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

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< cpp‎ | numeric‎ | special functions

Defined in header `<cmath>` |  |
| (1) |
double hermite ( unsigned int n, double x ); float hermite ( unsigned int n, float x ); long double hermite ( unsigned int n, long double x ); |  | (since C++17) (until C++23)
/* floating-point-type */ hermite( unsigned int n, /* floating-point-type */ x ); |  | (since C++23)
float hermitef( unsigned int n, float x ); | (2) | (since C++17)
long double hermitel( unsigned int n, long double x ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Integer > double hermite ( unsigned int n, Integer x ); | (A) | (since C++17)

1-3) Computes the (physicist's) Hermite polynomials of the degree n and argument x. The library provides overloads of `std::hermite` for all cv-unqualified floating-point types as the type of the parameter x.(since C++23)
A) Additional overloads are provided for all integer types, which are treated as double.

### Parameters

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

### 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 C++17, but support ISO 29124:2010, provide this function 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.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), 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:

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

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::hermite(int_num, num) has the same effect as std::hermite(int_num, static_cast<double>(num)).

### Example

Run this code

    #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 (C++17)(C++17)(C++17) | Laguerre polynomials (function)
legendrelegendreflegendrel (C++17)(C++17)(C++17) | Legendre polynomials (function)

### External links

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

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