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scipy.special.

# hermitenorm

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

  Probabilist’s Hermite polynomial.

  Defined by

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

  \(He_n\) 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 \(He_n\) 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

      >>> p_monic = hermitenorm(3)
      >>> p_monic
      poly1d([ 1.,  0., -3.,  0.])

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

      >>> p_monic(1)
      np.float64(-2.0)

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

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