SpinWeightedSpheroidalHarmonics`
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, sλlm, 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 sλlm 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 AutomaticPossible 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 :