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

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< cpp‎ | experimental‎ | special functions

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.
2) 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-nHermite 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

laguerrelaguerreflaguerrel | Laguerre polynomials (function)
legendrelegendreflegendrel | Legendre polynomials (function)

### External links

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

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