Two Fourth-Order Rotational Laplace

Parabolic 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

CLICK TEXT TO READ PDF COPY OF PAPER

In parabolic coordinates -- where ${\bf x}:\bigl\{\lambda\mu\cos\phi,\lambda\mu\sin\phi,(\lambda^2-\mu^2)/2\ \bigr\},$ goes from 0 to $\infty$, $\mu$ goes from 0 to $\infty$, and $\phi$ goes from 0 to $2\pi$ -- the surfaces $\lambda$ constant and $\lambda =$ constant are both paraboloids of revolution or parabolic bowls. The Green's function can be written as follows (eq. [10.3.68] in Morse & Feshbach 1953):

Consequently, the following two expressions must be valid definite integrals:

$$\frac{1}{\vert{\bf x} - {\bf x^\prime}\vert}= \frac{2}{a}\int... ...) I_m(k\mu_<) K_m(k\mu_>) \ \mathrm{e}^{im(\phi-\phi^\prime)}.$$ (41)

and

$$\int_0^\infty dk k\ J_m(k\lambda) J_m(k\lambda^\prime) I_m(k... ...\sqrt{\lambda\lambda^\prime\mu\mu\prime}}, \ \ \ \mathrm{and,}$$ (42)

where

$$\int_0^\infty dk k\ J_m(k\lambda) J_m(k\lambda^\prime) I_m(k... ...sqrt{\chi^2-1})} {E_m(\lambda,\lambda^\prime,\mu,\mu^\prime)},$$ (43)

and, $$\chi=\frac{4\lambda^2\mu^2+4{\lambda^\prime}^2{\mu^\prime}^2 ... ...\mu^\prime}^2-\mu^2)^2} {8\lambda\lambda^\prime\mu\mu^\prime},$$