Green's function for 3D Laplace equation : Cohl et. al. ( 1999a, 1999b, 2000, 2001, 2002)

Gravitational potential : Cohl et. al. ( 1999a, 1999b, 2000, 2001, 2002)

Quadrics & cyclides : Cohl et. al. ( 1999a, 1999b, 2000, 2001, 2002)

Heine identity : Cohl et. al. ( 1999a, 1999b, 2000, 2001, 2002)

Cylindrical, toroidal, oblate and prolate spheroidal, parabolic, bispherical coordinate systems : Cohl et. al. (1999a, 2000)

New addition theorems for rotationally invariant coordinate systems which R-separate 3D Laplace equation: Cohl et. al. (1999a, 2000)

3D biharmonic, 3D triharmonic, and 3D higher harmonic Green's functions : Cohl (2002)

Spherical coordinate system : Cohl et. al. (2001)

Coulomb direct (classical) and exchange (quantum) integrals/interactions : Cohl et. al. (2001)

Two-electron interactions : Cohl et. al. (2001)

Spherical azimuthal and separation angle Fourier expansions : Cohl et. al. (2001)

Magnetic field of an infinitesimally thin circular current loop : Cohl & Tohline (1999)

Symmetry properties of associated Legendre/toroidal functions : Cohl et. al. (2000)

Whipple formulae for toroidal/associated Legendre functions : Cohl et. al. (2000)

New addition theorem for spherical coordinates : Cohl et. al. (2001)

Solar White Light Flares : Neidig et. al. (1993)

"Parallel Implementation of a Data-Transpose Technique for the Solution of Poisson's Equation in Cylindrical Coordinates"

Proceedings of the 8th SIAM Conference on Parallel Processing for Scientific Computing, Minneapolis, Minnesota, March.

1999a - Cohl, H. S.

"On the numerical solution of the cylindrical Poisson equation for isolated self-gravitating systems"

The Louisiana State University and Agricultural and Mechanical College, 122 pages

1999b - Cohl, H. S. and J. E. Tohline

"A Compact Cylindrical Green's Function Expansion for the Solution of Potential Problems"

The Astrophysical Journal,

2000 - Cohl, H. S., J. E. Tohline, A. R. P. Rau, H. M. Srivastava

"Developments in determining the gravitational potential using toroidal functions"

Astronomische Nachrichten,

2001 - Cohl, H. S., Rau, A. R. P., Tohline, J. E., Browne, D. A., Cazes, J. E. and Barnes, E. I.

"Useful alternative to the multipole expansion of 1/r potentials"

Physical Review A: Atomic and Molecular Physics and Dynamics,

2002 - Cohl, H. S.

"Portent of Heine's Reciprocal Square Root Identity"

Proceedings of the 3D Stellar Evolution Workshop, ed. R. Cavallo, S. Keller, S. Turcotte, Livermore, California

Introduction

In an astrophysical context (cf., §2.1 in Binney & Tremaine 1987),
the Newtonian potential is determined
by solving the appropriate boundary-value problem that is given by
the three-variable Poisson equation,

(1) |

where is the Laplacian, and is the gravitational constant. Exterior to an isolated mass distribution, physically correct boundary conditions may be imposed on Poisson's equation through the following integral formulation of the potential problem,

(2) |

Due to the long-range nature of the Newtonian/Coulomb
interaction, a physically correct solution of Poisson's equation
and, hence, an accurate determination of the potential
``interior'' to the boundary can be *highly*
boundary value dependent. If inaccurate potential values are given on
the boundary, the interior solution will reflect these defects.

In numerical simulations of astrophysical systems, a standard procedure for
determining the value of the gravitational
potential along the boundary of a computational mesh has
been the multipole method. In this method,
an approximate solution to equation (2) is obtained by
summing up higher and higher order spherical multipole moments
(monopole, dipole, quadrupole, etc.) of the mass distribution.
The multipole moments themselves are normally cast
in terms of two quantum numbers -- one meridional
and one azimuthal
*m* -- and for practical reasons the series summation is truncated
at a finite value of
and *m*.
An alternative to the standard multipole expansion has been given by
Cohl & Tohline (1999) in terms of a single sum over the azimuthal quantum
number *m*. This new expression yields the pure azimuthal mode contribution
to the Newtonian potential. In terms of the standard multipole description,
this new expansion is equivalent to completing the infinite sum over
the quantum number
for each given value of the quantum number *m*
(Cohl *et al.* 2000). It is now clear to us that this new expansion can
be extended to every rotationally invariant coordinate system
which separates the three-variable Laplace equation.

In §2 we describe the rotationally invariant coordinate systems which
separate Laplace's equation and briefly describe some of the properties
of these systems related to the double summation/integration expressions
for the reciprocal distance between two points (hereafter, Green's function
for Laplace's equation, or just Green's function) in these coordinate systems.
In §3 we describe the key expressions for the Green's function in circular
cylindrical coordinates, and show how these expressions are consistent with
an azimuthal Fourier series expansion for the Green's function whose
coefficients are given in terms of toroidal functions. In §4 we
describe some of the highly symmetric properties of toroidal functions,
such as their behavior for negative degree and order. We also present
toroidal function implications of the Whipple formulae for associated Legendre
functions. Using these Whipple formulae for toroidal functions, we then
obtain a new expansion which is shown to be equivalent to the expansion
given in Cohl & Tohline (1999). In §5 we briefly summarize several
key mathematical implications of these expansions as they apply to
prolate spheroidal, oblate spheroidal, and, parabolic coordinates. In
§6 we present some key mathematical implications of these expansions
as they apply to bispherical, and toroidal coordinates. All of the coordinate
systems presented in §§5 and 6 have known double integration/summation
expansions for the Green's function and therefore can be easily related to
these new alternative expansions. A detailed treatment in spherical
coordinates is being presented elsewhere (Cohl *et al.* 2000). The
mathematical relations we derive below are either in the form of an
infinite integral or an infinite series expansion over the set
of basis functions which separate Laplace's equation.

Since toroidal functions can be used to unify all of the rotationally invariant coordinate systems which separate Laplace's equation, we suspect that they represent a basis set that is better suited for general studies of nonaxisymmetric mass/charge distributions than, for example, spherical harmonics. It is with this in mind, that the relations in this paper are presented for use in numerical and analytical solutions to any theoretical physics problem which requires accurate evaluation of the gravitational and/or Coulomb potential.

Cohl, H. S., J. E. Tohline, A. R. P. Rau, H. M. Srivastava (2000)