NAME¶

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

Synopsis¶

double hermite( unsigned int n, double x );

double hermite( unsigned int n, float x );
double hermite( unsigned int n, long double x ); (1)
float hermitef( unsigned int n, float x );

long double hermitel( unsigned int n, long double x );
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
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 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¶

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

External links¶

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