scipy.special.y0_zeros

scipy.special.y0_zeros(nt, complex=False)

Compute nt zeros of Bessel function Y0(z), and derivative at each zero.

The derivatives are given by Y0’(z0) = -Y1(z0) at each zero z0.

Parameters:

ntint

Number of zeros to return

complexbool, 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:

z0nndarray

Location of nth zero of Y0(z)

y0pz0nndarray

Value of derivative Y0’(z0) 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 4 real roots and the derivatives at the roots of \(Y_0\):

>>> import numpy as np
>>> from scipy.special import y0_zeros
>>> zeros, grads = y0_zeros(4)
>>> with np.printoptions(precision=5):
...     print(f"Roots: {zeros}")
...     print(f"Gradients: {grads}")
Roots: [ 0.89358+0.j  3.95768+0.j  7.08605+0.j 10.22235+0.j]
Gradients: [-0.87942+0.j  0.40254+0.j -0.3001 +0.j  0.2497 +0.j]

Plot the real part of \(Y_0\) and the first four computed roots.

>>> import matplotlib.pyplot as plt
>>> from scipy.special import y0
>>> xmin = 0
>>> xmax = 11
>>> x = np.linspace(xmin, xmax, 500)
>>> fig, ax = plt.subplots()
>>> ax.hlines(0, xmin, xmax, color='k')
>>> ax.plot(x, y0(x), label=r'$Y_0$')
>>> zeros, grads = y0_zeros(4)
>>> ax.scatter(zeros.real, np.zeros((4, )), s=30, c='r',
...            label=r'$Y_0$_zeros', zorder=5)
>>> ax.set_ylim(-0.5, 0.6)
>>> ax.set_xlim(xmin, xmax)
>>> plt.legend(ncol=2)
>>> plt.show()

Compute the first 4 complex roots and the derivatives at the roots of \(Y_0\) by setting complex=True:

>>> y0_zeros(4, True)
(array([ -2.40301663+0.53988231j,  -5.5198767 +0.54718001j,
         -8.6536724 +0.54841207j, -11.79151203+0.54881912j]),
 array([ 0.10074769-0.88196771j, -0.02924642+0.5871695j ,
         0.01490806-0.46945875j, -0.00937368+0.40230454j]))