scipy.special.eval_legendre#

scipy.special.eval_legendre(n, x, out=None) = <ufunc 'eval_legendre'>#

Evaluate Legendre polynomial at a point.

The Legendre polynomials can be defined via the Gauss hypergeometric function \({}_2F_1\) as

\[P_n(x) = {}_2F_1(-n, n + 1; 1; (1 - x)/2).\]

When \(n\) is an integer the result is a polynomial of degree \(n\). See 22.5.49 in [AS] for details.

Parameters
narray_like

Degree of the polynomial. If not an integer, the result is determined via the relation to the Gauss hypergeometric function.

xarray_like

Points at which to evaluate the Legendre polynomial

outndarray, optional

Optional output array for the function values

Returns
Pscalar or ndarray

Values of the Legendre polynomial

See also

roots_legendre

roots and quadrature weights of Legendre polynomials

legendre

Legendre polynomial object

hyp2f1

Gauss hypergeometric function

numpy.polynomial.legendre.Legendre

Legendre series

References

AS

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples

>>> from scipy.special import eval_legendre

Evaluate the zero-order Legendre polynomial at x = 0

>>> eval_legendre(0, 0)
1.0

Evaluate the first-order Legendre polynomial between -1 and 1

>>> X = np.linspace(-1, 1, 5)  # Domain of Legendre polynomials
>>> eval_legendre(1, X)
array([-1. , -0.5,  0. ,  0.5,  1. ])

Evaluate Legendre polynomials of order 0 through 4 at x = 0

>>> N = range(0, 5)
>>> eval_legendre(N, 0)
array([ 1.   ,  0.   , -0.5  ,  0.   ,  0.375])

Plot Legendre polynomials of order 0 through 4

>>> X = np.linspace(-1, 1)

>>> import matplotlib.pyplot as plt
>>> for n in range(0, 5):
...     y = eval_legendre(n, X)
...     plt.plot(X, y, label=r'$P_{}(x)$'.format(n))

>>> plt.title("Legendre Polynomials")
>>> plt.xlabel("x")
>>> plt.ylabel(r'$P_n(x)$')
>>> plt.legend(loc='lower right')
>>> plt.show()
../../_images/scipy-special-eval_legendre-1.png