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

Spherical Bessel function of the second kind or its derivative.

Defined as [R529],

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

where \(Y_n\) is the 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.


yn : ndarray


For real arguments, the function is computed using the ascending recurrence [R530]. For complex arguments, the definitional relation to the cylindrical Bessel function of the second kind is used.

The derivative is computed using the relations [R531],

\[ \begin{align}\begin{aligned}y_n' = y_{n-1} - \frac{n + 1}{2} y_n.\\y_0' = -y_1\end{aligned}\end{align} \]

New in version 0.18.0.


[R529](1, 2)
[R530](1, 2)
[R531](1, 2)