bifurcation%20set

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… ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. … ►In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). …In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. …

3: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from

\mathbf{x}=\boldsymbol{{0}}

, the uniform asymptotic approximations of §36.12 can be used. …

4: 36.14 Other Physical Applications

… ►The physical manifestations of bifurcation sets are caustics. …

5: 36.7 Zeros

… ►Inside the cusp, that is, for

x^{2}<8|y|^{3}/27

, the zeros form pairs lying in curved rows. … ►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. There are also three sets of zero lines in the plane

z=0

related by

2\pi/3

rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates

(x=r\cos\theta,\;y=r\sin\theta)

is given by …

6: 29.19 Physical Applications

… ►Brack et al. (2001) shows that Lamé functions occur at bifurcations in chaotic Hamiltonian systems. …

7: 36.11 Leading-Order Asymptotics

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

36.11.3

\Psi_{2}\left(0,y\right)=\begin{cases}\sqrt{\ifrac{\pi}{y}}\left(\exp\left(% \tfrac{1}{4}i\pi\right)+o\left(1\right)\right),&y\to+\infty,\\ \sqrt{\ifrac{\pi}{|y|}}\exp\left(-\tfrac{1}{4}i\pi\right)\left(1+i\sqrt{2}\exp% \left(-\frac{1}{4}iy^{2}\right)+o\left(1\right)\right),&y\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, $\exp\NVar{z}$ : 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.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §36.11, §36.11 and Ch.36

►

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

►

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

►

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}

ⓘ

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

…

8: Bibliography B

… ►

G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

ⓘ

External Links:: Document
Referenced by:: §33.22(iv).
Encodings:: BibTeX

… ►

A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

ⓘ

Notes:: Double-precision Fortran, maximum accuracy 10S.
External Links:: Document, Code, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: 1st item, 4th item.
Encodings:: BibTeX

… ►

W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §10.73(iv).
Encodings:: BibTeX

… ►

M. Brack, M. Mehta, and K. Tanaka (2001) Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §29.19(i).
Encodings:: BibTeX

…

9: 20 Theta Functions

Chapter 20 Theta Functions

…

10: William P. Reinhardt

… ►Reinhardt firmly believes that the Mandelbrot set is a special function, and notes with interest that the natural boundaries of analyticity of many “more normal” special functions are also fractals. … ►

20 Theta Functions

… ►In November 2015, Reinhardt was named Senior Associate Editor of the DLMF and Associate Editor for Chapters 20, 22, and 23.

bifurcation%20set

1—10 of 606 matching pages

1: 36.4 Bifurcation Sets

§36.4 Bifurcation Sets

Bifurcation (Catastrophe) Set for Cuspoids

Bifurcation (Catastrophe) Set for Umbilics

§36.4(ii) Visualizations

2: 36.5 Stokes Sets