Skip to main content
`Ctrl`+`K`
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:
    n int

      Degree of the polynomial.

    monic bool, optional

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

  Returns:
    T orthopoly1d

      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

  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()