`scipy.special._orthogonal:hermitenorm`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:1857

Signature

def hermitenorm(n, monic = False)

Summary

Probabilist's Hermite polynomial.

Extended Summary

Defined by

H e_{n} (x) = (- 1)^{n} e^{x^{2} /2} \frac{d ^{n}}{d x ^{n}} e^{- x^{2} /2};

$H e_{n}$ is a polynomial of degree $n$ .

Parameters

n : int: Degree of the polynomial.
monic : bool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns

He : orthopoly1d: Probabilist's Hermite polynomial.

Notes

The polynomials $H e_{n}$ are orthogonal over $(- \infty, \infty)$ with weight function $e^{- x^{2} /2}$ .

Examples

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

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p_monic = hermitenorm(3)
p_monic

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poly1d([ 1.,  0., -3.,  0.])

Evaluate the probabilist's Hermite polynomial of degree 3 at x = 1:

p_monic(1)

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np.float64(-2.0)

Plot probabilist's Hermite polynomials of degree 0 to 4:

x = np.linspace(-3, 3, 100)
fig, ax = plt.subplots()

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for i in range(5):
    ax.plot(x, hermitenorm(i)(x), label=f"n={i}")
plt.title(f"Probabilist's Hermite polynomials $He_n$")
plt.xlabel("x")
plt.ylabel(rf"$He_n(x)$")
plt.legend(loc="best")

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<matplotlib.legend.Legend object at 0x7f3f684f63c0>

plt.show()

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Aliases

scipy.special.hermitenorm