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scipy.special.

# jnyn_zeros

scipy.special.jnyn_zeros(n, nt)

  Compute nt zeros of Bessel functions Jn(x), Jn’(x), Yn(x), and Yn’(x).

  Returns 4 arrays of length nt, corresponding to the first nt zeros of Jn(x), Jn’(x), Yn(x), and Yn’(x), respectively. The zeros are returned in ascending order.

  Parameters:
    n int

      Order of the Bessel functions

    nt int

      Number (<=1200) of zeros to compute

  Returns:
    Jn ndarray

      First nt zeros of Jn

    Jnp ndarray

      First nt zeros of Jn’

    Yn ndarray

      First nt zeros of Yn

    Ynp ndarray

      First nt zeros of Yn’

  See also

  `jn_zeros`, `jnp_zeros`, `yn_zeros`, `ynp_zeros`

  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 three roots of \(J_1\), \(J_1'\), \(Y_1\) and \(Y_1'\).

      >>> from scipy.special import jnyn_zeros
      >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
      >>> jn_roots, yn_roots
      (array([ 3.83170597,  7.01558667, 10.17346814]),
       array([2.19714133, 5.42968104, 8.59600587]))

  Plot \(J_1\), \(J_1'\), \(Y_1\), \(Y_1'\) and their roots.

      >>> import numpy as np
      >>> import matplotlib.pyplot as plt
      >>> from scipy.special import jnyn_zeros, jvp, jn, yvp, yn
      >>> jn_roots, jnp_roots, yn_roots, ynp_roots = jnyn_zeros(1, 3)
      >>> fig, ax = plt.subplots()
      >>> xmax= 11
      >>> x = np.linspace(0, xmax)
      >>> x[0] += 1e-15
      >>> ax.plot(x, jn(1, x), label=r"$J_1$", c='r')
      >>> ax.plot(x, jvp(1, x, 1), label=r"$J_1'$", c='b')
      >>> ax.plot(x, yn(1, x), label=r"$Y_1$", c='y')
      >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$", c='c')
      >>> zeros = np.zeros((3, ))
      >>> ax.scatter(jn_roots, zeros, s=30, c='r', zorder=5,
      ...            label=r"$J_1$ roots")
      >>> ax.scatter(jnp_roots, zeros, s=30, c='b', zorder=5,
      ...            label=r"$J_1'$ roots")
      >>> ax.scatter(yn_roots, zeros, s=30, c='y', zorder=5,
      ...            label=r"$Y_1$ roots")
      >>> ax.scatter(ynp_roots, zeros, s=30, c='c', zorder=5,
      ...            label=r"$Y_1'$ roots")
      >>> ax.hlines(0, 0, xmax, color='k')
      >>> ax.set_ylim(-0.6, 0.6)
      >>> ax.set_xlim(0, xmax)
      >>> ax.legend(ncol=2, bbox_to_anchor=(1., 0.75))
      >>> plt.tight_layout()
      >>> plt.show()