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.

See also

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.

Array API Standard Support

y0 has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	✅
JAX	⚠️ no JIT	⛔
Dask	✅	n/a

See Support for the array API standard for more information.

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()