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()