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

From cppreference.com
C++
Experimental
Mathematical 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- n Hermite polynomial of x , that is (-1) n e x 2
d n
dx n
e -x 2 , 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) = 4x 2 - 2 .
  • hermite(3, x) = 8x 3 - 12x .
  • hermite(4, x) = 16x 4 - 48x 2 + 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 polynomials
(function)
Legendre polynomials
(function)

External links

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

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