treams.special.vsh_Y

treams.special.vsh_Y(l, m, theta, phi) = <ufunc 'vsh_Y'>

Vector spherical harmonic Y in spherical coordinates

The vector spherical harmonics can be defined by \(\boldsymbol Y_{lm}(\theta, \varphi) = \boldsymbol{\hat r} \times \boldsymbol X_{lm}(\theta, \varphi)\) using treams.special.vsh_X(). Alternatively, it can can be expressed as

\[\boldsymbol Y_{lm}(\theta, \varphi) = \mathrm i \sqrt{\frac{2 l + 1}{4 \pi l (l + 1)} \frac{(l - m)!}{(l + m)!}} \left(\tau_l^m (\cos\theta) \boldsymbol{\hat\theta} + \mathrm i \pi_l^m (\cos\theta) \boldsymbol{\hat\varphi}\right) \mathrm e^{\mathrm i m \varphi}\]

with the angular functions treams.special.pi_fun() and treams.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