scipy.special.
gegenbauer#
- 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 - pas a Gegenbauer polynomial using the- gegenbauerfunction 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 - pat various points- xin the interval- (-3, 3), simply pass an array- xto- pas follows:- >>> x = np.linspace(-3, 3, 400) >>> y = p(x) - We can then visualize - x, yusing- 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() 