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Module « scipy.special »

Fonction y1p_zeros - module scipy.special

Signature de la fonction y1p_zeros

def y1p_zeros(nt, complex=False) 

Description

help(scipy.special.y1p_zeros)

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

The values are given by Y1(z1) at each z1 where Y1'(z1)=0.

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
-------
z1pn : ndarray
    Location of nth zero of Y1'(z)
y1z1pn : 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 four roots of :math:`Y_1'` and the values of
:math:`Y_1` at these roots.

>>> import numpy as np
>>> from scipy.special import y1p_zeros
>>> y1grad_roots, y1_values = y1p_zeros(4)
>>> with np.printoptions(precision=5):
...     print(f"Y1' Roots: {y1grad_roots.real}")
...     print(f"Y1 values: {y1_values.real}")
Y1' Roots: [ 3.68302  6.9415  10.1234  13.28576]
Y1 values: [ 0.41673 -0.30317  0.25091 -0.21897]

`y1p_zeros` can be used to calculate the extremal points of :math:`Y_1`
directly. Here we plot :math:`Y_1` and the first four extrema.

>>> import matplotlib.pyplot as plt
>>> from scipy.special import y1, yvp
>>> y1_roots, y1_values_at_roots = y1p_zeros(4)
>>> real_roots = y1_roots.real
>>> xmax = 15
>>> x = np.linspace(0, xmax, 500)
>>> x[0] += 1e-15
>>> fig, ax = plt.subplots()
>>> ax.plot(x, y1(x), label=r'$Y_1$')
>>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
>>> ax.scatter(real_roots, np.zeros((4, )), s=30, c='r',
...            label=r"Roots of $Y_1'$", zorder=5)
>>> ax.scatter(real_roots, y1_values_at_roots.real, s=30, c='k',
...            label=r"Extrema of $Y_1$", zorder=5)
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.5, 0.5)
>>> ax.set_xlim(0, xmax)
>>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
>>> plt.tight_layout()
>>> plt.show()

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