SpinWeightedSpheroidalHarmonics`

SpinWeightedSpheroidalEigenvalue

SpinWeightedSpheroidalEigenvalue[s,l,m,γ]

gives the spin-weighted spheroidal eigenvalue with spheroidicity γ, spin-weight s, degree l and order m.

Details and Options

Mathematical function, suitable for both symbolic and numerical manipulation.
For real γ, SpinWeightedSpheroidalEigenvalue computes the eigenvalue, _sλ_lm, for the oblate spin-weighted spheroidal equation with oblateness γ: .
For imaginary γ, SpinWeightedSpheroidalEigenvalue computes the eigenvalue for the prolate spin-weighted spheroidal equation.
SpinWeightedSpheroidalEigenvalue can be evaluated for arbitrary complex spheroidicity.
The definition for _sλ_lm is consistent with that of [Teukolsky, S. A., Astrophys. J. 185, 635–647 (1973)] and of [Sasaki, M. & Tagoshi, H. Living Rev. Relativ. (2003) 6: 6].
For γ=0, SpinWeightedSpheroidalEigenvalue reduces to (l(l+1) - s(s+1)).
For s=0, SpinWeightedSpheroidalEigenvalue[0, l, m, γ] is equivalent to SpheroidalEigenvalue[l, m, I γ] - 2 m γ.
For numerical values of γ, SpinWeightedSpheroidalEigenvalue aims to produce a numerical value with the maximum possible precision given the precision of γ.
In most cases, the precision of the result will be approximately equal to the precision of γ.
By default, numerical values are computed using an expansion based on spin-weighted spherical harmonics for an initial guess followed by Leaver's method to obtain an accurate answer.
Series expansions about γ=0 can be computed to arbitrary order for generic s, l and m.
Series expansions about γ = ∞ can be computed to arbitrary order for specific s, l and m.
For Methods that support suboptions, they may be passed as a list. For example Method  {"Leaver", "InitialGuess"  0.1} sets the "InitialGuess" suboption to have value 0.1.
The possible suboptions that the "Lever" method accepts are: "InitialGuess"  λ and "NumInversions"  n.
The possible suboptions that the "SphericalExpansion" method accepts are: "NumTerms"  n.
The following options can be given:
Method Automatic Possible choices include "Leaver" and "SphericalExpansion".

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a range of real values for γ:

Plot over a subset of the complexes:

Series expansion in the spherical limit as γ approaches 0:

Series expansion at Infinity:

Scope (9)

Numerical Evaluation (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex spheroidicity:

Evaluate efficiently at high precision:

Specific Values (1)

SpinWeightedSpheroidalEigenvalue evaluates exactly if γ=0:

Visualisation (2)

Plot the SpinWeightedSpheroidalEigenvalue function for integer orders:

Plot the real part of :

Plot the imaginary part of :