swallowtail bifurcation set

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2: 36.5 Stokes Sets

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►

See accompanying text — Figure 36.5.7: Sheets of the Stokes surface for the swallowtail catastrophe (colored and with mesh) and the bifurcation set (gray). … Magnify

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3: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

… ►and far from the bifurcation set, the cuspoid canonical integrals are approximated by … ►

36.11.4

\Psi_{3}\left(x,0,0\right)=\frac{\sqrt{2\pi}}{(5|x|^{3})^{1/8}}\begin{cases}% \exp\left(-2\sqrt{2}(\ifrac{x}{5})^{5/4}\right)\left(\cos\left(2\sqrt{2}(% \ifrac{x}{5})^{5/4}-\tfrac{1}{8}\pi\right)+o\left(1\right)\right),&x\to+\infty% ,\\ \cos\left(4(\ifrac{|x|}{5})^{5/4}-\tfrac{1}{4}\pi\right)+o\left(1\right),&x\to% -\infty.\end{cases}

ⓘ

Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : 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, $o\left(\NVar{x}\right)$ : order less than and $x$ : real parameter
Referenced by:: §36.11
Permalink:: http://dlmf.nist.gov/36.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

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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

ⓘ

Symbols:: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $y$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

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36.11.6

\Psi_{3}\left(0,0,z\right)=\frac{\Gamma\left(\tfrac{1}{3}\right)}{|z|^{1/3}% \sqrt{3}}+\begin{cases}o\left(1\right),&z\to+\infty,\\ \dfrac{2\sqrt{\pi}5^{1/4}}{(3|z|)^{3/4}}\left(\cos\left(\dfrac{2}{3}\left(% \dfrac{3|z|}{5}\right)^{5/2}-\dfrac{1}{4}\pi\right)+o\left(1\right)\right),&z% \to-\infty.\end{cases}

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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, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter
Permalink:: http://dlmf.nist.gov/36.11.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

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4: 36.7 Zeros

… ►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 …The rings are almost circular (radii close to

(\Delta x)/9

and varying by less than 1%), and almost flat (deviating from the planes

z_{n}

by at most

(\Delta z)/36

). …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. … ►

§36.7(iv) Swallowtail and Hyperbolic Umbilic Canonical Integrals

►The zeros of these functions are curves in

\mathbf{x}=(x,y,z)

space; see Nye (2007) for

\Phi_{3}

and Nye (2006) for

\Phi^{(\mathrm{H})}