SpinWeightedSpheroidalHarmonics`

SpinWeightedSpheroidalHarmonicSFunction

SpinWeightedSpheroidalHarmonicSFunction[…]

represents a solution to the spin-weighted spheroidal equation.

Details and Options

A SpinWeightedSpheroidalHarmonicSFunction is created when using SpinSpinWeightedSpheroidalHarmonicS without specifying values for θ and ϕ.
A SpinSpinWeightedSpheroidalHarmonicSFunction may be treated as a numeric function of two arguments, θ and ϕ.
The parameters s, l, m and γ, the method used, and pre-computed expansion coefficients are stored inside the SpinWeightedSpheroidalHarmonicSFunction.
Numerical evaluation for specific θ and ϕ is done by summing the pre-computed expansion coefficients times the basis functions (the particular basis depends on the Method).

Examples

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Basic Examples (5)

Generate a SpinWeightedSpheroidalHarmonicSFunction for a specific s, l, m and γ:

Evaluate numerically:

Plot over a range of values for θ:

Plot over a subset of the complexes:

Extract expansion coefficients:

Scope (6)

Numerical Evaluation (3)

Evaluate numerically:

The precision of the output tracks the precision of the spheroidicity used when creating the SpinWeightedSpheroidalHarmonicSFunction:

When evaluating for multiple values of θ and ϕ it is significantly more efficient to generate a SpinWeightedSpheroidalHarmonicSFunction once, and then evaluate that for the desired values of θ and ϕ:

Differentiation (2)

First derivative with respect to θ:

First derivative with respect to ϕ:

Higher derivatives:

Expansion Coefficients (1)

Extract expansion coefficients for expansion in spin-weighed spherical harmonics

Extract expansion coefficients for Leaver's expansion in powers of cos²:

Tech Notes

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SpinWeightedSpheroidalHarmonics