std::legendre, std::legendref, std::legendrel
| double      legendre( unsigned int n, double x ); double      legendre( unsigned int n, float x ); | (1) | (since C++17) | 
| double      legendre( unsigned int n, Integral x ); | (2) | (since C++17) | 
| Contents | 
[edit] Parameters
| n | - | the degree of the polynomial | 
| x | - | the argument, a value of a floating-point or integral type | 
[edit] Return value
If no errors occur, value of the order-n unassociated Legendre polynomial of x, that is | 1 | 
| 2n n! | 
| dn | 
| dxn | 
-1)n
, is returned.
[edit] 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 nis greater or equal than 128, the behavior is implementation-defined
[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
The first few Legendre polynomials are:
- legendre(0, x) = 1
- legendre(1, x) = x
-  legendre(2, x) = 
 (3x21 2 
 -1)
-  legendre(3, x) = 
 (5x31 2 
 -3x)
-  legendre(4, x) = 
 (35x41 8 
 -30x2
 +3)
[edit] 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
[edit] See also
| (C++17)(C++17)(C++17) | Laguerre polynomials (function) | 
| (C++17)(C++17)(C++17) | Hermite polynomials (function) | 
[edit] External links
Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource.


