gegenbauer(n, alpha, monic=False)
Defined to be the solution of
$$(1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)} - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)} + n(n + 2\alpha)C_n^{(\alpha)} = 0$$for $\alpha > -1/2$ ; $C_n^{(\alpha)}$ is a polynomial of degree $n$ .
The polynomials $C_n^{(\alpha)}$ are orthogonal over $[-1,1]$ with weight function $(1 - x^2)^{(\alpha - 1/2)}$ .
Degree of the polynomial.
Parameter, must be greater than -0.5.
If :None:None:`True`, scale the leading coefficient to be 1. Default is :None:None:`False`.
Gegenbauer polynomial.
Gegenbauer (ultraspherical) polynomial.
>>> from scipy import special
... import matplotlib.pyplot as plt
We can initialize a variable p
as a Gegenbauer polynomial using the gegenbauer
function and evaluate at a point x = 1
.
>>> p = special.gegenbauer(3, 0.5, monic=False)
... p poly1d([ 2.5, 0. , -1.5, 0. ])
>>> p(1) 1.0
To evaluate p
at various points x
in the interval (-3, 3)
, simply pass an array x
to p
as follows:
>>> x = np.linspace(-3, 3, 400)
... y = p(x)
We can then visualize x, y
using matplotlib.pyplot
.
>>> fig, ax = plt.subplots()
... ax.plot(x, y)
... ax.set_title("Gegenbauer (ultraspherical) polynomial of degree 3")
... ax.set_xlabel("x")
... ax.set_ylabel("G_3(x)")
... plt.show()
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.special._orthogonal.gegenbauer
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