# umbilics

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##### 1: 36.2 Catastrophes and Canonical Integrals
###### Normal Forms for Umbilic Catastrophes with Codimension $K=3$
(elliptic umbilic). …(hyperbolic umbilic).
###### Canonical Integrals
Addendum: For further special cases see §36.2(iv)
##### 2: 36.1 Special Notation
The main functions covered in this chapter are cuspoid catastrophes $\Phi_{K}\left(t;\mathbf{x}\right)$; umbilic catastrophes with codimension three $\Phi^{(\mathrm{E})}\left(s,t;\mathbf{x}\right)$, $\Phi^{(\mathrm{H})}\left(s,t;\mathbf{x}\right)$; canonical integrals $\Psi_{K}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$, $\Psi^{(\mathrm{H})}\left(\mathbf{x}\right)$; diffraction catastrophes $\Psi_{K}(\mathbf{x};k)$, $\Psi^{(\mathrm{E})}(\mathbf{x};k)$, $\Psi^{(\mathrm{H})}(\mathbf{x};k)$ generated by the catastrophes. …
##### 4: 36.10 Differential Equations
In terms of the normal forms (36.2.2) and (36.2.3), the $\Psi^{(\mathrm{U})}\left(\mathbf{x}\right)$ satisfy the following operator equations …
36.10.13 $6\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{\partial x\partial y}-2iz\frac{% \partial\Psi^{(\mathrm{E})}}{\partial y}+y\Psi^{(\mathrm{E})}=0,$
36.10.15 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial x}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial y}-x\Psi^{(\mathrm{H})}=0,$
36.10.16 $3\frac{{\partial}^{2}\Psi^{(\mathrm{H})}}{{\partial y}^{2}}+iz\frac{\partial% \Psi^{(\mathrm{H})}}{\partial x}-y\Psi^{(\mathrm{H})}=0.$
36.10.17 $i\frac{\partial\Psi^{(\mathrm{E})}}{\partial z}=\frac{{\partial}^{2}\Psi^{(% \mathrm{E})}}{{\partial x}^{2}}+\frac{{\partial}^{2}\Psi^{(\mathrm{E})}}{{% \partial y}^{2}},$
##### 5: 36.6 Scaling Relations
###### §36.6 Scaling Relations
$\Psi^{(\mathrm{U})}(\mathbf{x};k)=k^{\beta^{(\mathrm{U})}}\Psi^{(\mathrm{U})}% \left(\mathbf{y}^{(\mathrm{U})}(k)\right),$
###### Indices for $k$-Scaling of Magnitude of $\Psi_{K}$ or $\Psi^{(\mathrm{U})}$ (Singularity Index)
$\text{umbilics: }\beta^{(\mathrm{U})}=\frac{1}{3}.$
$\text{umbilics: }\gamma_{x}^{(\mathrm{U})}=\tfrac{2}{3},$
##### 6: 36.4 Bifurcation Sets
###### Bifurcation (Catastrophe) Set for Umbilics
Elliptic umbilic cusp lines (ribs): … Hyperbolic umbilic bifurcation set (codimension three): … Hyperbolic umbilic cusp line (rib): …
##### 7: 36.5 Stokes Sets
Stokes sets are surfaces (codimension one) in $\mathbf{x}$ space, across which $\Psi_{K}(\mathbf{x};k)$ or $\Psi^{(\mathrm{U})}(\mathbf{x};k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of $\Phi_{K}$ or $\Phi^{(\mathrm{U})}$. …
###### Hyperbolic Umbilic Stokes Set (Codimension three) Figure 36.5.5: Elliptic umbilic catastrophe with z = constant . … Magnify
##### 8: 36.8 Convergent Series Expansions
###### §36.8 Convergent Series Expansions
36.8.3 $\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum\limits_{n=0}^{\infty}(-% 3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\mathrm{Ai}\left(x\right)\mathrm{% Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)d_% {n}(y)}{n!}+\mathrm{Ai}'\left(x\right)\mathrm{Ai}\left(y\right)\sum\limits_{n=% 2}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)c_{n}(y)}{n!}+\mathrm{Ai}'\left(x% \right)\mathrm{Ai}'\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)d_{n}(y)}{n!},$
36.8.4 $\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$
##### 9: 36.9 Integral Identities
###### §36.9 Integral Identities
36.9.8 $\left|\Psi^{(\mathrm{H})}\left(x,y,z\right)\right|^{2}=8\pi^{2}\left(\frac{2}{% 9}\right)^{1/3}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{Ai}\left(% \left(\frac{4}{3}\right)^{1/3}(x+zv+3u^{2})\right)\mathrm{Ai}\left(\left(\frac% {4}{3}\right)^{1/3}(y+zu+3v^{2})\right)\mathrm{d}u\mathrm{d}v.$
36.9.9 $\left|\Psi^{(\mathrm{E})}\left(x,y,z\right)\right|^{2}=\frac{8\pi^{2}}{3^{2/3}% }\int_{0}^{\infty}\int_{0}^{2\pi}\Re\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}% \left(x+iy+2zu\exp\left(i\theta\right)+3u^{2}\exp\left(-2i\theta\right)\right)% \right)\*\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-iy+2zu\exp\left(-i\theta% \right)+3u^{2}\exp\left(2i\theta\right)\right)\right)\right)u\mathrm{d}u% \mathrm{d}\theta.{}$
36.11.7 $\Psi^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(% \frac{4}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$,
36.11.8 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.