter Referenced by: §36.2(ii) Permalink: http://dlmf.nist.gov/36.2.E21 Encodings: TeX, pMML, png See also: Annotations for §36.2(ii), §36.2 and Ch.36

Addendum: For further special cases see §36.2(iv)

§36.2(iii) Symmetries

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Keywords:: canonical integrals, of canonical integrals, symmetries
Notes:: These results follow from the definitions given in §36.2(i).
Referenced by:: Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/36.2.iii
See also:: Annotations for §36.2 and Ch.36

36.2.22		$\Psi_{2K}\left(\mathbf{x}^{\prime}\right)=\Psi_{2K}\left(\mathbf{x}\right),$
$x_{2m+1}^{\prime}=-x_{2m+1}$ , $x_{2m}^{\prime}=x_{2m}$ .
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $n$ : integer, $K$ : codimension and $x_{i}$ : real parameter Permalink: http://dlmf.nist.gov/36.2.E22 Encodings: TeX, pMML, png See also: Annotations for §36.2(iii), §36.2 and Ch.36

36.2.23		$\Psi_{2K+1}\left(\mathbf{x}^{\prime}\right)=\overline{\Psi_{2K+1}\left(\mathbf% {x}\right)},$
$x_{2m+1}^{\prime}=x_{2m+1}$ , $x_{2m}^{\prime}=-x_{2m}$ .
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\overline{\NVar{z}}$ : complex conjugate, $n$ : integer, $K$ : codimension and $x_{i}$ : real parameter Permalink: http://dlmf.nist.gov/36.2.E23 Encodings: TeX, pMML, png See also: Annotations for §36.2(iii), §36.2 and Ch.36

36.2.24		$\Psi^{(\mathrm{U})}\left(x,y,z\right)=\overline{\Psi^{(\mathrm{U})}\left(x,y,-% z\right)},$
$\mathrm{U=E,H}$ .
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)$ : umbilic canonical integral function, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.2.E24 Encodings: TeX, pMML, png See also: Annotations for §36.2(iii), §36.2 and Ch.36

36.2.25		$\Psi^{(\mathrm{E})}\left(x,-y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right).$
ⓘ Symbols: $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.2.E25 Encodings: TeX, pMML, png See also: Annotations for §36.2(iii), §36.2 and Ch.36

36.2.26		$\Psi^{(\mathrm{E})}\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3}}{2}y,\pm\tfrac{\sqrt% {3}}{2}x-\tfrac{1}{2}y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right),$
ⓘ Symbols: $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.2.E26 Encodings: TeX, pMML, png See also: Annotations for §36.2(iii), §36.2 and Ch.36

(rotation by $\pm\tfrac{2}{3}\pi$ in $x, y$ plane).

36.2.27		$\Psi^{(\mathrm{H})}\left(x,y,z\right)=\Psi^{(\mathrm{H})}\left(y,x,z\right).$
ⓘ Symbols: $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $y$ : real parameter, $z$ : real parameter and $x$ : real parameter Permalink: http://dlmf.nist.gov/36.2.E27 Encodings: TeX, pMML, png See also: Annotations for §36.2(iii), §36.2 and Ch.36

§36.2(iv) Addendum to 36.2(ii) Special Cases

ⓘ

Keywords:: canonical integrals, special cases
Notes:: See Berry and Howls (2010). (The material in this subsection was added in Version 1.0.5. It will be incorporated in the next print edition.)
Referenced by:: §36.2(ii), Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/36.2.iv
See also:: Annotations for §36.2 and Ch.36

36.2.28		$\Psi^{(\mathrm{E})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{E})}\left(0,0,-% z\right)}\\ =2\pi\sqrt{\frac{\pi z}{27}}\exp\left(\frac{2}{27}iz^{3}\right)\*\left(J_{-1/6% }\left(\frac{2}{27}z^{3}\right)+iJ_{1/6}\left(\frac{2}{27}z^{3}\right)\right),$
$z\geq 0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit and $z$ : real parameter Referenced by: Erratum (V1.0.5) for Chapters 8, 20, 36 Permalink: http://dlmf.nist.gov/36.2.E28 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation has been added. For the proof see Berry and Howls (2010). See also: Annotations for §36.2(iv), §36.2 and Ch.36

36.2.29		$\Psi^{(\mathrm{H})}\left(0,0,z\right)=\overline{\Psi^{(\mathrm{H})}\left(0,0,-% z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp\left(\frac{1}{27}iz^{3}\right)\Psi^{(% \mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$
$-\infty<z<\infty$ .
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit and $z$ : real parameter Referenced by: Erratum (V1.0.5) for Chapters 8, 20, 36 Permalink: http://dlmf.nist.gov/36.2.E29 Encodings: TeX, pMML, png Addition (effective with 1.0.5): This equation has been added. For the proof see Berry and Howls (2010). See also: Annotations for §36.2(iv), §36.2 and Ch.36

36.1 Special Notation 36.3 Visualizations of Canonical Integrals

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