`scipy.special._orthogonal:chebyt`

source: /scipy/special/_orthogonal.py :1701

Signature

def chebyt ( n , monic = False )

Summary

Chebyshev polynomial of the first kind.

Extended Summary

Defined to be the solution of

(1 - x^{2}) \frac{d ^{2}}{d x ^{2}} T_{n} - x \frac{d}{d x} T_{n} + n^{2} T_{n} = 0;

$T_{n}$ is a polynomial of degree $n$ .

Parameters

n : int: Degree of the polynomial.
monic : bool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns

T : orthopoly1d: Chebyshev polynomial of the first kind.

Notes

The polynomials $T_{n}$ are orthogonal over $[- 1, 1]$ with weight function $(1 - x^{2})^{- 1/2}$ .

Examples

Chebyshev polynomials of the first kind of order :math:`n` can be obtained as the determinant of specific :math:`n \times n` matrices. As an example we can check how the points obtained from the determinant of the following :math:`3 \times 3` matrix lay exactly on :math:`T_3`:

import numpy as np
import matplotlib.pyplot as plt
from scipy.linalg import det
from scipy.special import chebyt
x = np.arange(-1.0, 1.0, 0.01)
fig, ax = plt.subplots()

✓

ax.set_ylim(-2.0, 2.0)
ax.set_title(r'Chebyshev polynomial $T_3$')
ax.plot(x, chebyt(3)(x), label=rf'$T_3$')
for p in np.arange(-1.0, 1.0, 0.1):
    ax.plot(p,
            det(np.array([[p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])),
            'rx')
plt.legend(loc='best')

✗

plt.show()

✓

They are also related to the Jacobi Polynomials :math:`P_n^{(-0.5, -0.5)}` through the relation: .. math:: P_n^{(-0.5, -0.5)}(x) = \frac{1}{4^n} \binom{2n}{n} T_n(x) Let's verify it for :math:`n = 3`:

from scipy.special import binom
from scipy.special import jacobi
x = np.arange(-1.0, 1.0, 0.01)
np.allclose(jacobi(3, -0.5, -0.5)(x),
            1/64 * binom(6, 3) * chebyt(3)(x))

✓

We can plot the Chebyshev polynomials :math:`T_n` for some values of :math:`n`:

x = np.arange(-1.5, 1.5, 0.01)
fig, ax = plt.subplots()

✓

ax.set_ylim(-4.0, 4.0)
ax.set_title(r'Chebyshev polynomials $T_n$')
for n in np.arange(2,5):
    ax.plot(x, chebyt(n)(x), label=rf'$T_n={n}$')
plt.legend(loc='best')

✗

plt.show()

✓

Aliases

scipy.special.chebyt