`scipy.special._spherical_bessel:spherical_in`

source: /scipy/special/_spherical_bessel.py :217

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

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

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)

✗

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

✓

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))

✓

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')

✗

plt.show()

✓

Aliases

scipy.special.spherical_in