scipy.special.

# spherical_yn#

scipy.special.spherical_yn(n, z, derivative=False)[source]#

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}

References

[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 $$[1, 2]$$:

>>> import numpy as np
>>> 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
>>> 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()