Developments in determining the gravitational potential using toroidal functions

Abstract

Cohl & Tohline (1999) have shown how the integration/summation expression for the Green's function in cylindrical coordinates can be written as an azimuthal Fourier series expansion, with toroidal functions as expansion coefficients. In this paper, we show how this compact representation can be extended to other rotationally invariant coordinate systems which are known to admit separable solutions for Laplace's equation.

Equilibrium Sequences and Gravitational Instability of Rotating Isothermal Rings

“…f , , and ò k is equal to unity for k=1 and two otherwise (e.g., Morse & Feshbach 1953;Cohl & Tohline 1999;Cohl et al 2000). 6 The gravitational potential is then given by (3) give the semimajor axis and ellipticity of nuclear rings adopted from Comerón et al (2010).…”

Multipole Analysis of Circular Cylindircal Magnetic Systems

“…This mathematical description employs the use of Q-functions, and these functions are predominantly used for problems which exhibit toroidal sym-metry. However, they have not been extensively applied to cylindrical geometries in the engineering world [32,33], and in fact, recent literature only sparsely mentions the restricted class of toroidal functions which are used for cylindrical geometries [34,35,36]. Other formulations, specifically those due to Kildishev [37,38,39], have made contributions to this problem by employing a spheroidal harmonic analysis.…”

Electrostatic Excitation of a Conducting Toroid: Exact Solution and Thin-Wire Approximation

“…6 R. W. Scharstein and H. B. Wilson This result can be transformed to Smythe's (1968Smythe's ( , 1974 answers using Whipple's identity (Cohl et al, 2000). Charge density on the surface ξ = ξ 0 is…”

“…and Whipple's relation (Cohl et al, 2000) has been used to transform the ratio of firstkind P m −1/2 functions of varying order to a ratio of second-kind Q m−1/2 functions of varying degree and order zero. In this limiting case, the argument of the denominator function is…”