std::assoc_legendre,std::assoc_legendref,std::assoc_legendrel(3)

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

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

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 \(\mathsf{P}_n^m\)Pm
   n of x, that is \((1 - x^2) ^ {m/2} \: \frac{ \mathsf{d} ^ m}{ \mathsf{d}x ^ m} \,
   \mathsf{P}_n(x)\)(1-x2
   )m/2
   dm
   dxm
   P
   n(x), is returned (where \(\mathsf{P}_n(x)\)P
   n(x) is the unassociated Legendre polynomial, std::legendre(n, x)).
   Note that the Condon-Shortley phase term \((-1)^m\)(-1)m
   is omitted from this definition.

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 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 as
   boost::math::legendre_p, except that the boost.math definition includes the
   Condon-Shortley phase term.
   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
// Run this code
 #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
   legendref
   legendrel Legendre polynomials
   (C++17) (function)
   (C++17)
   (C++17)

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