`scipy.special._basic:lqn`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_basic.py:2215

Signature

def lqn(n, z)

Summary

Legendre functions of the second kind.

Extended Summary

Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive). Returns two arrays of size (n+1,) + z.shape containing Qn(z) and Qn'(z).

Parameters

n : int: Maximum degree of the Legendre functions.
z : array_like, complex: Real or complex input values.

Returns

Qn_z : ndarray, shape (n+1,) + shape(z): Values for all degrees 0..n
Qn_d_z : ndarray, shape (n+1,) + shape(z): Derivatives for all degrees 0..n

Examples

Compute :math:`Q_n(x)` and its derivatives on an interval.

import numpy as np
from scipy.special import lqn
import matplotlib.pyplot as plt

✓

xs = np.linspace(-2, 2, 200)
n_max = 3
Qn, dQn = lqn(n_max, xs)

✓

Plot the Legendre functions of the second kind :math:`Q_n(x)`.

fig, ax = plt.subplots()

✓

ax.plot(xs, Qn.T, "-")
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$Q_n(x)$")
ax.legend([fr"$n={n}$" for n in range(n_max + 1)])

✗

<matplotlib.legend.Legend object at 0x7f3f592f1940>

plt.show()

✓

Plot the derivatives :math:`Q_n'(x)`.

fig, ax = plt.subplots()

✓

ax.plot(xs, dQn.T, "-")
ax.set_xlabel(r"$x$")
ax.set_ylabel(r"$Q_n'(x)$")
ax.legend([fr"$n={n}$" for n in range(n_max + 1)])

✗

<matplotlib.legend.Legend object at 0x7f3f6854aa50>

plt.show()

✓

Aliases

scipy.special.lqn