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

 

Defined in header <cmath>
	(1)
float       legendre ( unsigned int n, float x ); double      legendre ( unsigned int n, double x ); long double legendre ( unsigned int n, long double x );			(since C++17) (until C++23)
/* floating-point-type */ legendre( unsigned int n,                                     /* floating-point-type */ x );			(since C++23)
float       legendref( unsigned int n, float x );		(2)	(since C++17)
long double legendrel( unsigned int n, long double x );		(3)	(since C++17)
Additional overloads
Defined in header <cmath>
template< class Integer > double      legendre ( unsigned int n, Integer x );		(A)	(since C++17)

1-3) Computes the unassociated Legendre polynomials of the degree n and argument x. The library provides overloads of std::legendre for all cv-unqualified floating-point types as the type of the parameter x. (since C++23)

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

Parameters

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

Return value

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n \)

1

2n n!

dn

dxn

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

Function	Polynomial
legendre(0, x)	1
legendre(1, x)	x
legendre(2, x)	1 2 (3x2 - 1)
legendre(3, x)	1 2 (5x3 - 3x)
legendre(4, x)	1 8 (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 argument num of integer type, std::legendre(int_num, num) has the same effect as std::legendre(int_num, static_cast<double>(num)).

Example

Run this code

#include <cmath>
#include <iostream>
 
double P3(double x)
{
    return 0.5 * (5 * std::pow(x, 3) - 3 * x);
}
 
double P4(double x)
{
    return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
}
 
int main()
{
    // spot-checks
    std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
              << std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
}

Output:

-0.335938=-0.335938
0.157715=0.157715

See also

laguerrelaguerreflaguerrel (C++17)(C++17)(C++17)	Laguerre polynomials (function)
hermitehermitefhermitel (C++17)(C++17)(C++17)	Hermite polynomials (function)

External links

Weisstein, Eric W. "Legendre Polynomial." From MathWorld — A Wolfram Web Resource.