std::legendre,std::legendref,std::legendrel

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

1-3) Computes the unassociatedLegendre polynomials of the degreen and argumentx. The library provides overloads ofstd::legendre 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 unassociated Legendre polynomial ofx, that is\(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)

2n
n!

dxn

(x2
-1)n
, 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
The function is not required to be defined for|x|>1
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 first few Legendre polynomials are:

Function

Polynomial

legendre(0, x)

legendre(1, x)

legendre(2, x)

(3x2
- 1)

legendre(3, x)

(5x3
- 3x)

legendre(4, x)

(35x4
- 30x2
+ 3)

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::legendre(int_num, num) has the same effect asstd::legendre(int_num,static_cast<double>(num)).

[edit]Example

Run this code

#include <cmath>#include <iostream> double P3(double x){return0.5*(5*std::pow(x,3)-3* x);} double P4(double x){return0.125*(35*std::pow(x,4)-30* x* x+3);} int main(){// spot-checksstd::cout<< std::legendre(3,0.25)<<'='<< P3(0.25)<<'\n'<< std::legendre(4,0.25)<<'='<< P4(0.25)<<'\n';}

Output:

-0.335938=-0.3359380.157715=0.157715

[edit]See also

laguerrelaguerreflaguerrel (C++17)(C++17)(C++17)	Laguerre polynomials (function)[edit]
hermitehermitefhermitel (C++17)(C++17)(C++17)	Hermite polynomials (function)[edit]