# scipy.special.y0#

scipy.special.y0(x, out=None) = <ufunc 'y0'>#

Bessel function of the second kind of order 0.

Parameters:
xarray_like

Argument (float).

outndarray, optional

Optional output array for the function results

Returns:
Yscalar or ndarray

Value of the Bessel function of the second kind of order 0 at x.

j0

Bessel function of the first kind of order 0

yv

Bessel function of the first kind

Notes

The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval a rational approximation $$R(x)$$ is employed to compute,

$Y_0(x) = R(x) + \frac{2 \log(x) J_0(x)}{\pi},$

where $$J_0$$ is the Bessel function of the first kind of order 0.

In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7.

This function is a wrapper for the Cephes [1] routine y0.

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

Calculate the function at one point:

>>> from scipy.special import y0
>>> y0(1.)
0.08825696421567697


Calculate at several points:

>>> import numpy as np
>>> y0(np.array([0.5, 2., 3.]))
array([-0.44451873,  0.51037567,  0.37685001])


Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = y0(x)
>>> ax.plot(x, y)
>>> plt.show()