spherical_yn — SciPy v1.15.2 Manual (original) (raw)

scipy.special.

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],

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

Added in version 0.18.0.

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 yny_nyn​') for n in np.arange(0, 4): ... ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$') plt.legend(loc='best') plt.show()