scipy.special.eval_hermite

scipy.special.eval_hermite(n, x, out=None) = <ufunc 'eval_hermite'>

Evaluate physicist’s Hermite polynomial at a point.

Defined by

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

\(H_n\) is a polynomial of degree \(n\). See 22.11.7 in [AS] or [DLMF] for details.

Parameters:

narray_like

Degree of the polynomial

xarray_like

Points at which to evaluate the Hermite polynomial

outndarray, optional

Optional output array for the function values

Returns:

Hscalar or ndarray

Values of the Hermite polynomial

See also

roots_hermite

roots and quadrature weights of physicist’s Hermite polynomials

hermite

physicist’s Hermite polynomial object

numpy.polynomial.hermite.Hermite

Physicist’s Hermite series

eval_hermitenorm

evaluate Probabilist’s Hermite polynomials

References

[AS]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

[DLMF]

NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/18.5.T1

Examples

Try it in your browser!

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

Evaluate the order 3 polynomial at a few values of x.

>>> eval_hermite(3, [0, 0.25, 0.5])
array([-0.   , -2.875, -5.   ])

Plot the polynomials of order 0 through 4.

>>> x = np.linspace(-1.6, 1.6, 800)
>>> fig, ax = plt.subplots(1, 1, constrained_layout=True)
>>> for n in range(5):
...     y = eval_hermite(n, x)
...     ax.plot(x, y, label=f'{n = }')
>>> ax.grid()
>>> ax.set_title("Physicist's Hermite Polynomials")
>>> ax.set_xlabel("$x$")
>>> ax.set_ylabel("$H_n(x)$")
>>> ax.legend(loc='upper left', bbox_to_anchor=(1, 1), shadow=True)
>>> plt.show()

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