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Module « scipy.special »

Fonction jacobi - module scipy.special

Signature de la fonction jacobi

def jacobi(n, alpha, beta, monic=False) 

Description

help(scipy.special.jacobi)

Jacobi polynomial.

Defined to be the solution of

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

for :math:`\alpha, \beta > -1`; :math:`P_n^{(\alpha, \beta)}` is a
polynomial of degree :math:`n`.

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

Returns
-------
P : orthopoly1d
    Jacobi polynomial.

Notes
-----
For fixed :math:`\alpha, \beta`, the polynomials
:math:`P_n^{(\alpha, \beta)}` are orthogonal over :math:`[-1, 1]`
with weight function :math:`(1 - x)^\alpha(1 + x)^\beta`.

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 Jacobi polynomials satisfy the recurrence relation:

.. math::
    P_n^{(\alpha, \beta-1)}(x) - P_n^{(\alpha-1, \beta)}(x)
      = P_{n-1}^{(\alpha, \beta)}(x)

This can be verified, for example, for :math:`\alpha = \beta = 2`
and :math:`n = 1` over the interval :math:`[-1, 1]`:

>>> import numpy as np
>>> from scipy.special import jacobi
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> np.allclose(jacobi(0, 2, 2)(x),
...             jacobi(1, 2, 1)(x) - jacobi(1, 1, 2)(x))
True

Plot of the Jacobi polynomial :math:`P_5^{(\alpha, -0.5)}` for
different values of :math:`\alpha`:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(-1.0, 1.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-2.0, 2.0)
>>> ax.set_title(r'Jacobi polynomials $P_5^{(\alpha, -0.5)}$')
>>> for alpha in np.arange(0, 4, 1):
...     ax.plot(x, jacobi(5, alpha, -0.5)(x), label=rf'$\alpha={alpha}$')
>>> plt.legend(loc='best')
>>> plt.show()

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