treams.special.vsh_X¶
- treams.special.vsh_X(l, m, theta, phi) = <ufunc 'vsh_X'>¶
Vector spherical harmonic X in spherical coordinates
The vector spherical harmonics can be defined via the (scalar) spherical harmonics (
treams.special.sph_harm())\[\boldsymbol X_{lm}(\theta, \varphi) = \frac{\nabla \times \boldsymbol r}{\sqrt{l (l + 1)}} Y_{lm}(\theta, \varphi)\]which can be expressed as
\[\boldsymbol X_{lm}(\theta, \varphi) = \mathrm i \sqrt{\frac{2 l + 1}{4 \pi l (l + 1)} \frac{(l - m)!}{(l + m)!}} \left(\mathrm i \pi_{lm}(\theta, \varphi) \boldsymbol{\hat\theta} - \tau_{lm}(\theta, \varphi) \boldsymbol{\hat\varphi}\right)\]with the angular functions
treams.special.pi_fun()andtreams.special.tau_fun().- Parameters:
l (int, array_like) – Degree \(l \geq 0\)
m (int, array_like) – Order \(|m| \leq l\)
theta (float or complex, array_like) – Polar angle
phi (float, array_like) – Azimuthal angle
- Returns:
complex, 3-array