treams.special.lpmv

treams.special.lpmv(m, v, z) = <ufunc 'lpmv'>

Associated legendre polynomials of real and complex argument

For complex arguments the branch cut is at |x| > 1. Additionally, only integer orders and degrees are possible when using a complex argument. The computation is using a recursion relation in this case. For a real argument scipy.special.lpmv is called.

The function is defined as

\[P_v^m (z) = \frac{(-1)^m}{2^\nu \nu!} (1 - z^2)^\frac{m}{2} \frac{\mathrm d^{\nu + m}}{\mathrm d z^{\nu + m}} (z^2 - 1)^\nu\]

for integer degree and order.

Warning

Please note the order of the arguments. It is kept this way to stay in line with Scipy. Most of the other functions, with sph_harm as a notable exception for the same reason, use degree and order swapped around.

Parameters:

  • m (float, array_like) – Order

  • v (float, array_like) – Degree

  • z (float or complex, array_like) – Argument

Returns:

float or complex

References