SpinWeightedSpheroidalHarmonics`

SpinWeightedSphericalHarmonicY

SpinWeightedSphericalHarmonicY[s, l, m, θ, ϕ] gives the spin-weighted spherical harmonic of spin-weight s, degree l and order m.

Details and Options

SpinWeightedSphericalHarmonicY satisfies the spin-weighted spherical equation,
The eigenvalue, λ for the spin-weighted spherical equation is given by λ = l(l+1) - s(s+1).
SpinWeightedSphericalHarmonicY is unit-normalised on the two-sphere, .
For s=0, SpinWeightedSphericalHarmonicY is equivalent to SphericalHarmonicY.
For certain special arguments, SpinWeightedSphericalHarmonicY automatically evaluates to exact values.
SpinWeightedSphericalHarmonicY can be evaluated to arbitrary numerical precision.
SpinWeightedSphericalHarmonicY automatically threads over lists.

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Evaluate symbolically:

Evaluate numerically

Plot over a range of values for θ

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

Evaluate SpinWeightedSphericalHarmonicY symbolically for and specific s, l and m:

Evaluate SpinWeightedSphericalHarmonicY symbolically for symbolic s, l and m:

Plot _sY_lm(θ,ϕ) as a function of θ for various values of s, l and m

Plot the real part of _-2Y₂₂(x+ y,π;0.5):

Plot the imaginary part of _-2Y₂₂(x+ y,π;0.5):

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

Plot _-2Y_2m(θ,0) as a function of θ for various values of m:

Plot _-2Y_2m(θ,ϕ) as a function of θ and ϕ for all allowed values of m: