SciPy

scipy.special.lpmv

scipy.special.lpmv(m, v, x) = <ufunc 'lpmv'>

Associated Legendre function of integer order and real degree.

Defined as

\[P_v^m = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_v(x)\]

where

\[P_v = \sum_{k = 0}^\infty \frac{(-v)_k (v + 1)_k}{(k!)^2} \left(\frac{1 - x}{2}\right)^k\]

is the Legendre function of the first kind. Here \((\cdot)_k\) is the Pochhammer symbol; see poch.

Parameters:

m : array_like

Order (int or float). If passed a float not equal to an integer the function returns NaN.

v : array_like

Degree (float).

x : array_like

Argument (float). Must have |x| <= 1.

Returns:

pmv : ndarray

Value of the associated Legendre function.

See also

lpmn
Compute the associated Legendre function for all orders 0, ..., m and degrees 0, ..., n.
clpmn
Compute the associated Legendre function at complex arguments.

Notes

Note that this implementation includes the Condon-Shortley phase.

References

[R476]Zhang, Jin, “Computation of Special Functions”, John Wiley and Sons, Inc, 1996.