scipy.special.spherical_yn

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

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],

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

New in version 0.18.0.

References

1(1,2)

https://dlmf.nist.gov/10.47.E4

2(1,2)

https://dlmf.nist.gov/10.51.E1

3(1,2)

https://dlmf.nist.gov/10.51.E2