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Ultraspherical Harmonics

Abstract

First a recursive method is devised in order to find out the expressions for spherical coordinates and the Laplace-operator in n (any number of) dimensions. This result is then applied in solving of a Poisson's (or Helmholtz') differential equation in n dimensions. This is done by means of separation of coordinates, first separation of the radial part of the differential equation, and then recursively the angular arguments. The general separated angular equation then turns out to be independent of the dimension of the original problem, so that the whole solution can be composed of a radial equation and one angular equation, the other angular equations already being known from the lower dimensional case.

Mathematical presentation:

1. Ultraspherical Coordinates
2. Laplace Operator in Ultraspherical coordinates
3. Poisson equation, separating the arguments
4. Radial equation and solution.
5. Recursive angular equations and solutions.

Examples and Graphical illustrations:

• Orthogonal Functions (1-dimensional introduction)
• Polar Harmonics (2-dimensional case) coming soon.
• Spherical Harmonics (3-dimensional case) comig soon.
• Four-spherical Harmonics (4-dimensional case) coming soon.

Tip: take a glance at the examples and graphical illustrations first!

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