NAME¶

std::legendre,std::legendref,std::legendrel - std::legendre,std::legendref,std::legendrel

Synopsis¶

double legendre( unsigned int n, double x );

double legendre( unsigned int n, float x );
double legendre( unsigned int n, long double x ); (1)
float legendref( unsigned int n, float x );

long double legendrel( unsigned int n, long double x );
double legendre( unsigned int n, IntegralType x ); (2)

1) Computes the unassociated Legendre polynomials of the degree n 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, 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
x - the argument, a value of a floating-point or integral type

Return value¶

If no errors occur, value of the order-n unassociated Legendre polynomial of x, that
is

1
2n
n!

dn
dxn

(x2
- 1)n
, is returned.

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.
* The function is not required to be defined for |x| > 1.
* If n is greater or equal than 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 Legendre polynomials are:

* legendre(0, x) = 1.
* legendre(1, x) = x.
* legendre(2, x) =

1
2

(3x2
- 1).
* legendre(3, x) =

1
2

(5x3
- 3x).
* legendre(4, x) =

1
8

(35x4
- 30x2
+ 3).

Example¶

(works as shown with gcc 6.0)

// Run this code

#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#include <cmath>
#include <iostream>

double P3(double x)
{
return 0.5 * (5 * std::pow(x, 3) - 3 * x);
}

double P4(double x)
{
return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
}

int main()
{
// spot-checks
std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\n'
<< std::legendre(4, 0.25) << '=' << P4(0.25) << '\n';
}

Output:¶

-0.335938=-0.335938
0.157715=0.157715

See also¶

laguerre Laguerre polynomials
laguerref (function)
laguerrel
hermite Hermite polynomials
hermitef (function)
hermitel

External links¶

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