chebyt(n, monic=False)
Defined to be the solution of
$$(1 - x^2)\frac{d^2}{dx^2}T_n - x\frac{d}{dx}T_n + n^2T_n = 0;$$$T_n$ is a polynomial of degree $n$ .
The polynomials $T_n$ are orthogonal over $[-1, 1]$ with weight function $(1 - x^2)^{-1/2}$ .
Degree of the polynomial.
If :None:None:`True`, scale the leading coefficient to be 1. Default is :None:None:`False`.
Chebyshev polynomial of the first kind.
Chebyshev polynomial of the first kind.
chebyu
Chebyshev polynomial of the second kind.
Chebyshev polynomials of the first kind of order $n$ can be obtained as the determinant of specific $n \times n$ matrices. As an example we can check how the points obtained from the determinant of the following $3 \times 3$ matrix lay exacty on $T_3$ :
>>> 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 $P_n^{(-0.5, -0.5)}$ through the relation:
$$P_n^{(-0.5, -0.5)}(x) = \frac{1}{4^n} \binom{2n}{n} T_n(x)$$Let's verify it for $n = 3$ :
>>> from scipy.special import binom
... from scipy.special import chebyt
... 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)) True
We can plot the Chebyshev polynomials $T_n$ for some values of $n$ :
>>> import matplotlib.pyplot as plt
... from scipy.special import chebyt
... 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()
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.special._orthogonal.chebyu
scipy.special._orthogonal.chebyc
scipy.special._orthogonal.chebyt
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