jacobi#
- scipy.special.jacobi(n, alpha, beta, monic=False)[source]#
- Jacobi polynomial. - Defined to be the solution of \[(1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)} + (\beta - \alpha - (\alpha + \beta + 2)x) \frac{d}{dx}P_n^{(\alpha, \beta)} + n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0\]- for \(\alpha, \beta > -1\); \(P_n^{(\alpha, \beta)}\) is a polynomial of degree \(n\). - Parameters:
- nint
- Degree of the polynomial. 
- alphafloat
- Parameter, must be greater than -1. 
- betafloat
- Parameter, must be greater than -1. 
- monicbool, optional
- If True, scale the leading coefficient to be 1. Default is False. 
 
- Returns:
- Porthopoly1d
- Jacobi polynomial. 
 
 - Notes - For fixed \(\alpha, \beta\), the polynomials \(P_n^{(\alpha, \beta)}\) are orthogonal over \([-1, 1]\) with weight function \((1 - x)^\alpha(1 + x)^\beta\). - References [AS]- Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. - Examples - The Jacobi polynomials satisfy the recurrence relation: \[P_n^{(\alpha, \beta-1)}(x) - P_n^{(\alpha-1, \beta)}(x) = P_{n-1}^{(\alpha, \beta)}(x)\]- This can be verified, for example, for \(\alpha = \beta = 2\) and \(n = 1\) over the interval \([-1, 1]\): - >>> import numpy as np >>> from scipy.special import jacobi >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(jacobi(0, 2, 2)(x), ... jacobi(1, 2, 1)(x) - jacobi(1, 1, 2)(x)) True - Plot of the Jacobi polynomial \(P_5^{(\alpha, -0.5)}\) for different values of \(\alpha\): - >>> import matplotlib.pyplot as plt >>> x = np.arange(-1.0, 1.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-2.0, 2.0) >>> ax.set_title(r'Jacobi polynomials $P_5^{(\alpha, -0.5)}$') >>> for alpha in np.arange(0, 4, 1): ... ax.plot(x, jacobi(5, alpha, -0.5)(x), label=rf'$\alpha={alpha}$') >>> plt.legend(loc='best') >>> plt.show() 