scipy.special.spherical_yn

scipy.special.spherical_yn(n, z, derivative=False)

Spherical Bessel function of the second kind or its derivative.

Defined as [1],

\[y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z),\]

where \(Y_n\) is the Bessel function of the second kind.

Parameters

nint, array_like

Order of the Bessel function (n >= 0).

zcomplex or float, array_like

Argument of the Bessel function.

derivativebool, optional

If True, the value of the derivative (rather than the function itself) is returned.

Returns

ynndarray

Notes

For real arguments, the function is computed using the ascending recurrence [2]. For complex arguments, the definitional relation to the cylindrical Bessel function of the second kind is used.

The derivative is computed using the relations [3],

\[ \begin{align}\begin{aligned}y_n' = y_{n-1} - \frac{n + 1}{z} y_n.\\y_0' = -y_1\end{aligned}\end{align} \]

New in version 0.18.0.

References

1

https://dlmf.nist.gov/10.47.E4

2

https://dlmf.nist.gov/10.51.E1

3

https://dlmf.nist.gov/10.51.E2

AS

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples

The spherical Bessel functions of the second kind \(y_n\) accept both real and complex second argument. They can return a complex type:

>>> from scipy.special import spherical_yn
>>> spherical_yn(0, 3+5j)
(8.022343088587197-9.880052589376795j)
>>> type(spherical_yn(0, 3+5j))
<class 'numpy.complex128'>

We can verify the relation for the derivative from the Notes for \(n=3\) in the interval \([1, 2]\):

>>> from scipy.special import spherical_yn
>>> x = np.arange(1.0, 2.0, 0.01)
>>> np.allclose(spherical_yn(3, x, True),
...             spherical_yn(2, x) - 4/x * spherical_yn(3, x))
True

The first few \(y_n\) with real argument:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import spherical_yn
>>> x = np.arange(0.0, 10.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-2.0, 1.0)
>>> ax.set_title(r'Spherical Bessel functions $y_n$')
>>> for n in np.arange(0, 4):
...     ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$')
>>> plt.legend(loc='best')
>>> plt.show()