scipy.special.

sh_chebyt

scipy.special.sh_chebyt(n, monic=False)

Shifted Chebyshev polynomial of the first kind.

Defined as \(T^*_n(x) = T_n(2x - 1)\) for \(T_n\) the nth Chebyshev polynomial of the first kind.

Parameters:

nint

Degree of the polynomial.

monicbool, optional

If True, scale the leading coefficient to be 1. Default is False.

Returns:

Torthopoly1d

Shifted Chebyshev polynomial of the first kind.

Notes

The polynomials \(T^*_n\) are orthogonal over \([0, 1]\) with weight function \((x - x^2)^{-1/2}\).

Examples

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Evaluate the shifted Chebyshev polynomial of the first kind \(T^*_3\) at \(x = 0.75\):

>>> import numpy as np
>>> from scipy.special import chebyt, sh_chebyt
>>> np.isclose(sh_chebyt(3)(0.75), -1.0)
True

The polynomial \(T^*_n\) is the Chebyshev polynomial \(T_n\) shifted from \([-1, 1]\) to \([0, 1]\):

>>> x = np.linspace(0, 1, 5)
>>> np.allclose(sh_chebyt(3)(x), chebyt(3)(2*x - 1))
True

Plot \(T^*_n\) for several values of \(n\):

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 1, 400)
>>> fig, ax = plt.subplots()
>>> for n in range(4):
...     ax.plot(x, sh_chebyt(n)(x), label=rf"$T^*_{n}$")
>>> ax.set_title(r"Shifted Chebyshev polynomials $T^*_n$")
>>> ax.set_xlabel("x")
>>> ax.legend(loc="best")
>>> plt.show()

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