std::legendre,std::legendref,std::legendrel(3)

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

Synopsis
   Defined in header <cmath>
   double legendre( unsigned int n, double x );
   float legendre( unsigned int n, float x );
   long double legendre( unsigned int n, long double x ); (1) (since C++17)
   float legendref( unsigned int n, float x );
   long double legendrel( unsigned int n, long double x );
   double legendre( unsigned int n, IntegralType x ); (2) (since C++17)
   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.

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 \(\mathsf{P}_n(x) = \frac{1}{2^n n!} \frac{\mathsf{d}^n}{\mathsf{d}x^n} (x^2-1)^n
   \)
   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 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
// Run this code
 #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
   laguerref
   laguerrel Laguerre polynomials
   (C++17) (function)
   (C++17)
   (C++17)
   hermite
   hermitef
   hermitel Hermite polynomials
   (C++17) (function)
   (C++17)
   (C++17)

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