scipy.special.ynp_zeros#
- scipy.special.ynp_zeros(n, nt)[source]#
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
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 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()