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hermiteH

Hermite polynomials

Description

example

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

Examples

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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]

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 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

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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.

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

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Hermite Polynomials

Hermite polynomials are defined by this recursion formula.

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

Hermite polynomials in MATLAB® satisfy this normalization.

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

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.

 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.

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