Spherical functions

Various fundamental systems of spaces of functions defined on the unit sphere. Equivalently we can perceive them as functions of the unit directional vector. This leads to two natural means of their visualisation.

Explanation

• Complex-valued spherical harmonic functions Y_l,m form the most commonly used orthonormal basis of L²(S₂, d²Ω). They are also eigenfunctions of L_z.
• Functions Y_l,m a Y_l,-m only differ in sign of their real or imaginary parts, so using their sums and differences we can find a real-valued ON basis.
• Each Y_l,m is some simple polynomial in the components of the directional vector. We can alternatively find spherical functions in this form, without a preferred rotation axis. This forms an overcomplete set.
• The last button generates a random superposition of various m but constant l.
• Attention: all normalization factors are left out from the functions above.

See also

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