Separation and Azimuthal Angle Fourier eigenmode descriptions

Green's functions for the inhomogeneous linear PDE's of mathematical physics

Compact integral representations of these PDE's

azimuthal and separation angle Fourier geometric descriptions

compute physically appropriate boundary value solutions

3D Laplace's equation

3D Helmholtz equation

3D Diffusion Equation

3D Wave Equation

3D Schroedinger Equation

3D Biharmonic Equation

Search for azimuthal eigenmodes was introduced by Laplace

5 volume Mechanique Celeste (1799-1825)

Laplace coefficients for the planetary disturbing function function in celestial mechanics

are azimuthal Fourier cosine representation of the reciprocal distance between two points

which are given by a toroidal distribution in space.

Heine (1881) gave Fourier decomposition of this function

dormant ever since...

m is a better quantum number than l, even classically

l=0 corresponds to all spherically symmetric geometries

m=0 corresponds to all axisymmetric geometries including spherical

classical all-space moment integrals given purely by the integral of the axisymmetric component

total mass

average vector and tensor quantities, angular momentum, moment of inertia

classical self-energy of a mass/charge distribution

quantum direct and exchange integrals - single azimuthal mode

Global underlying Fourier symmetries are contained
within analytic series and integral expressions for the Green's functions
obtainable within the coordinate systems which are known to R-separably solve these PDE's

Analytic survey is underway

identify the inherent geometries which describe Green's functions

in terms of an azimuthal or separation angle eigenmode description

these represent new underlying geometries which are more amenable to efficient

computations of physical quantities

algorithmic implementations

helioseismology

scattering theory

...

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