Spin-weighted spheroidal harmonic module

The spin-weighted spheroidal harmonics \(S_{sjm\gamma}(\theta)\) satisfy the equation

\[ \left[\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta \frac{d}{d\theta} \right) - \left(\gamma^2\sin^2\theta+\frac{(m+s\cos\theta)^2}{\sin^2\theta} +2\gamma s\cos\theta-s-2m\gamma-\lambda_{sjm\gamma} \right)\right]S_{sjm\gamma} = 0, \]

with eigenvalues \(\lambda_{sjm\gamma}\) and spheroidicity \(\gamma\). For the Teukolsky equation, \(\gamma = a\omega\).

API

pybhpt.swsh.fac(n)

Computes the factorial of a non-negative integer n.

Parameters:

n (int) – A non-negative integer.

Returns:

The factorial of n.

Return type:

float

pybhpt.swsh.Yslm(s, l, m, th, ph=None)

Evaluate the spin-weighted spherical harmonic $Y_{s}^{lm}$ at a given angle theta.

Parameters:

• s (int) – The spin weight of the harmonic.

• l (int) – The angular number of the spherical harmonic.

• m (int) – The azimuthal number of the spherical harmonic.

• th (array_like) – The polar angle(s) at which to evaluate the spherical harmonic.

Returns:

The values of the spherical harmonic at the specified angles.

Return type:

array_like

pybhpt.swsh.Yslm_derivative(s, l, m, th, ph=None)

Evaluate the derivative of the spin-weighted spherical harmonic $Y_{s}^{lm}$ at a given angle theta.

Parameters:

• s (int) – The spin weight of the harmonic.

• l (int) – The angular number of the spherical harmonic.

• m (int) – The azimuthal number of the spherical harmonic.

• th (array_like) – The polar angle(s) at which to evaluate the spherical harmonic.

Returns:

The derivatives of the spherical harmonic at the specified angles.

Return type:

array_like

pybhpt.swsh.Yslm_derivative2(s, l, m, th, ph=None)

Evaluate the second derivative of the spin-weighted spherical harmonic $Y_{s}^{lm}$ at a given angle theta.

Parameters:

• s (int) – The spin weight of the harmonic.

• l (int) – The angular number of the spherical harmonic.

• m (int) – The azimuthal number of the spherical harmonic.

• th (array_like) – The polar angle(s) at which to evaluate the spherical harmonic.

Returns:

The second derivatives of the spherical harmonic at the specified angles.

Return type:

array_like

pybhpt.swsh.Yslm_eigenvalue(s, l, *args)

pybhpt.swsh.clebsch(l1, l2, l3, m1, m2, m3)

Compute the Clebsch-Gordon coefficient <l1,m1,l2,m2|l3,m3>.

Parameters:

• l1 (int) – The angular number of the first state.

• l2 (int) – The angular number of the second state.

• l3 (int) – The angular number of the combined state.

• m1 (int) – The azimuthal number of the first state.

• m2 (int) – The azimuthal number of the second state.

• m3 (int) – The azimuthal number of the combined state.

Returns:

The Clebsch-Gordon coefficient <l1,m1,l2,m2|l3,m3>.

Return type:

float

pybhpt.swsh.w3j(l1, l2, l3, m1, m2, m3)

Compute the Wigner 3j-symbol

l1 l2 l3 |

m1 m2 m3 |

Parameters:

• l1 (int) – The angular number of the first state.

• l2 (int) – The angular number of the second state.

• l3 (int) – The angular number of the combined state.

• m1 (int) – The azimuthal number of the first state.

• m2 (int) – The azimuthal number of the second state.

• m3 (int) – The azimuthal number of the combined state.

Returns:

The Wigner 3j-symbol $ egin{pmatrix} l1 & l2 & l3 m1 & m2 & m3 end{pmatrix} $

Return type:

float

pybhpt.swsh.k1(s, l, j, m)

pybhpt.swsh.k2(s, l, j, m)

pybhpt.swsh.k2m2(s, l, m)

pybhpt.swsh.k2m1(s, l, m)

pybhpt.swsh.k2p0(s, l, m)

pybhpt.swsh.k2p1(s, l, m)

pybhpt.swsh.k2p2(s, l, m)

pybhpt.swsh.k1m1(s, l, m)

pybhpt.swsh.k1p0(s, l, m)

pybhpt.swsh.k1p1(s, l, m)

pybhpt.swsh.akm2(s, l, m, g)

pybhpt.swsh.akm1(s, l, m, g)

pybhpt.swsh.akp0(s, l, m, g)

pybhpt.swsh.akp1(s, l, m, g)

pybhpt.swsh.akp2(s, l, m, g)

pybhpt.swsh.spectral_sparse_matrix(s, m, g, nmax)

pybhpt.swsh.swsh_eigs(s, l, m, g, nmax=None, return_eigenvectors=True)

pybhpt.swsh.swsh_coeffs(s, l, m, g, th)

pybhpt.swsh.swsh_eigenvalue(s, l, m, g, nmax=None)

Compute the eigenvalue of the spin-weighted spheroidal harmonic.

