gegenbauer — SciPy v1.15.3 Manual (original) (raw)

scipy.special.

scipy.special.gegenbauer(n, alpha, monic=False)[source]#

Gegenbauer (ultraspherical) polynomial.

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 \(n\).

Parameters:

nint

Degree of the polynomial.

alphafloat

Parameter, must be greater than -0.5.

monicbool, optional

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

Returns:

Corthopoly1d

Gegenbauer polynomial.

Notes

The polynomials \(C_n^{(\alpha)}\) are orthogonal over\([-1,1]\) with weight function \((1 - x^2)^{(\alpha - 1/2)}\).

Examples

import numpy as np from scipy import special import matplotlib.pyplot as plt

We can initialize a variable p as a Gegenbauer polynomial using thegegenbauer 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()