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Module « scipy.special »

Fonction genlaguerre - module scipy.special

Signature de la fonction genlaguerre

def genlaguerre(n, alpha, monic=False) 

Description

help(scipy.special.genlaguerre)

Generalized (associated) Laguerre polynomial.

Defined to be the solution of

.. math::
    x\frac{d^2}{dx^2}L_n^{(\alpha)}
      + (\alpha + 1 - x)\frac{d}{dx}L_n^{(\alpha)}
      + nL_n^{(\alpha)} = 0,

where :math:`\alpha > -1`; :math:`L_n^{(\alpha)}` is a polynomial
of degree :math:`n`.

Parameters
----------
n : int
    Degree of the polynomial.
alpha : float
    Parameter, must be greater than -1.
monic : bool, optional
    If `True`, scale the leading coefficient to be 1. Default is
    `False`.

Returns
-------
L : orthopoly1d
    Generalized Laguerre polynomial.

See Also
--------
laguerre : Laguerre polynomial.
hyp1f1 : confluent hypergeometric function

Notes
-----
For fixed :math:`\alpha`, the polynomials :math:`L_n^{(\alpha)}`
are orthogonal over :math:`[0, \infty)` with weight function
:math:`e^{-x}x^\alpha`.

The Laguerre polynomials are the special case where :math:`\alpha
= 0`.

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

Examples
--------
The generalized Laguerre polynomials are closely related to the confluent
hypergeometric function :math:`{}_1F_1`:

    .. math::
        L_n^{(\alpha)} = \binom{n + \alpha}{n} {}_1F_1(-n, \alpha +1, x)

This can be verified, for example,  for :math:`n = \alpha = 3` over the
interval :math:`[-1, 1]`:

>>> import numpy as np
>>> from scipy.special import binom
>>> from scipy.special import genlaguerre
>>> from scipy.special import hyp1f1
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> np.allclose(genlaguerre(3, 3)(x), binom(6, 3) * hyp1f1(-3, 4, x))
True

This is the plot of the generalized Laguerre polynomials
:math:`L_3^{(\alpha)}` for some values of :math:`\alpha`:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(-4.0, 12.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-5.0, 10.0)
>>> ax.set_title(r'Generalized Laguerre polynomials $L_3^{\alpha}$')
>>> for alpha in np.arange(0, 5):
...     ax.plot(x, genlaguerre(3, alpha)(x), label=rf'$L_3^{(alpha)}$')
>>> plt.legend(loc='best')
>>> plt.show()

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