Bispherical Coordinates

Please feel free to enjoy my contributions to these interesting fields:

Numerical solution for 3D Poisson equation in circular cylindrical coordinates : Cohl et. al. ( 1997, 1999a)
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)

1997 - Cohl, H. S., Xian-He Sun and J. E. Tohline
"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, 527, 86-101.
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, 321, 5/6, 363-372.
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, 64, 5, 52509.
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

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In bispherical coordinates --

where $\bigl\{a\sin\theta\cos\phi/(s-\tau), a\sin\theta\sin\phi/(s-\tau), a\sinh\sigma/(s-\tau)\bigr\},$ goes from 1 to $\infty$ , $s\equiv\cosh\sigma$ goes from -1 to +1, and

goes from 0 to $2\pi$ -- the surfaces $\sigma =$ constant are spheres and the surface $\theta =$ constant is a spindle-shaped cyclide. According to eq. (10.3.74) of Morse & Feshbach (1953),

(44)

Consequently, the following two expressions must be valid addition theorems:

$\begin{displaymath}\frac{1}{\vert{\bf x} - {\bf x^\prime}\vert} = \frac{1}{a}\b... ...\ell^m(\cos\theta^\prime) \ \mathrm{e}^{im(\phi-\phi^\prime)}, \end{displaymath}$

(45)

$\begin{displaymath}\sum_{\ell=\vert m\vert}^{\infty} \frac{\Gamma(\ell-m+1)}{\Ga... ...eta^\prime(s-\tau)(s^\prime-\tau^\prime)}}, \ \ \mathrm{and}, \end{displaymath}$

(46)

where

$\begin{displaymath}\sum_{\ell=\vert m\vert}^{\infty} \frac{\Gamma(\ell-m+1)}{\Ga... ...{\chi^2-1})} {F_m(\sigma,\sigma^\prime,\theta,\theta^\prime)}, \end{displaymath}$