scipy.special.lpmv

scipy.special.lpmv(m, v, x, out=None) = <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

marray_like

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

varray_like

Degree (float).

xarray_like

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

outndarray, optional

Optional output array for the function results

Returns

pmvscalar or 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

1

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