NAME¶

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

Synopsis¶

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 ); (1)
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 ); (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 Pm
n of x, that is (1 - x2
)m/2

dm
dxm

P
n(x), is returned (where P
n(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¶

legendre Legendre polynomials
legendref (function)
legendrel

External links¶

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