scipy.special.

lpmv

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

Associated Legendre function of integer order and real degree.

Defined as

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

where

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

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.

Notes

Note that this implementation includes the Condon-Shortley phase.

Array API Standard Support

lpmv has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	⚠️ no JIT	⛔
Dask	✅	n/a

See Support for the array API standard for more information.

References

[1]

19. Zhang and J.M. Jin, “Computation of Special Functions”, Wiley 1996.

Examples

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Compute a single associated Legendre function value:

>>> from scipy.special import lpmv
>>> lpmv(1, 2.5, 0.3)
-0.1292299

Plot the associated Legendre functions for orders m = 1 and m = 2 and degrees v = 0, ..., 5:

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> nus = np.arange(0, 6)
>>> xs = np.linspace(-1+1e-5, 1-1e-5, 101)

Order 1.

>>> fig, ax = plt.subplots()
>>> ax.plot(xs, lpmv(1, nus, xs[:, np.newaxis]), "-")
>>> ax.legend([rf"$\nu={nu}$" for nu in nus])
>>> ax.set_xlabel(r"$x$")
>>> ax.set_ylabel(r"$P_\nu^1(x)$")
>>> plt.show()

Order 2.

>>> fig, ax = plt.subplots()
>>> ax.plot(xs, lpmv(2, nus, xs[:, np.newaxis]), "-")
>>> ax.legend([rf"$\nu={nu}$" for nu in nus])
>>> ax.set_xlabel(r"$x$")
>>> ax.set_ylabel(r"$P_\nu^2(x)$")
>>> plt.show()

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