scipy.special.hermite

scipy.special.hermite(n, monic=False)

Physicist’s Hermite polynomial.

Defined by

\[H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2};\]

\(H_n\) 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:

Horthopoly1d

Hermite polynomial.

Notes

The polynomials \(H_n\) are orthogonal over \((-\infty, \infty)\) with weight function \(e^{-x^2}\).

Examples

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

>>> p_monic = special.hermite(3, monic=True)
>>> p_monic
poly1d([ 1. ,  0. , -1.5,  0. ])
>>> p_monic(1)
-0.49999999999999983
>>> x = np.linspace(-3, 3, 400)
>>> y = p_monic(x)
>>> plt.plot(x, y)
>>> plt.title("Monic Hermite polynomial of degree 3")
>>> plt.xlabel("x")
>>> plt.ylabel("H_3(x)")
>>> plt.show()