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

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Numerics library

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C-style checked integer arithmetic

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Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel
assoc_legendreassoc_legendrefassoc_legendrel
betabetafbetal
comp_ellint_1comp_ellint_1fcomp_ellint_1l
comp_ellint_2comp_ellint_2fcomp_ellint_2l
comp_ellint_3comp_ellint_3fcomp_ellint_3l
cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl
cyl_bessel_kcyl_bessel_kfcyl_bessel_kl
cyl_neumanncyl_neumannfcyl_neumannl
ellint_1ellint_1fellint_1l
ellint_2ellint_2fellint_2l
ellint_3ellint_3fellint_3l
expintexpintfexpintl

hermitehermitefhermitel
laguerrelaguerreflaguerrel
legendrelegendreflegendrel
riemann_zetariemann_zetafriemann_zetal
sph_besselsph_besselfsph_bessell
sph_legendresph_legendrefsph_legendrel
sph_neumannsph_neumannfsph_neumannl

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

1-3) Computes the (physicist's)Hermite polynomials of the degreen and argumentx. The library provides overloads ofstd::hermite for all cv-unqualified floating-point types as the type of the parameterx.(since C++23)

A) Additional overloads are provided for all integer types, which are treated asdouble.

[edit]Parameters

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

[edit]Return value

If no errors occur, value of the order-n Hermite polynomial ofx, that is(-1)n
e^x2

dxn

e^-x2, is returned.

[edit]Error handling

Errors may be reported as specified inmath_errhandling.

If the argument is NaN, NaN is returned and domain error is not reported.
Ifn is greater or equal than 128, the behavior is implementation-defined.

[edit]Notes

Implementations that do not support C++17, but supportISO 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 headertr1/cmath and namespacestd::tr1.

An implementation of this function is alsoavailable in boost.math.

The Hermite polynomials are the polynomial solutions of the equationu,,
-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 argumentnum of integer type,std::hermite(int_num, num) has the same effect asstd::hermite(int_num,static_cast<double>(num)).

[edit]Example

Run this code

#include <cmath>#include <iostream> double H3(double x){return8*std::pow(x,3)-12* x;} double H4(double x){return16*std::pow(x,4)-48* x* x+12;} int main(){// spot-checksstd::cout<< std::hermite(3,10)<<'='<< H3(10)<<'\n'<< std::hermite(4,10)<<'='<< H4(10)<<'\n';}

Output:

7880=7880155212=155212

[edit]See also

laguerrelaguerreflaguerrel (C++17)(C++17)(C++17)	Laguerre polynomials (function)[edit]
legendrelegendreflegendrel (C++17)(C++17)(C++17)	Legendre polynomials (function)[edit]