`scipy.special._spherical_bessel:spherical_jn`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_spherical_bessel.py:7

Signature

def spherical_jn(n, z, derivative = False)

Summary

Spherical Bessel function of the first kind or its derivative.

Extended Summary

Defined as ^[1],

j_{n} (z) = \frac{π}{2 z} J_{n + 1/2} (z),

where $J_{n}$ is the 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

jn : ndarray: Value or derivative of spherical Bessel function of the first kind.

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],

j_{n}^{'} (z) j_{0}^{'} (z) = j_{n - 1} (z) - \frac{n + 1}{z} j_{n} (z), = - j_{1} (z) .

Examples

The spherical Bessel functions of the first kind :math:`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)

✗

np.complex128(-9.878987731663194-8.021894345785999j)

type(spherical_jn(0, 3+5j))

✓

<class 'numpy.complex128'>

We can verify the relation for the derivative from the Notes for :math:`n=3` in the interval :math:`[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 :math:`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')

✗

<matplotlib.legend.Legend object at 0x7f3f58bbb380>

plt.show()

✓

Aliases

scipy.special.spherical_jn