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

 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) (since C++17) double      hermite( unsigned int n, Integral x ); (2) (since C++17)
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.

## Contents

### 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 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:

• 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

#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