scipy.special.jnp_zeros

scipy.special.jnp_zeros(n, nt)

Compute zeros of integer-order Bessel function derivatives Jn’.

Compute nt zeros of the functions \(J_n'(x)\) on the interval \((0, \infty)\). The zeros are returned in ascending order. Note that this interval excludes the zero at \(x = 0\) that exists for \(n > 1\).

Parameters:

nint

Order of Bessel function

ntint

Number of zeros to return

Returns:

ndarray

First nt zeros of the Bessel function.

See also

jvp

Derivatives of integer-order Bessel functions of the first kind

jv

Float-order Bessel functions of the first kind

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 \(J_2'\).

>>> from scipy.special import jnp_zeros
>>> jnp_zeros(2, 4)
array([ 3.05423693,  6.70613319,  9.96946782, 13.17037086])

As jnp_zeros yields the roots of \(J_n'\), it can be used to compute the locations of the peaks of \(J_n\). Plot \(J_2\), \(J_2'\) and the locations of the roots of \(J_2'\).

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import jn, jnp_zeros, jvp
>>> j2_roots = jnp_zeros(2, 4)
>>> xmax = 15
>>> x = np.linspace(0, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.plot(x, jn(2, x), label=r'$J_2$')
>>> ax.plot(x, jvp(2, x, 1), label=r"$J_2'$")
>>> ax.hlines(0, 0, xmax, color='k')
>>> ax.scatter(j2_roots, np.zeros((4, )), s=30, c='r',
...            label=r"Roots of $J_2'$", zorder=5)
>>> ax.set_ylim(-0.4, 0.8)
>>> ax.set_xlim(0, xmax)
>>> plt.legend()
>>> plt.show()

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