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Module « scipy.special »

Fonction y1_zeros - module scipy.special

Signature de la fonction y1_zeros

def y1_zeros(nt, complex=False) 

Description

help(scipy.special.y1_zeros)

Compute nt zeros of Bessel function Y1(z), and derivative at each zero.

The derivatives are given by Y1'(z1) = Y0(z1) at each zero z1.

Parameters
----------
nt : int
    Number of zeros to return
complex : bool, default False
    Set to False to return only the real zeros; set to True to return only
    the complex zeros with negative real part and positive imaginary part.
    Note that the complex conjugates of the latter are also zeros of the
    function, but are not returned by this routine.

Returns
-------
z1n : ndarray
    Location of nth zero of Y1(z)
y1pz1n : ndarray
    Value of derivative Y1'(z1) for nth zero

References
----------
.. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special
       Functions", John Wiley and Sons, 1996, chapter 5.
       https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

Examples
--------
Compute the first 4 real roots and the derivatives at the roots of
:math:`Y_1`:

>>> import numpy as np
>>> from scipy.special import y1_zeros
>>> zeros, grads = y1_zeros(4)
>>> with np.printoptions(precision=5):
...     print(f"Roots: {zeros}")
...     print(f"Gradients: {grads}")
Roots: [ 2.19714+0.j  5.42968+0.j  8.59601+0.j 11.74915+0.j]
Gradients: [ 0.52079+0.j -0.34032+0.j  0.27146+0.j -0.23246+0.j]

Extract the real parts:

>>> realzeros = zeros.real
>>> realzeros
array([ 2.19714133,  5.42968104,  8.59600587, 11.74915483])

Plot :math:`Y_1` and the first four computed roots.

>>> import matplotlib.pyplot as plt
>>> from scipy.special import y1
>>> xmin = 0
>>> xmax = 13
>>> x = np.linspace(xmin, xmax, 500)
>>> zeros, grads = y1_zeros(4)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, y1(x), label=r'$Y_1$')
>>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
...            label=r'$Y_1$_zeros', zorder=5)
>>> ax.set_ylim(-0.5, 0.5)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend()
>>> plt.show()

Compute the first 4 complex roots and the derivatives at the roots of
:math:`Y_1` by setting ``complex=True``:

>>> y1_zeros(4, True)
(array([ -0.50274327+0.78624371j,  -3.83353519+0.56235654j,
         -7.01590368+0.55339305j, -10.17357383+0.55127339j]),
 array([-0.45952768+1.31710194j,  0.04830191-0.69251288j,
        -0.02012695+0.51864253j,  0.011614  -0.43203296j]))

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