scipy.special.legendre_p#

scipy.special.legendre_p(n, z, *, diff_n=0) = <scipy.special._multiufuncs.MultiUFunc object>[source]#

Legendre polynomial of the first kind.

Parameters:

narray_like of ints: Degree of the Legendre polynomial. Must have n >= 0.
zarray_like: Input value.
diff_nint, optional: A non-negative integer. Compute and return all derivatives up to order diff_n. Default is 0.

Returns:

pndarray or tuple of ndarray: Legendre polynomial with diff_n derivatives.

See also

References

[1]

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

Examples

Evaluate the Legendre polynomial \(P_3\) at \(z = 0.5\):

>>> import numpy as np
>>> from scipy.special import legendre_p
>>> np.allclose(legendre_p(3, 0.5), -0.4375)
True

Compute the value and first derivative with respect to z:

>>> p, dp = legendre_p(3, 0.5, diff_n=1)
>>> np.allclose([p, dp], [-0.4375, 0.375])
True