scipy.special.

chebys#

scipy.special.chebys(n, monic=False)[source]#

Chebyshev polynomial of the second kind on \([-2, 2]\).

Defined as \(S_n(x) = U_n(x/2)\) where \(U_n\) is the nth Chebyshev polynomial of the second kind.

Parameters:

nint: Degree of the polynomial.
monicbool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns:

Sorthopoly1d: Chebyshev polynomial of the second kind on \([-2, 2]\).

See also

chebyu: Chebyshev polynomial of the second kind

Notes

The polynomials \(S_n(x)\) are orthogonal over \([-2, 2]\) with weight function \(\sqrt{1 - (x/2)^2}\).

References

[1]

Abramowitz and Stegun, “Handbook of Mathematical Functions” Section 22. National Bureau of Standards, 1972.

Examples

Evaluate the Chebyshev polynomial of the second kind \(S_3\) at \(x = 1\):

>>> import numpy as np
>>> from scipy.special import chebys, chebyu
>>> np.isclose(chebys(3)(1), -1.0)
True

The polynomial \(S_n\) is a scaled Chebyshev polynomial of the second kind:

>>> x = np.linspace(-2, 2, 5)
>>> np.allclose(chebys(3)(x), chebyu(3)(x/2))
True

Plot \(S_n\) for several values of \(n\):

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(-2, 2, 400)
>>> fig, ax = plt.subplots()
>>> for n in range(4):
...     ax.plot(x, chebys(n)(x), label=rf"$S_{n}$")
>>> ax.set_title(r"Chebyshev polynomials $S_n$")
>>> ax.set_xlabel("x")
>>> ax.legend(loc="best")
>>> plt.show()

../../_images/scipy-special-chebys-1.png