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

Modified spherical Bessel function of the second kind or its derivative.

Defined as [R527],

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

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


n : int, array_like

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

z : complex or float, array_like

Argument of the Bessel function.

derivative : bool, optional

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


kn : ndarray


The function is computed using its definitional relation to the modified cylindrical Bessel function of the second kind.

The derivative is computed using the relations [R528],

\[ \begin{align}\begin{aligned}k_n' = -k_{n-1} - \frac{n + 1}{2} k_n.\\k_0' = -k_1\end{aligned}\end{align} \]

New in version 0.18.0.


[R527](1, 2)
[R528](1, 2)