# ring functions

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## 7 matching pages

##### 1: 14.1 Special Notation

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##### 2: 14.19 Toroidal (or Ring) Functions

###### §14.19 Toroidal (or Ring) Functions

… ►When $\nu =n-\frac{1}{2}$, $n=0,1,2,\mathrm{\dots}$, $\mu \in \mathbb{R}$, and $x\in (1,\mathrm{\infty})$ solutions of (14.2.2) are known as*toroidal*or

*ring functions*. …

##### 3: 20.13 Physical Applications

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►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

##### 4: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

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►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena.
Among these are the formation of vortex rings in Bose Einstein condensates.
…For details see the NIST news item

*Decay of a dark soliton into vortex rings in a Bose–Einstein condensate*. … ►Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose–Einstein Condensate, Phys. Rev. Lett. 86, 2926–2929 (2001)##### 5: 28.33 Physical Applications

###### §28.33 Physical Applications

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho $ per unit area, and radial tension $\tau $ per unit arc length. … ►###### §28.33(iii) Stability and Initial-Value Problems

… ►Torres-Vega et al. (1998) for Mathieu functions in phase space.

##### 6: Bibliography B

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions.
Cambridge University Press, Cambridge.
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Automatic computation of zeros of Bessel functions and other special functions.
SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.
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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V.
Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
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Quasinormal ringing of Kerr black holes: The excitation factors.
Phys. Rev. D 74 (104020), pp. 1–27.
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Theoretical and experimental investigation of the elliptical annual ring antenna.
IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.
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##### 7: 36.7 Zeros

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►This is the Airy function
$\mathrm{Ai}$ (§9.2).
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►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 $(\Delta z)/36$).
Away from the $z$-axis and approaching the cusp lines (ribs) (36.4.11), the lattice becomes distorted and the rings are deformed, eventually joining to form “hairpins” whose arms become the pairs of zeros (36.7.1) of the cusp canonical integral.
…, $y=0$), the number of rings in the $m$th row, measured from the origin and before the transition to hairpins, is given by
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