`scipy.special._spherical_bessel:spherical_yn`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_spherical_bessel.py:98

Signature

def spherical_yn(n, z, derivative = False)

Summary

Spherical Bessel function of the second kind or its derivative.

Extended Summary

Defined as ^[1],

y_{n} (z) = \frac{π}{2 z} Y_{n + 1/2} (z),

where $Y_{n}$ is the Bessel function of the second 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

yn : ndarray: Value or derivative of spherical Bessel function of the second kind.

Notes

For real arguments, the function is computed using the ascending recurrence ^[2]. For complex arguments, the definitional relation to the cylindrical Bessel function of the second kind is used.

The derivative is computed using the relations ^[3],

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

Examples

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

from scipy.special import spherical_yn

✓

spherical_yn(0, 3+5j)

✗

np.complex128(8.022343088587196-9.880052589376795j)

type(spherical_yn(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_yn(3, x, True),
            spherical_yn(2, x) - 4/x * spherical_yn(3, x))

✓

True

The first few :math:`y_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(-2.0, 1.0)
ax.set_title(r'Spherical Bessel functions $y_n$')
for n in np.arange(0, 4):
    ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$')
plt.legend(loc='best')

✗

<matplotlib.legend.Legend object at 0x7f3f58a04c20>

plt.show()

✓

Aliases

scipy.special.spherical_yn