# §36.2 Catastrophes and Canonical Integrals

## §36.2(i) Definitions

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

 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 $K=3$

 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

 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, $\exp\NVar{z}$: 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, $\exp\NVar{z}$: 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 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\sqrt{\ifrac{\pi}{3}}\,\exp\left(i% \left(\tfrac{4}{27}z^{3}+\tfrac{1}{3}xz-\tfrac{1}{4}\pi\right)\right)\int_{% \infty\exp\left(-7\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp\left(% i\left(u^{6}+2zu^{4}+(z^{2}+x)u^{2}+\frac{y^{2}}{12u^{2}}\right)\right)\mathrm% {d}u,$

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

 36.2.7 $\displaystyle\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$ $\displaystyle=\dfrac{4\pi}{3^{1/3}}\exp\left(i\left(\tfrac{2}{27}z^{3}-\tfrac{% 1}{3}xz\right)\right)\left(\exp\left(-i\dfrac{\pi}{6}\right)\mathrm{F}_{+}(% \mathbf{x})+\exp\left(i\dfrac{\pi}{6}\right)\mathrm{F}_{-}(\mathbf{x})\right),$ $\displaystyle\mathrm{F}_{\pm}(\mathbf{x})$ $\displaystyle=\int_{0}^{\infty}\cos\left(ry\exp\left(\pm i\dfrac{\pi}{6}\right% )\right)\exp\left(2ir^{2}z\exp\left(\pm i\dfrac{\pi}{3}\right)\right)\mathrm{% Ai}\left(3^{2/3}r^{2}+3^{-1/3}\exp\left(\mp i\dfrac{\pi}{3}\right)\left(\tfrac% {1}{3}z^{2}-x\right)\right)\mathrm{d}r.$
 36.2.8 $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=4\sqrt{\ifrac{\pi}{6}}\,\exp\left(i% \left(\tfrac{1}{27}z^{3}+\tfrac{1}{6}z(y+x)+\tfrac{1}{4}\pi\right)\right)\*% \int_{\infty\exp\left(5\pi i/12\right)}^{\infty\exp\left(\pi i/12\right)}\exp% \left(i\left(2u^{6}+2zu^{4}+\left(\tfrac{1}{2}z^{2}+x+y\right)u^{2}-\frac{(y-x% )^{2}}{24u^{2}}\right)\right)\mathrm{d}u,$

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

 36.2.9 $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)=\frac{2\pi}{3^{1/3}}\int_{\infty% \exp\left(5\pi i/6\right)}^{\infty\exp\left(\pi i/6\right)}\exp\left(i(s^{3}+% xs)\right)\mathrm{Ai}\left(\frac{zs+y}{3^{1/3}}\right)\mathrm{d}s.$

### Diffraction Catastrophes

 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, $\exp\NVar{z}$: 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, $\exp\NVar{z}$: 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

 36.2.12 $\Psi_{0}=\sqrt{\pi}\exp\left(i\frac{\pi}{4}\right).$

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

 36.2.13 $\Psi_{1}\left(x\right)=\frac{2\pi}{3^{1/3}}\mathrm{Ai}\left(\frac{x}{3^{1/3}}% \right).$

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

 36.2.14 $\Psi_{2}\left(\mathbf{x}\right)=P(x_{2},x_{1})=\int_{-\infty}^{\infty}\exp% \left(\mathrm{i}(t^{4}+x_{2}t^{2}+x_{1}t)\right)\mathrm{d}t.$

(Other notations also appear in the literature.)

 36.2.15 $\Psi_{K}\left(\boldsymbol{{0}}\right)=\frac{2}{K+2}\Gamma\left(\frac{1}{K+2}% \right)\*\begin{cases}\exp\left(i\dfrac{\pi}{2(K+2)}\right),&K\text{ even,}\\ \cos\left(\dfrac{\pi}{2(K+2)}\right),&K\text{ odd}.\end{cases}$
 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.
 36.2.18 $\displaystyle\Psi^{(\mathrm{E})}\left(\boldsymbol{{0}}\right)$ $\displaystyle=\tfrac{1}{3}\sqrt{\pi}\Gamma\left(\tfrac{1}{6}\right)=3.28868,$ $\displaystyle\Psi^{(\mathrm{H})}\left(\boldsymbol{{0}}\right)$ $\displaystyle=\tfrac{1}{3}{\Gamma}^{2}\left(\tfrac{1}{3}\right)=2.39224.$ ⓘ 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 $\Psi_{2}\left(0,y\right)=\frac{\pi}{2}\sqrt{\frac{|y|}{2}}\exp\left(-i\frac{y^% {2}}{8}\right)\left(\exp\left(i\frac{\pi}{8}\right)J_{-\ifrac{1}{4}}\left(% \frac{y^{2}}{8}\right)-\operatorname{sign}\left(y\right)\exp\left(-i\frac{\pi}% {8}\right)J_{\ifrac{1}{4}}\left(\frac{y^{2}}{8}\right)\right).$

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

 36.2.20 $\Psi^{(\mathrm{E})}\left(x,y,0\right)=2\pi^{2}(\tfrac{2}{3})^{2/3}\Re\left(% \mathrm{Ai}\left(\frac{x+iy}{12^{1/3}}\right)\mathrm{Bi}\left(\frac{x-iy}{12^{% 1/3}}\right)\right),$
 36.2.21 $\Psi^{(\mathrm{H})}\left(x,y,0\right)=\frac{4\pi^{2}}{3^{2/3}}\mathrm{Ai}\left% (\frac{x}{3^{1/3}}\right)\mathrm{Ai}\left(\frac{y}{3^{1/3}}\right).$

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

## §36.2(iii) Symmetries

 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 $\Psi_{2K+1}\left(\mathbf{x}^{\prime}\right)=\overline{\Psi_{2K+1}\left(\mathbf% {x}\right)},$ $x_{2m+1}^{\prime}=x_{2m+1}$, $x_{2m}^{\prime}=-x_{2m}$.
 36.2.24 $\Psi^{(\mathrm{U})}\left(x,y,z\right)=\overline{\Psi^{(\mathrm{U})}\left(x,y,-% z\right)},$ $\mathrm{U=E,H}$.
 36.2.25 $\Psi^{(\mathrm{E})}\left(x,-y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right).$
 36.2.26 $\Psi^{(\mathrm{E})}\left(-\tfrac{1}{2}x\mp\tfrac{\sqrt{3}}{2}y,\pm\tfrac{\sqrt% {3}}{2}x-\tfrac{1}{2}y,z\right)=\Psi^{(\mathrm{E})}\left(x,y,z\right),$

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

 36.2.27 $\Psi^{(\mathrm{H})}\left(x,y,z\right)=\Psi^{(\mathrm{H})}\left(y,x,z\right).$

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

 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. ⓘ 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