scipy.special.

ynp_zeros

scipy.special.ynp_zeros(n, nt)

Compute zeros of integer-order Bessel function derivatives Yn’(x).

Compute nt zeros of the functions \(Y_n'(x)\) on the interval \((0, \infty)\). The zeros are returned in ascending order.

Parameters:

nint

Order of Bessel function

ntint

Number of zeros to return

Returns:

ndarray

First nt zeros of the Bessel derivative function.

See also

yvp

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

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Compute the first four roots of the first derivative of the Bessel function of second kind for order 0 \(Y_0'\).

>>> from scipy.special import ynp_zeros
>>> ynp_zeros(0, 4)
array([ 2.19714133,  5.42968104,  8.59600587, 11.74915483])

Plot \(Y_0\), \(Y_0'\) and confirm visually that the roots of \(Y_0'\) are located at local extrema of \(Y_0\).

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import yn, ynp_zeros, yvp
>>> zeros = ynp_zeros(0, 4)
>>> xmax = 13
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, yn(0, x), label=r'$Y_0$')
>>> ax.plot(x, yvp(0, x, 1), label=r"$Y_0'$")
>>> ax.scatter(zeros, np.zeros((4, )), s=30, c='r',
...            label=r"Roots of $Y_0'$", zorder=5)
>>> for root in zeros:
...     y0_extremum =  yn(0, root)
...     lower = min(0, y0_extremum)
...     upper = max(0, y0_extremum)
...     ax.vlines(root, lower, upper, color='r')
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.set_ylim(-0.6, 0.6)
>>> ax.set_xlim(0, xmax)
>>> plt.legend()
>>> plt.show()

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