std::assoc_legendre, std::assoc_legendref, std::assoc_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

 

double      assoc_legendre( unsigned int n, unsigned int m, double x ); double      assoc_legendre( unsigned int n, unsigned int m, float x ); double      assoc_legendre( unsigned int n, unsigned int m, long double x ); float       assoc_legendref( unsigned int n, unsigned int m, float x ); long double assoc_legendrel( unsigned int n, unsigned int m, long double x );	(1)
double      assoc_legendre( unsigned int n, unsigned int m, IntegralType x );	(2)

1) Computes the associated Legendre polynomials of the degree n, order m, 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, assoc_legendre 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, a value of unsigned integer type
m	-	the order of the polynomial, a value of unsigned integer type
x	-	the argument, a value of a floating-point or integral type

Return value

If no errors occur, value of the associated Legendre polynomial Pmn of x, that is (1 - x2)m/2

dm

dxm

Pn(x), is returned (where Pn(x) is the unassociated Legendre polynomial, std::legendre(n, x)).

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 |x| > 1, a domain error may occur.
• If n is greater or equal to 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 first few associated Legendre polynomials are:

• assoc_legendre(0, 0, x) = 1.
• assoc_legendre(1, 0, x) = x.
• assoc_legendre(1, 1, x) = -(1 - x2)1/2.
• assoc_legendre(2, 0, x) =

  1

  2

  (3x2 - 1).
• assoc_legendre(2, 1, x) = -3x(1 - x2)1/2.
• assoc_legendre(2, 2, x) = 3(1 - x2).

Example

(works as shown with gcc 6.0)

Run this code

#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#include <cmath>
#include <iostream>
 
double P20(double x)
{
    return 0.5 * (3 * x * x - 1);
}
 
double P21(double x)
{
    return -3.0 * x * std::sqrt(1 - x * x);
}
 
double P22(double x)
{
    return 3 * (1 - x * x);
}
 
int main()
{
    // spot-checks
    std::cout << std::assoc_legendre(2, 0, 0.5) << '=' << P20(0.5) << '\n'
              << std::assoc_legendre(2, 1, 0.5) << '=' << P21(0.5) << '\n'
              << std::assoc_legendre(2, 2, 0.5) << '=' << P22(0.5) << '\n';
}

Output:

-0.125=-0.125
-1.29904=-1.29904
2.25=2.25

See also

legendrelegendreflegendrel

Legendre polynomials (function)

External links

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