`scipy.special._orthogonal:chebyc`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:2703

Signature

def chebyc(n, monic = False)

Summary

Chebyshev polynomial of the first kind on $[- 2, 2]$ .

Extended Summary

Defined as $C_{n} (x) = 2 T_{n} (x /2)$ , where $T_{n}$ is the nth Chebyshev polynomial of the first kind.

Parameters

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

Returns

C : orthopoly1d: Chebyshev polynomial of the first kind on $[- 2, 2]$ .

Notes

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

Examples

Evaluate the Chebyshev polynomial :math:`C_3` at :math:`x = 1`:

import numpy as np
from scipy.special import chebyc, chebyt

✓

np.isclose(chebyc(3)(1), -2.0)

✗

np.True_

The polynomial :math:`C_n` is a scaled Chebyshev polynomial of the first kind:

x = np.linspace(-2, 2, 5)
np.allclose(chebyc(3)(x), 2 * chebyt(3)(x/2))

✓

True

Plot :math:`C_n` for several values of :math:`n`:

import matplotlib.pyplot as plt
x = np.linspace(-2, 2, 400)
fig, ax = plt.subplots()

✓

for n in range(4):
    ax.plot(x, chebyc(n)(x), label=rf"$C_{n}$")
ax.set_title(r"Chebyshev polynomials $C_n$")
ax.set_xlabel("x")
ax.legend(loc="best")

✗

<matplotlib.legend.Legend object at 0x7f3f60ccf380>

plt.show()

✓

Aliases

scipy.special.chebyc