chebyu(n, monic=False)
Defined to be the solution of
$$(1 - x^2)\frac{d^2}{dx^2}U_n - 3x\frac{d}{dx}U_n + n(n + 2)U_n = 0;$$$U_n$ is a polynomial of degree $n$ .
The polynomials $U_n$ 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 second kind.
Chebyshev polynomial of the second kind.
chebyt
Chebyshev polynomial of the first kind.
Chebyshev polynomials of the second 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 $U_3$ :
>>> import matplotlib.pyplot as plt
... from scipy.linalg import det
... from scipy.special import chebyu
... 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 $U_3$')
... ax.plot(x, chebyu(3)(x), label=rf'$U_3$')
... for p in np.arange(-1.0, 1.0, 0.1):
... ax.plot(p,
... det(np.array([[2*p, 1, 0], [1, 2*p, 1], [0, 1, 2*p]])),
... 'rx')
... plt.legend(loc='best')
... plt.show()
They satisfy the recurrence relation:
$$U_{2n-1}(x) = 2 T_n(x)U_{n-1}(x)$$where the $T_n$ are the Chebyshev polynomial of the first kind. Let's verify it for $n = 2$ :
>>> from scipy.special import chebyt
... from scipy.special import chebyu
... x = np.arange(-1.0, 1.0, 0.01)
... np.allclose(chebyu(3)(x), 2 * chebyt(2)(x) * chebyu(1)(x)) True
We can plot the Chebyshev polynomials $U_n$ for some values of $n$ :
>>> import matplotlib.pyplot as plt
... from scipy.special import chebyu
... x = np.arange(-1.0, 1.0, 0.01)
... fig, ax = plt.subplots()
... ax.set_ylim(-1.5, 1.5)
... ax.set_title(r'Chebyshev polynomials $U_n$')
... for n in np.arange(1,5):
... ax.plot(x, chebyu(n)(x), label=rf'$U_n={n}$')
... 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.chebys
scipy.special._orthogonal.chebyt
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