scipy.special.spherical_in

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

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

Defined as [R543],

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

where \(I_n\) is the modified Bessel function of the first kind.

Parameters:

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.

Returns:

in : ndarray

Notes

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

The derivative is computed using the relations [R544],

\[ \begin{align}\begin{aligned}i_n' = i_{n-1} - \frac{n + 1}{2} i_n.\\i_1' = i_0\end{aligned}\end{align} \]

New in version 0.18.0.

References

[R543]

(1, 2) http://dlmf.nist.gov/10.47.E7

[R544]

(1, 2) http://dlmf.nist.gov/10.51.E5