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

The Rotational Cylindrical System

(8) |

where *J*_{0} is the order zero Bessel function of the first kind.
Alternatively, this same quantity can be written in a form of the
Lipschitz-Hankel integral (§3.11 in Jackson 1975; §13.21 in Watson 1944),

(9) |

where *K*_{0} is the order zero modified Bessel function of the
second kind. According to Neumann's *addition theorem* for Bessel
functions, the order zero Bessel function of the first kind can be written
as a Fourier series expansion over products of Bessel functions of varying
order, namely (§11.1 in Watson 1944),

(10) |

where *J*_{m} is the order *m* Bessel function of the first kind.
In the same vein, the order zero modified Bessel function of the second
kind can be expanded as follows using Graf's generalization of Neumann's
*addition theorem* (§11.3 in Watson 1944):

(11) |

where *I*_{m} and *K*_{m} are the order *m* modified
Bessel functions of the first and second kind, respectively.
Substituting the two
*addition theorems* (10) and (11) into the integral expressions
(8) and (9), respectively, yields the following two double
integration/summation expressions
for the Green's function (problem 3.14 of Jackson 1975)

(12) |

and (§3.11 of Jackson 1975),

(13) |

We can further simplify both of these double summation/integration expressions through the use of known transcendental function solutions to the integrals in both (12) and (13) as given, for example, by eq. (13.22.2) in Watson (1944) and eq. (6.672.4) in Gradshteyn & Ryzhik (1994), namely,

(14) |

and,

(15) |

where
is the odd-half-integer
degree Legendre function of the second kind. Hence, we may rewrite
both eqs. (12) and (13) in the following compact azimuthal Fourier
series form:

(16) |

where

(17) |

Equation (16) is the Green's function expansion that was presented in Cohl & Tohline (1999) as an alternative to the standard multipole expansion technique.

In fact, this result can be easily derived by starting with the algebraic expression for the Green's function in cylindrical coordinates,

(18) |

and using the Heine identity (§10.2 in Bateman 1959; §74 in Heine 1881; see also §4.5.4 of Magnus, Oberhettinger, and Soni 1966),

(19) |

where and Using (19) to expand (18) immediately yields (16).

Although the use of Heine's identity leads to the main result (16) more
quickly, *addition theorems* such as (10) and (11) can be
extremely useful when casting various expressions for the Green's function
for Laplace's equation in different coordinate systems in terms of single
summation/integration expressions like (8), (9), and (16). Such expressions
encapsulate the symmetries manifested within these diverse forms. In what
follows we offer an interesting twist on the development in the theory of
*addition theorems.* We note that the essential ingredient which leads
to our new development, eq. (19), was given by Heine (1881) over a century ago
in, ``Handbuch der Kugelfunctionen,''^{1}^{*} but it appears not
to have been utilized much in practical astrophysics applications over the past
century.

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