hermiteH

Hermite polynomials

Syntax

hermiteH(n,x)

Description

hermiteH(n,x) represents the nth-degree Hermite polynomial at the point x.

Examples

First Five Hermite Polynomials

Find the first five Hermite polynomials for the variable x.

syms x
hermiteH([0 1 2 3 4], x)

ans =
[ 1, 2*x, 4*x^2 - 2, 8*x^3 - 12*x, 16*x^4 - 48*x^2 + 12]

Hermite Polynomials for Numeric and Symbolic Arguments

Depending on whether the input is numeric or symbolic, hermiteH returns numeric or exact symbolic results.

Find the value of the fifth-degree Hermite polynomial at 1/3. Because the input is numeric, hermiteH returns numeric results.

hermiteH(5,1/3)

ans =
   34.2058

Find the same result for exact symbolic input. hermiteH returns an exact symbolic result.

hermiteH(5,sym(1/3))

ans =
8312/243

Plot Hermite Polynomials

Plot the first five Hermite polynomials.

syms x y
fplot(hermiteH(0:4,x))
axis([-2 2 -30 30])
grid on

ylabel('H_n(x)')
legend('H_0(x)', 'H_1(x)', 'H_2(x)', 'H_3(x)', 'H_4(x)', 'Location', 'Best')
title('Hermite polynomials')

Input Arguments

n — Degree of polynomial

nonnegative integer | symbolic variable | symbolic expression | symbolic function | vector | matrix

Degree of the polynomial, specified as a nonnegative integer, symbolic variable, expression, or function, or as a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.

x — Input

number | vector | matrix | array | symbolic number | symbolic variable | symbolic array | symbolic function | symbolic expression

Input, specified as a number, vector, matrix, or array, or a symbolic number, variable, array, function, or expression.

More About

Hermite Polynomials

Hermite polynomials are defined by this recursion formula.

$$H\left(0,x\right)=1,H\left(1,x\right)=2x,H\left(n,x\right)=2xH\left(n-1,x\right)-2\left(n-1\right)H\left(n-2,x\right)$$

Hermite polynomials in MATLAB^® satisfy this normalization.

$${\displaystyle {\int }_{-\infty }^{\infty }{\left(H_n(x)\right)}^2}{\text{e}}^{-x^2}dx=2^n\sqrt{\pi }n!$$

Tips

• hermiteH returns floating-point results for numeric arguments that are not symbolic objects.

• hermiteH acts element-wise on nonscalar inputs.

• At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then hermiteH expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

References

[1] Hochstrasser, U. W. “Orthogonal Polynomials.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.