Vous êtes un professionnel et vous avez besoin d'une formation ? Mise en oeuvre d'IHM
avec Qt et PySide6 Voir le programme détaillé

Module « scipy.special »

Fonction laguerre - module scipy.special

Signature de la fonction laguerre

def laguerre(n, monic=False) 

Description

help(scipy.special.laguerre)

Laguerre polynomial.

Defined to be the solution of

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

:math:`L_n` is a polynomial of degree :math:`n`.

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

Returns
-------
L : orthopoly1d
    Laguerre Polynomial.

See Also
--------
genlaguerre : Generalized (associated) Laguerre polynomial.

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

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 Laguerre polynomials :math:`L_n` are the special case
:math:`\alpha = 0` of the generalized Laguerre polynomials
:math:`L_n^{(\alpha)}`.
Let's verify it on the interval :math:`[-1, 1]`:

>>> import numpy as np
>>> from scipy.special import genlaguerre
>>> from scipy.special import laguerre
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> np.allclose(genlaguerre(3, 0)(x), laguerre(3)(x))
True

The polynomials :math:`L_n` also satisfy the recurrence relation:

.. math::
    (n + 1)L_{n+1}(x) = (2n +1 -x)L_n(x) - nL_{n-1}(x)

This can be easily checked on :math:`[0, 1]` for :math:`n = 3`:

>>> x = np.arange(0.0, 1.0, 0.01)
>>> np.allclose(4 * laguerre(4)(x),
...             (7 - x) * laguerre(3)(x) - 3 * laguerre(2)(x))
True

This is the plot of the first few Laguerre polynomials :math:`L_n`:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(-1.0, 5.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-5.0, 5.0)
>>> ax.set_title(r'Laguerre polynomials $L_n$')
>>> for n in np.arange(0, 5):
...     ax.plot(x, laguerre(n)(x), label=rf'$L_{n}$')
>>> plt.legend(loc='best')
>>> plt.show()

Vous êtes un professionnel et vous avez besoin d'une formation ? RAG (Retrieval-Augmented Generation)
et Fine Tuning d'un LLM Voir le programme détaillé