std::beta,std::betaf,std::betal(3)

NAME
std::beta,std::betaf,std::betal - std::beta,std::betaf,std::betal

Synopsis
   double beta( double x, double y );
   float betaf( float x, float y ); (1)
   long double betal( long double x, long double y );
   Promoted beta( Arithmetic x, Arithmetic y ); (2)
   1) Computes the beta function of x and y.
   2) A set of overloads or a function template for all combinations of arguments of
   arithmetic type not covered by (1). If any argument has integral type, it is cast to
   double. If any argument is long double, then the return type Promoted is also long
   double, otherwise the return type is always double.
   As all special functions, beta 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
   x, y - values of a floating-point or integral type

Return value
   If no errors occur, value of the beta function of x and y, that is ∫1
   0tx-1
   (1 - t)(y-1)
   dt, or, equivalently,
   Γ(x)Γ(y)
   Γ(x + y)
   is returned.

Error handling
   Errors may be reported as specified in math_errhandling.
     * If any argument is NaN, NaN is returned and domain error is not reported.
     * The function is only required to be defined where both x and y are greater than
       zero, and is allowed to report a domain error otherwise.

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.
   beta(x, y) equals beta(y, x).
   When x and y are positive integers, beta(x, y) equals \(\frac{(x - 1)!(y - 1)!}{(x +
   y - 1)!}\)
   (x - 1)!(y - 1)!
   (x + y - 1)!
   . Binomial coefficients can be expressed in terms of the beta function:
   \(\binom{n}{k} = \frac{1}{(n + 1)B(n - k + 1, k + 1)}\)⎛
   ⎜
   ⎝n
   k⎞
   ⎟
   ⎠=
   1
   (n + 1)Β(n - k + 1, k + 1)
   .

Example
   (works as shown with gcc 6.0) // Run this code
 #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
 #include <cmath>
 #include <iomanip>
 #include <iostream>
 #include <string>
 double binom(int n, int k)
 {
     return 1 / ((n + 1) * std::beta(n - k + 1, k + 1));
 }
 int main()
 {
     std::cout << "Pascal's triangle:\n";
     for (int n = 1; n < 10; ++n)
     {
         std::cout << std::string(20 - n * 2, ' ');
         for (int k = 1; k < n; ++k)
             std::cout << std::setw(3) << binom(n, k) << ' ';
         std::cout << '\n';
     }
 }

Output:
 Pascal's triangle:
                   2
                 3 3
               4 6 4
             5 10 10 5
           6 15 20 15 6
         7 21 35 35 21 7
       8 28 56 70 56 28 8
     9 36 84 126 126 84 36 9

See also
   tgamma
   tgammaf
   tgammal gamma function
   (C++11) (function)
   (C++11)
   (C++11)

External links
   Weisstein, Eric W. "Beta Function." From MathWorld--A Wolfram Web Resource.