# std::legendre, std::legendref, std::legendrel

 double      legendre( unsigned int n, double x ); double      legendre( unsigned int n, float x ); double      legendre( unsigned int n, long double x ); float       legendref( unsigned int n, float x ); long double legendrel( unsigned int n, long double x ); (1) (since C++17) double      legendre( unsigned int n, Integral x ); (2) (since C++17)
1) Computes the unassociated Legendre polynomials of the degree n and argument x
4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (1) after casting the argument to double.

## Contents

### 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 2nn!
 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 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

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

#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

 laguerrelaguerreflaguerrel(C++17)(C++17)(C++17) Laguerre polynomials (function)  hermitehermitefhermitel(C++17)(C++17)(C++17) Hermite polynomials (function)