std:: beta, std:: betaf, std:: betal

From cppreference.com
C++
Experimental
Mathematical special functions
double beta ( double x, double y ) ;

float betaf ( float x, float y ) ;

long double betal ( long double x, long double y ) ;
(1)
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 ∫ 10 t x-1 (1 - t) (y-1) d t , 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
(x - 1)!(y - 1)!
(x + y - 1)!
. Binomial coefficients can be expressed in terms of the beta function: ⎛
⎜
⎝ 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

(C++11) (C++11) (C++11)
gamma function
(function)

External links

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

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