gumerov_expansion_coefficients package¶
- gumerov_expansion_coefficients.RS_all(kr: Array, theta: Array, phi: Array, *, n_end: int, type: Literal['regular', 'singular']) Array[source]¶
Regular / Singular elementary solution of 3D Helmholtz equation.
\[ \begin{align}\begin{aligned}Y_n^m (\theta, \phi) &:= (-1)^m \sqrt{\frac{(2n+1)(n-\left|m\right|)!}{4 \pi (n+\left|m\right|)!}} P_n^{\left|m\right|} (\cos \theta) e^{i m \phi}\\R_n^m (kr, \theta, \phi) &:= j_n(kr) Y_n^m (\theta, \phi)\\S_n^m (kr, \theta, \phi) &:= h_n^{(1)}(kr) Y_n^m (\theta, \phi)\end{aligned}\end{align} \]Parameters¶
- krArray
k * r of shape (…,)
- thetaArray
polar angle of shape (…,)
- phiArray
azimuthal angle of shape (…,)
- n_endint
Maximum degree of spherical harmonics.
Returns¶
- Array
Regular / Singular elementary solution of 3D Helmholtz equation of shape (…, n_end**2)
- gumerov_expansion_coefficients.idx_all(n_end: int, /, xp: ArrayNamespace, dtype: Any, device: Any) tuple[Array, Array][source]¶
Get all quantum numbers (n, m) where n < n_end.
Parameters¶
- n_endint
Maximum degree of spherical harmonics.
- xpArrayNamespace
The array namespace.
- dtypeAny
The data type of the output arrays.
- deviceAny
The device of the output arrays.
Returns¶
- tuple[Array, Array]
Arrays of quantum numbers n and m of shape (n_end**2,).
Examples¶
>>> n, m = idx_all(3, xp=np, dtype=np.int32, device=None) >>> n array([0, 1, 1, 1, 2, 2, 2, 2, 2], dtype=int32) >>> m array([ 0, 0, 1, -1, 0, 1, 2, -2, -1], dtype=int32)
- gumerov_expansion_coefficients.translational_coefficients(kr: Array, theta: Array, phi: Array, *, same: bool, n_end: int) Array[source]¶
Translational coefficients (E|F)^{m’,m}_{n’,n}.
\[ \begin{align}\begin{aligned}Y_n^m (\theta, \phi) &:= (-1)^m \sqrt{\frac{(2n+1)(n-\left|m\right|)!}{4 \pi (n+\left|m\right|)!}} P_n^{\left|m\right|} (\cos \theta) e^{i m \phi}\\R_n^m (kr, \theta, \phi) &:= j_n(kr) Y_n^m (\theta, \phi)\\S_n^m (kr, \theta, \phi) &:= h_n^{(1)}(kr) Y_n^m (\theta, \phi)\end{aligned}\end{align} \]Parameters¶
- krArray
k * r of shape (…,)
- thetaArray
polar angle of shape (…,)
- phiArray
azimuthal angle of shape (…,)
- samebool
If True, return (R|R) = (S|S). If False, return (S|R).
- n_endint
Maximum degree of spherical harmonics.
Returns¶
- Array
Initial sectorial translational coefficients of shape (…, n_end**2)