NAME

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

Synopsis

Defined in header <cmath>
double hermite( unsigned int n, double x );

float hermite( unsigned int n, float x );
long double hermite( unsigned int n, long double x ); (1) (since C++17)
float hermitef( unsigned int n, float x );

long double hermitel( unsigned int n, long double x );
double hermite( unsigned int n, IntegralType x ); (2) (since C++17)

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.

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^x2

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

// 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

laguerre
laguerref
laguerrel Laguerre polynomials
(C++17) (function)
(C++17)
(C++17)
legendre
legendref
legendrel Legendre polynomials
(C++17) (function)
(C++17)
(C++17)

External links

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