Parameters:

• s (int) – The spin weight of the harmonic.

• l (int) – The angular number of the harmonic.

• m (int) – The azimuthal number of the harmonic.

• g (float or complex) – The spheroidicity parameter.

• nmax (int, optional) – The maximum number of basis functions to use in the computation. If None, a default value is chosen.

Returns:

The eigenvalue of the spin-weighted spheroidal harmonic.

Return type:

float or complex

class pybhpt.swsh.SWSHBase(*args)

Bases: object

class pybhpt.swsh.SWSHSeriesBase(s, j, m, g)

Bases: SWSHBase

sparse_matrix(nmax)

eigs(nmax=None, **kwargs)

generate_eigenvalue()

generate_eigs()

class pybhpt.swsh.SpinWeightedSpheroidalHarmonic(s, j, m, g)

Bases: SWSHSeriesBase

A class for generating a spin-weighted spheroidal harmonic.

Parameters:

• s (int) – The spin weight of the harmonic.

• j (int) – The angular number of the harmonic.

• m (int) – The azimuthal number of the harmonic.

• g (float or complex) – The spheroidicity parameter.

Variables:

couplingcoefficients (array_like) – The coupling coefficients between the spin-weighted spheroidal harmonic and the spin-weighted spherical

property couplingcoefficients

couplingcoefficient(l)

Yslm(l, th)

Evaluate the spin-weighted spherical harmonic $Y_{s}^{lm}(theta)$ at a given angle theta.

Parameters:

• l (int) – The angular number of the spherical harmonic.

• th (array_like) – The polar angle(s) at which to evaluate the spherical harmonic.

Returns:

The values of the spherical harmonic at the specified angles.

Return type:

array_like

Yslm_derivative(l, th)

Evaluate the derivative of the spin-weighted spherical harmonic $Y_{s}^{lm}(theta)$ at a given angle theta.

Parameters:

• l (int) – The angular number of the spherical harmonic.

• th (array_like) – The polar angle(s) at which to evaluate the derivative of the spherical harmonic.

Returns:

The values of the derivative of the spherical harmonic at the specified angles.

Return type:

array_like

Yslm_derivative2(l, th)

Evaluate the second derivative of the spin-weighted spherical harmonic $Y_{s}^{lm}(theta)$ at a given angle theta.

Parameters:

• l (int) – The angular number of the spherical harmonic.

• th (array_like) – The polar angle(s) at which to evaluate the second derivative of the spherical harmonic.

Returns:

The values of the second derivative of the spherical harmonic at the specified angles.

Return type:

array_like

Sslm(*args)

Evaluate the spin-weighted spheroidal harmonic $S_{s}^{lm}(theta)$ at a given angle theta.

Parameters:

th (array_like) – The polar angle(s) at which to evaluate the spheroidal harmonic.

Returns:

The values of the spheroidal harmonic at the specified angles.

Return type:

array_like

Sslm_derivative(*args)

Evaluate the derivative of the spin-weighted spheroidal harmonic $S_{s}^{lm}(theta)$ at a given angle theta.

Parameters:

th (array_like) – The polar angle(s) at which to evaluate the spheroidal harmonic.

Returns:

The derivatives of the spheroidal harmonic at the specified angles.

Return type:

array_like

Sslm_derivative2(*args)

Evaluate the second derivative of the spin-weighted spheroidal harmonic $S_{s}^{lm}(theta)$ at a given angle theta.

Parameters:

th (array_like) – The polar angle(s) at which to evaluate the spheroidal harmonic.

Returns:

The second derivatives of the spheroidal harmonic at the specified angles.

Return type:

array_like

__call__(th, ph=None, deriv=None)

Call self as a function.

pybhpt.swsh.muCoupling(s, l)

Eigenvalue for the spin-weighted spherical harmonic lowering operator Setting s -> -s gives the negative of the eigenvalue for the raising operator

pybhpt.swsh.Asjlm(s, j, l, m)

Coupling coefficient between scalar and spin-weighted spherical harmonics

Parameters:

• s (int) – The spin weight of the harmonic.

• j (int) – The angular number of the scalar harmonic.

• l (int) – The angular number of the spin-weighted harmonic.

• m (int) – The azimuthal number of the harmonics.

Returns:

The coupling coefficient $A_{s}^{jlm}$

Return type:

float

pybhpt.swsh.spin_operator_normalization(s, ns, l)