std::assoc_legendre, std::assoc_legendref, std::assoc_legendrel

Computes the associated Legendre polynomials of the degree n, order m, and argument x.

# Declarations

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 );

double assoc_legendre( unsigned int n, unsigned int m, IntegralType x );

# 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

# 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:

# Example

#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';
}

# See also

legendrelegendreflegendrel