SpinWeightedSpheroidalHarmonics`
SpinWeightedSpheroidalHarmonics`

SpinWeightedSpheroidalHarmonicS

SpinWeightedSpheroidalHarmonicS[s,l,m,γ][θ,ϕ]

gives the spin-weighted spheroidal harmonic sSlm(θ,ϕ;γ).

SpinWeightedSpheroidalHarmonicS[s,l,m,γ]

constructs a SpinWeightedSpheroidalHarmonicSFunction which can be evaluated for specific values of θ and ϕ.

Details and Options

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For real γ, SpinWeightedSpheroidalHarmonicS computes the oblate spin-weighted spheroidal harmonic with oblateness γ.
  • SpinWeightedSpheroidalHarmonicS satisfies the spin-weighted spheroidal equation .
  • SpinWeightedSpheroidalHarmonicS is unit-normalised on the two-sphere, . This is consistent with the Meixner-Schäfke convention for the spheroidal Legendre functions in the same sense that the normalisation of SphericalHarmonicY is consistent with the Legendre functions.
  • For γ=0, SpinWeightedSpheroidalHarmonicS reduces to SpinWeightedSphericalHarmonicY.
  • For s=0 and γ=0, SpinWeightedSpheroidalHarmonicS is equivalent to SphericalHarmonicY.
  • For s=0, SpinWeightedSpheroidalHarmonicS[0, l, m, γ, θ, 0] is equivalent to SpheroidalPS[l,m, γ,Cos[θ]].
  • For certain special arguments (in particular, with γ=0), SpinWeightedSpheroidalHarmonicS automatically evaluates to exact values.
  • SpinWeightedSpheroidalHarmonicS can be evaluated to arbitrary numerical precision.
  • SpinWeightedSpheroidalHarmonicS automatically threads over lists.
  • For numerical values of γ, SpinWeightedSpheroidalHarmonicS aims to produce a numerical value with the maximum possible precision given the precision of γ.
  • Numerical values are computed using either Leaver's method or an expansion in terms of spin-weighted spherical harmonics.
  • The number of terms used for numerical evaluation is such that the normalisation is determined as accurately as possible given the precision of the input γ.
  • 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.
  • The following options can be given:
  • Method AutomaticPossible choices include "Leaver" and "SphericalExpansion".
  • For Methods that support suboptions, they may be passed as a list. For example Method {"Leaver", "NumTerms" 10} sets the "NumTerms" suboption to have value 10.
  • The possible suboptions that the "Leaver" method accepts are: "NumTerms" n.
  • The possible suboptions that the "SphericalExpansion" method accepts are: "NumTerms" n.

Examples

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Basic Examples  (6)

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:

Series expansion at γ=0:

Scope  (16)

Numerical Evaluation  (3)

Evaluate numerically:

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

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:

Specific Values  (2)

Evaluate SpinWeightedSpheroidalHarmonicS symbolically for γ=0 and specific s, l and m:

Evaluate SpinWeightedSpheroidalHarmonicS symbolically for γ=0 and symbolic s, l and m:

Visualisation  (4)

Plot sSlm(θ,ϕ;γ) as a function of θ for various values of s, l and m:

Plot the real part of -2S22(x+ y,π;0.5):

Plot the imaginary part of -2S22(x+ y,π;0.5):

Plot the absolute value of the SpinWeightedSpheroidalHarmonicS function in three dimensions:

Plot -2S2m(θ,0;1.9) as a function of θ for various values of m:

Plot -2S2m(θ,ϕ;1.9) as a function of θ and ϕ for all allowed values of m: