# std::hermite, std::hermitef, std::hermitel

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< cpp‎ | experimental‎ | special math

double hermite( unsigned int n, double x ); double hermite( unsigned int n, float x ); double hermite( unsigned int n, long double x ); float hermitef( unsigned int n, float x ); long double hermitel( unsigned int n, long double x ); | (1) |
double hermite( unsigned int n, IntegralType x ); | (2) |

1) Computes the (physicist's) Hermite polynomials of the degree `n` and argument `x`
4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

As all special functions, `hermite` is only guaranteed to be available in `<cmath>` if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

### Parameters

n | - | the degree of the polynomial
x | - | the argument, a value of a floating-point or integral type

### Return value

If no errors occur, value of the order-`n`Hermite polynomial of `x`, that is (-1)n
ex2

dn
dxn

e-x2
, is returned.

### Error handling

Errors may be reported as specified in math_errhandling

- If the argument is NaN, NaN is returned and domain error is not reported
- If `n` is greater or equal than 128, the behavior is implementation-defined

### Notes

Implementations that do not support TR 29124 but support TR 19768, provide this function in the header `tr1/cmath` and namespace `std::tr1`

An implementation of this function is also available in boost.math

The Hermite polynomials are the polynomial solutions of the equation u,,
-2xu,
= -2nu

The first few are:

- hermite(0, x) = 1
- hermite(1, x) = 2x
- hermite(2, x) = 4x2
  -2
- hermite(3, x) = 8x3
  -12x
- hermite(4, x) = 16x4
  -48x2
  +12

### Example

(works as shown with gcc 6.0)

Run this code

    #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
    #include <cmath>
    #include <iostream>
    double H3(double x) { return 8*std::pow(x,3) - 12*x; }
    double H4(double x) { return 16*std::pow(x,4)-48*x*x+12; }
    int main()
    {
        // spot-checks
        std::cout << std::hermite(3, 10) << '=' << H3(10) << '\n'
                  << std::hermite(4, 10) << '=' << H4(10) << '\n';
    }

Output:

    7880=7880
    155212=155212

### See also

laguerre laguerref laguerrel | Laguerre polynomials (function)
legendre legendref legendrel | Legendre polynomials (function)

### External links

Weisstein, Eric W. ""Hermite Polynomial." From MathWorld--A Wolfram Web Resource.

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