Green's function for 3D Laplace equation or 3D Poisson Equation

Gravitational Potential, Quadrics, Cyclides, 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 Harmonics The: Separable Analytical Solutions to Separable Analytical Solutions to Laplace's Equation

The Quadrics and the Cyclides (Howard S. Cohl) 7/28/01

Those smooth

Elementary functions:

Exponential functions and Logarithmic functions:

Solutions to the differential equation and it's inverse:

Trigonometric functions (six kinds of solutions to the differential equation and their inverses)

Hyperbolic functions (six kinds of solutions to the differential equation and their inverses)

Jacobi Elliptic functions (twelve kinds of solutions to the differential equation

Dirac Delta functions

Legendre functions

Gauss Hypergeometric Legendre P Expressions:

(|1-z|<2)

(|z^2|<1)

(|1/z^2|<1)

(cosh eta>1)

Gauss Hypergeometric Legendre Q Expressions:

(|1-z|<2)

(|z^2|<1)

(cosh eta>1)

Complex arguments

Real arguments from -1 to 1: trigonometric arguments.

Real arguments from 1 to infinity: hyperbolic arguments.

Purely imaginary arguments from i*0 to i*infinity: hyperbolic arguments.

2he relations between the orders and the degrees.

Complex orders and complex degrees.

Integer orders and integer degrees.

Integer orders and odd-half-integer degrees.

Odd-half-integer orders and integer degrees.

Odd-half-integer orders and odd-half-integer degrees.

Assorted other rational orders and degrees.

Toroidal functions

- Introduction
- Rotationally Invariant Coordinate Systems which Separate Laplace's Equation
- The Rotational Cylindrical System
- The Highly Symmetric Nature of Toroidal Functions
- Three Second-Order Rotational Laplace Systems
- Two Fourth-Order Rotational Laplace Systems
- Conclusion
- Bibliography
- About this document ...

7/22/01 - The Whipple Formulae for Toroidal Functions (H.S. Cohl)

OK! This is one of the first basic entries. Let's all start thinking about this if you have any ideas, corrections, misconceptions, or comments in general, I will glad to hear what you have to say, if your even a slight bit interested in this, I would be glad to hear. Check out some of my ideas about the Whipple formulae as they are applied to toroidal or ring functions. These Whipple formulae have been presented previously in terms of a viewpoint aimed at spherical harmonics, now that we view the equations in terms of toroidal coordinates, whole new symmetries of Legendre functions arise.

Well, these interesting functions have the properties that they can relate Legendre functions of the first and second kinds directly in terms of each other. The only hitch is that you need a different argument to relate them. The way it works it as such. The Legendre functions of the first kind generally are well-behaved near the origin and blow up at positive infinity. Consequently the Legendre functions of the second kind blow up at unity and exponentially converges towards zero for large values of the argument. The relevant domain for toroidal functions is from 1 to infinity. The standard hyperbolic argument for these functions are naturally chosen to be the hyperbolic cosine since it ranges from 1 to infinity. The Whipple formulae relate the Legendre functions with argument 1 to infinity, cosh, to a reversed range given by the hyperbolic cotangent function. the hyperbolic cotangent function ranges from infinity at unity to unity at infinity. At what point alpha does cosh alpha equal coth alpha? The point alpha is given by

Spherical harmonics

Conical harmonics

Oblate spheroidal harmonics

Prolate spheroidal harmonics

Bispherical harmonics Site: H.S. Cohl 7/22/01 - The Potential of a Point : Laplace's equation

CHAPTER I: Symmetry in PDEs

CHAPTER II: Laplace's equation

CHAPTER III: Poisson's equation

CHAPTER IV: The quadrics and the cylcides

CHAPTER V: Symmetric classes

III.A : Translational class:

III.B : Rotational class:

III.C : Corporeal class:

BASIC TO COMPLEX WE REACH ALL LEVELS HOPEFULLY, ... eventually for sure. he he

INTRODUCTION: Symmetry in PDEs

Studies of the Solution of Laplace's equation and Poisson's equation.

Recent investigations pose questions regarding the form of known available analytic solutions of Laplace's equation. The technique of separation of variables has been widely employed in order to study the form and behavior of solutions to various partial differential equations. Furthermore there exists solutions to Laplace's equation which can be obtained using R-separation of varibales (R). Geometrical representations of these solutions cast them in the light of orthogonal curvilear coordinate systems. In the 9 coordinate systems whose solution to Laplace's equation can be obtained through the standard separation of variables (S) technique the family of surfaces obtained are quadrics. The "bonus" solutions which can be derived through R lead to solutions which are families of cyclides.

