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

Header: <cmath>

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

# Declarations

float assoc_legendre ( unsigned int n, unsigned int m, float x );
double assoc_legendre ( unsigned int n, unsigned int m, double x );
long double assoc_legendre ( unsigned int n, unsigned int m, long double x );

(since C++17) (until C++23)

/* floating-point-type */ assoc_legendre( unsigned int n, unsigned int m,
/* floating-point-type */ x );

(since C++23)

float assoc_legendref( unsigned int n, unsigned int m, float x );

(since C++17)

long double assoc_legendrel( unsigned int n, unsigned int m, long double x );

(since C++17)

Additional overloads

template< class Integer >
double assoc_legendre ( unsigned int n, unsigned int m, Integer x );

(since C++17)

# Parameters

• n: the degree of the polynomial, an unsigned integer value
• m: the order of the polynomial, an unsigned integer value
• x: the argument, a floating-point or integer value

# Return value

Note that the Condon-Shortley phase term ((-1)^m)(-1)m is omitted from this definition.

# 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 as boost::math::legendre_p, except that the boost.math definition includes the Condon-Shortley phase term.

The first few associated Legendre polynomials are:

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::assoc_legendre(int_num1, int_num2, num) has the same effect as std::assoc_legendre(int_num1, int_num2, static_cast(num)).

# Example

#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