`scipy.special._spherical_bessel:spherical_in`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_spherical_bessel.py:188

Signature

def spherical_in(n, z, derivative = False)

Summary

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

Extended Summary

Defined as ^[1],

i_{n} (z) = \frac{π}{2 z} 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: Value or derivative of modified spherical Bessel function of the first kind.

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 ^[2],

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

Examples

The modified spherical Bessel functions of the first kind :math:`i_n` accept both real and complex second argument. They can return a complex type:

from scipy.special import spherical_in

✓

spherical_in(0, 3+5j)

✗

np.complex128(-1.1689867793369184-1.2697305267234227j)

type(spherical_in(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_in(3, x, True),
            spherical_in(2, x) - 4/x * spherical_in(3, x))

✓

True

The first few :math:`i_n` with real argument:

import matplotlib.pyplot as plt
x = np.arange(0.0, 6.0, 0.01)
fig, ax = plt.subplots()

✓

ax.set_ylim(-0.5, 5.0)
ax.set_title(r'Modified spherical Bessel functions $i_n$')
for n in np.arange(0, 4):
    ax.plot(x, spherical_in(n, x), label=rf'$i_{n}$')
plt.legend(loc='best')

✗

<matplotlib.legend.Legend object at 0x7f3f5904ecf0>

plt.show()

✓

Aliases

scipy.special.spherical_in