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Showing content from https://en.cppreference.com/w/cpp/language/../../cpp/numeric/special_functions/assoc_legendre.html below:

std::assoc_legendre, std::assoc_legendref, std::assoc_legendrel - cppreference.com

(1) 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 );

(2) (since C++17)

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

(3) (since C++17)

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

(A) (since C++17) 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)

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

[edit] 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 [edit] Return value

If no errors occur, value of the associated Legendre polynomial

\(\mathsf{P}_n^m\)Pm
n

of

x

, that is

\((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \, \mathsf{P}_n(x)\)(1-x2
)m/2
 Pn(x)

, is returned (where

\(\mathsf{P}_n(x)\)Pn(x)

is the unassociated Legendre polynomial,

std::legendre(n, x)

).

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

[edit] Error handling

Errors may be reported as specified in math_errhandling

[edit] 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:

Function Polynomial     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
)

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<double>(num)).

[edit] 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';
}

Output:

-0.125=-0.125
1.29904=1.29904
2.25=2.25
[edit] See also [edit] External links

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