# elliptic umbilic bifurcation set

(0.002 seconds)

## 4 matching pages

##### 1: 36.4 Bifurcation Sets
Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid …
##### 3: 36.7 Zeros
###### §36.7(iii) EllipticUmbilic Canonical Integral
The zeros are lines in $\mathbf{x}=(x,y,z)$ space where $\operatorname{ph}\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)$ is undetermined. Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …Near $z=z_{n}$, and for small $x$ and $y$, the modulus $|\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)|$ has the symmetry of a lattice with a rhombohedral unit cell that has a mirror plane and an inverse threefold axis whose $z$ and $x$ repeat distances are given by …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. …
36.11.5 $\Psi_{3}\left(0,y,0\right)=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i% \pi\right)\sqrt{\ifrac{\pi}{y}}\left(1-(i/{\sqrt{3}})\exp\left(\tfrac{3}{2}i(% \ifrac{2y}{5})^{5/3}\right)+o\left(1\right)\right),$ $y\to+\infty$.
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 $\Psi^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}% \exp\left(\frac{1}{27}iz^{3}\right)+o\left(1\right)\right),$ $z\to\pm\infty$.