§36.2 Catastrophes and Canonical Integrals

36.2.1		$${\mathrm{\Phi }}_K(t;\mathbf{x})=t^{K+2}+\sum _{m=1}^K{x}_m{t}^m.$$
ⓘ Defines: ${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$ Symbols: $n$: integer, $t$: variable, $K$: codimension and $x_{\mathrm{i}}$: 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.

36.2.2		${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;\mathbf{x})$	$=s^3-3s{t}^2+z(s^2+t^2)+yt+xs,$
$\mathbf{x}=\{x,y,z\}$,
ⓘ Defines: ${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;\mathbf{x})$: 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		${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;\mathbf{x})$	$=s^3+t^3+zst+yt+xs,$
$\mathbf{x}=\{x,y,z\}$,
ⓘ Defines: ${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;\mathbf{x})$: 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).

36.2.4		$${\mathrm{\Psi }}_K\left(\mathbf{x}\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}{\mathrm{\Phi }}_K(t;\mathbf{x})\right)dt.$$
ⓘ Defines: ${\mathrm{\Psi }}_K\left(\mathbf{x}\right)$: canonical integral function Symbols: ${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$, $dx$: differential of $x$, $\exp z$: 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		$${\mathrm{\Psi }}^{(\mathrm{U})}\left(\mathbf{x}\right)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})\right)dsdt,$$
$\mathrm{U}=\mathrm{E},\mathrm{H}$.
ⓘ Defines: ${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$: hyperbolic umbilic canonical integral function and ${\mathrm{\Psi }}^{(\mathrm{U})}\left(\mathbf{x}\right)$: umbilic canonical integral function Symbols: $dx$: differential of $x$, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$, $t$: 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		$${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)=\begin{array}{l}2\sqrt{\pi /3}\exp \left(\mathrm{i}\left(\frac{4}{27}{z}^3+\frac{1}{3}xz-\frac{1}{4}\pi \right)\right)\\ \times {\int }_{\mathrm{\infty }\exp \left(-7\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(u^6+2z{u}^4+(z^2+x)u^2+\frac{y^2}{12u^2}\right)\right)du,\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, $\exp z$: 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		${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$	$={\displaystyle \frac{4\pi }{3^{1/3}}}\exp \left(\mathrm{i}\left(\frac{2}{27}{z}^3-\frac{1}{3}xz\right)\right)\left(\exp \left(-\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{+}(\mathbf{x})+\exp \left(\mathrm{i}{\displaystyle \frac{\pi }{6}}\right){\mathrm{F}}_{-}(\mathbf{x})\right),$
		${\mathrm{F}}_{\pm }(\mathbf{x})$	$={\displaystyle {\int }_0^{\mathrm{\infty }}}\begin{array}{l}\cos \left(ry\exp \left(\pm \mathrm{i}{\displaystyle \frac{\pi }{6}}\right)\right)\exp \left(2\mathrm{i}{r}^{2}z\exp \left(\pm \mathrm{i}{\displaystyle \frac{\pi }{3}}\right)\right)\\ \times \mathrm{Ai}\left(3^{2/3}{r}^2+3^{-1/3}\exp \left(\mp \mathrm{i}{\displaystyle \frac{\pi }{3}}\right)\left(\frac{1}{3}{z}^2-x\right)\right)dr.\end{array}$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(\mathbf{x}\right)$: elliptic umbilic canonical integral function, $\exp z$: 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		$${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)=\begin{array}{l}4\sqrt{\pi /6}\exp \left(\mathrm{i}\left(\frac{1}{27}{z}^3+\frac{1}{6}z(y+x)+\frac{1}{4}\pi \right)\right)\\ \times {\int }_{\mathrm{\infty }\exp \left(5\pi \mathrm{i}/12\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/12\right)}\exp \left(\mathrm{i}\left(2u^6+2z{u}^4+\left(\frac{1}{2}{z}^2+x+y\right)u^2-\frac{{(y-x)}^2}{24u^2}\right)\right)du,\end{array}$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$: 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		$${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)=\frac{2\pi }{3^{1/3}}{\int }_{\mathrm{\infty }\exp \left(5\pi \mathrm{i}/6\right)}^{\mathrm{\infty }\exp \left(\pi \mathrm{i}/6\right)}\exp \left(\mathrm{i}(s^3+xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}\right)ds.$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(\mathbf{x}\right)$: 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

36.2.10		$${\mathrm{\Psi }}_K(\mathbf{x};k)=\sqrt{k}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}k{\mathrm{\Phi }}_K(t;\mathbf{x})\right)dt,$$
$k>0$.
ⓘ Defines: ${\mathrm{\Psi }}_K(\mathbf{x};k)$: diffraction catastrophe Symbols: ${\mathrm{\Phi }}_K(t;\mathbf{x})$: cuspoid catastrophe of codimension $K$, $dx$: differential of $x$, $\exp z$: 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		$${\mathrm{\Psi }}^{(\mathrm{U})}(\mathbf{x};k)=k{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\exp \left(\mathrm{i}k{\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})\right)dsdt,$$
$\mathrm{U}=\mathrm{E},\mathrm{H}$; $k>0$.
ⓘ Defines: ${\mathrm{\Psi }}^{(\mathrm{E})}(\mathbf{x};k)$: elliptic umbilic canonical integral function, ${\mathrm{\Psi }}^{(\mathrm{H})}(\mathbf{x};k)$: hyperbolic umbilic canonical integral function and ${\mathrm{\Psi }}^{(\mathrm{U})}(\mathbf{x};k)$: umbilic canonical integral function Symbols: $dx$: differential of $x$, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{\Phi }}^{(\mathrm{U})}(s,t;\mathbf{x})$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$, $k$: 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).