scipy.special.eval_laguerre#

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

Evaluate Laguerre polynomial at a point.

The Laguerre polynomials can be defined via the confluent hypergeometric function \({}_1F_1\) as

\[L_n(x) = {}_1F_1(-n, 1, x).\]

See 22.5.16 and 22.5.54 in [AS] (or equivalently [DLMF1] and [DLMF2]) for details. When \(n\) is an integer the result is a polynomial of degree \(n\).

Parameters:

narray_like: Degree of the polynomial. If not an integer the result is determined via the relation to the confluent hypergeometric function.
xarray_like: Points at which to evaluate the Laguerre polynomial
outndarray, optional: Optional output array for the function values

Returns:

Lscalar or ndarray: Values of the Laguerre polynomial

See also

roots_laguerre: roots and quadrature weights of Laguerre polynomials
laguerre: Laguerre polynomial object
numpy.polynomial.laguerre.Laguerre: Laguerre series
eval_genlaguerre: evaluate generalized Laguerre polynomials

References

[AS]

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

[DLMF1]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.1#I1.ix7.p1

[DLMF2]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.5.E12

Examples

>>> import numpy as np
>>> from scipy.special import eval_laguerre
>>> import matplotlib.pyplot as plt

Evaluate the order 3 polynomial at a few values of x.

>>> eval_laguerre(3, [0, 0.25, 0.5])
array([ 1.        ,  0.34114583, -0.14583333])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(0.0, 8.0, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_laguerre(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Laguerre Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$L_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

../../_images/scipy-special-eval_laguerre-1.png