laguerre#
- scipy.special.laguerre(n, monic=False)[source]#
- Laguerre polynomial. - Defined to be the solution of \[x\frac{d^2}{dx^2}L_n + (1 - x)\frac{d}{dx}L_n + nL_n = 0;\]- \(L_n\) is a polynomial of degree \(n\). - Parameters:
- nint
- Degree of the polynomial. 
- monicbool, optional
- If True, scale the leading coefficient to be 1. Default is False. 
 
- Returns:
- Lorthopoly1d
- Laguerre Polynomial. 
 
 - See also - genlaguerre
- Generalized (associated) Laguerre polynomial. 
 - Notes - The polynomials \(L_n\) are orthogonal over \([0, \infty)\) with weight function \(e^{-x}\). - 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 Laguerre polynomials \(L_n\) are the special case \(\alpha = 0\) of the generalized Laguerre polynomials \(L_n^{(\alpha)}\). Let’s verify it on the interval \([-1, 1]\): - >>> import numpy as np >>> from scipy.special import genlaguerre >>> from scipy.special import laguerre >>> x = np.arange(-1.0, 1.0, 0.01) >>> np.allclose(genlaguerre(3, 0)(x), laguerre(3)(x)) True - The polynomials \(L_n\) also satisfy the recurrence relation: \[(n + 1)L_{n+1}(x) = (2n +1 -x)L_n(x) - nL_{n-1}(x)\]- This can be easily checked on \([0, 1]\) for \(n = 3\): - >>> x = np.arange(0.0, 1.0, 0.01) >>> np.allclose(4 * laguerre(4)(x), ... (7 - x) * laguerre(3)(x) - 3 * laguerre(2)(x)) True - This is the plot of the first few Laguerre polynomials \(L_n\): - >>> import matplotlib.pyplot as plt >>> x = np.arange(-1.0, 5.0, 0.01) >>> fig, ax = plt.subplots() >>> ax.set_ylim(-5.0, 5.0) >>> ax.set_title(r'Laguerre polynomials $L_n$') >>> for n in np.arange(0, 5): ... ax.plot(x, laguerre(n)(x), label=rf'$L_{n}$') >>> plt.legend(loc='best') >>> plt.show() 