scipy.special.spherical_jn

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

Spherical Bessel function of the first kind or its derivative.

Defined as [1],

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

where \(J_n\) is the Bessel function of the first 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:

jnndarray

Notes

For real arguments greater than the order, the function is computed using the ascending recurrence [2]. For small real or complex arguments, the definitional relation to the cylindrical Bessel function of the first kind is used.

The derivative is computed using the relations [3],

\[ \begin{align}\begin{aligned}j_n'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z).\\j_0'(z) = -j_1(z)\end{aligned}\end{align} \]

New in version 0.18.0.

References

[1]

https://dlmf.nist.gov/10.47.E3

[2]

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

[3]

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

[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 first kind \(j_n\) accept both real and complex second argument. They can return a complex type:

>>> from scipy.special import spherical_jn
>>> spherical_jn(0, 3+5j)
(-9.878987731663194-8.021894345786002j)
>>> type(spherical_jn(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_jn(3, x, True),
...             spherical_jn(2, x) - 4/x * spherical_jn(3, x))
True

The first few \(j_n\) with real argument:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(0.0, 10.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-0.5, 1.5)
>>> ax.set_title(r'Spherical Bessel functions $j_n$')
>>> for n in np.arange(0, 4):
...     ax.plot(x, spherical_jn(n, x), label=rf'$j_{n}$')
>>> plt.legend(loc='best')
>>> plt.show()