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],
\[ \begin{align}\begin{aligned}y_n' = y_{n-1} - \frac{n + 1}{z} y_n.\\y_0' = -y_1\end{aligned}\end{align} \]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 $y_n$') >>> for n in np.arange(0, 4): ... ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$') >>> plt.legend(loc='best') >>> plt.show()
