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orthpoly::hermite

The Hermite polynomials

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

```orthpoly::hermite(n, x)
```

Description

orthpoly::hermite(n,x) computes the value of the n-th degree Hermite polynomial at the point x.

These polynomials have integer coefficients.

Examples

Example 1

Polynomials of domain type DOM_POLY are returned, if identifiers or indexed identifiers are specified:

`orthpoly::hermite(2, x)`

`orthpoly::hermite(3, x[1])`

However, using arithmetical expressions as input the "values" of these polynomials are returned:

`orthpoly::hermite(2, 6*x)`

`orthpoly::hermite(3, x[1] + 2)`

"Arithmetical expressions" include numbers:

```orthpoly::hermite(2, sqrt(2)), orthpoly::hermite(3, 8 + I),
orthpoly::hermite(1000, 0.3);```

If the degree of the polynomial is a variable or expression, then orthpoly::hermite returns itself symbolically:

`orthpoly::hermite(n, x)`

Parameters

 n A nonnegative integer or an arithmetical expression representing a nonnegative integer: the degree of the polynomial. x An indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type DOM_IDENT) or an indexed identifier (of type "_index").

Return Values

If x is an indeterminate, then a polynomial of domain type DOM_POLY is returned. If x is an arithmetical expression, then the value of the Hermite polynomial at this point is returned as an arithmetical expression. If n is an arithmetical expression, then orthpoly::hermite returns itself symbolically.

Algorithms

The Hermite polynomials are given by the recursion formula

with H(0, x) = 1 and H(1, x) = 2 x.

These polynomials are orthogonal on the real line with respect to the weight function .