Recent studies of the analytic form of the infinite extent Green's function for Laplace's equation sheds new light on the reasons for the existance of R type solutions. These systems which can be obtained for coordinate systems which are axisymmetric and have known analytic solutions to Laplace's equation through separations of variables, exhibit a new symmetry in terms of the azimuthal Fourier components of their solutions.

Laplace's equation is:

The Laplacian del^2=d/dx^2 + d/dy^2 + d/dz^2 is an operator which is found all over the place whenever people try to describe physical systems with mathematics (This discipline is called Mathematical physics). It appears in eq. (1), Poisson's equation, the heat equation, the diffusion equations, Helmholtz's equation, the Schroedinger equation, plus more ... Very important operator.

Something interesting to think about. Remember the fundamental equation of Green's functions

The analytic method for Poisson's equation is called the Green's function method. It is an integral method. That is, the solution of the differential equation is given by an integral equation. This is an equation very similar to a differential equation but it contains integrals as well as derivatives. Hard to solve. But the boundary conditions are built right in. One then needs the potential boundary values on a cylindrical boundary, in order to solve Poisson's equation.

Eq. (4) contains a Dirac delta function. Dirac was the first mathematician/physicist who worked seriously with these types of functions. Actually in a strict mathematical sense, the Dirac delta function (which I will from here on refer to as Delta function) isn't a function. We can go into that more later if you are interested. Dirac used the delta function to describe the density of an electron, as a point source. It gives the right answer, which you can convince yourself of. The three-dimensional delta function is zero everywhere, except at xvec=xvec^\prime, there it is infinity, and if you integrate the three dimensional delta function over space volume*density gives you the mass (or charge) of the point source. Now, since we can approximate ANY mass (charge) distribution as a continuous collection of point masses (charges) we can integrate over all point sources (\rho) in order to obtain the Potential everywhere. This is just as good mathematically as solving Poisson's equation given the boundary conditions.

This is why the Dirac delta function appears in the derivation of Green's functions. So, we must understand how we calculate the delta function is all of our so called "interesting" coordinate systems. Jackson (1977) gives a very nice explanation of how this is done.

Elementary functions are absolutely crucial, and transcendental functions lead us through a realm of mathematical possibilities that we had not even considered before.

CHAPTER II: Laplace's equation

CHAPTER III: Poisson's equation

CHAPTER IV: The quadrics and the cylcides

CHAPTER V: Symmetric classes

III.A : Translational class:

III.B : Rotational class:

III.C : Corporeal class:

The Laplacians,Curls, Divergences, Gradients, and Elements:

The Laplacians: Here we attempt to give you the Laplacians in various coordinate systems.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

The Curls: Here we attempt to give you the Curls in various coordinate systems.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

The Divergences: Here we attempt to give you the Divergences in various coordinate systems.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

The Gradients : Here we attempt to give you the Gradient in various coordinate systems.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

The Elements: Here we attempt to give you the line, area, and volume elements in various coordinate systems.

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

If you have any ideas, corrections, misconceptions, or comments in general, I will glad to hear what you have to say. If your even a slight bit interested in this, I would be glad to hear from you. Check out some of my ideas about the Whipple formulae as they are applied to toroidal or ring functions. These Whipple formulae have been presented previously in terms of a viewpoint aimed at spherical harmonics, now that we view the equations in terms of toroidal coordinates, whole new symmetries of Legendre functions arise.

Some of the most amazing and interesting things that we can explore in our world is math. Some other things which interest us are in nature. We love the clouds, the skies, the plants, and the animals. There is a lot of beauty in the universe. When observational astronomers observe the celestial sphere, they witness an even greater diversity than we witness here on the earth. Here on the outskirts of the milky way, we are quite content but curious nonetheless.

Being an astronomer, like Newton or Kepler must have been quite exciting. I am sure they enjoyed their work with the satisfaction that comes along with discovery. Nowadays the Universe is constantly being observed and many puzzles arise and have arisen which are difficult to explain. Modern theoretical astrophysicists must work diligently in their efforts to explain the way things work. Still we must persevere, and search where no one has searched before to find answers to these questions. Remember, further generalizations usually exist. You might as ask what transcendental functions have to do with all of this?