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

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Mathematical special functions
Defined in header <cmath>
(1)
float beta ( float x, float y ) ;

double beta ( double x, double y ) ;

long double beta ( long double x, long double y ) ;
(since C++17)
(until C++23)
/* floating-point-type */ beta ( /* floating-point-type */ x,
/* floating-point-type */ y ) ;
(since C++23)
float betaf ( float x, float y ) ;
(2) (since C++17)
long double betal ( long double x, long double y ) ;
(3) (since C++17)
Defined in header <cmath>
template < class Arithmetic1, class Arithmetic2 >
/* common-floating-point-type */ beta ( Arithmetic1 x, Arithmetic2 y ) ;
(A) (since C++17)
1-3) Computes the Beta function of x and y . The library provides overloads of std::beta for all cv-unqualified floating-point types as the type of the parameters x and y . (since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters

x, y - floating-point or integer values

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 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 .

std :: beta ( x, y ) equals std :: beta ( y, x ) .

When x and y are positive integers, std :: 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)
.

The additional overloads are not required to be provided exactly as (A) . They only need to be sufficient to ensure that for their first argument num1 and second argument num2 :

  • If num1 or num2 has type long double , then std :: beta ( num1, num2 ) has the same effect as std :: beta ( static_cast < long double > ( num1 ) ,
    static_cast < long double > ( num2 ) )     .
  • Otherwise, if num1 and/or num2 has type double or an integer type, then std :: beta ( num1, num2 ) has the same effect as std :: beta ( static_cast < double > ( num1 ) ,
    static_cast < double > ( num2 ) )     .
  • Otherwise, if num1 or num2 has type float , then std :: beta ( num1, num2 ) has the same effect as std :: beta ( static_cast < float > ( num1 ) ,
    static_cast < float > ( num2 ) )     .
(until C++23)

If num1 and num2 have arithmetic types, then std :: beta ( num1, num2 ) has the same effect as std :: beta ( static_cast < /* common-floating-point-type */ > ( num1 ) ,
static_cast < /* common-floating-point-type */ > ( num2 ) ) , where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2 , arguments of integer type are considered to have the same floating-point conversion rank as double .

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

(since C++23)

Example

Run this code 
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <numbers>
#include <string>
 
long binom_via_beta(int n, int k)
{
    return std::lround(1 / ((n + 1) * std::beta(n - k + 1, k + 1)));
}
 
long binom_via_gamma(int n, int k)
{
    return std::lround(std::tgamma(n + 1) /
                      (std::tgamma(n - k + 1) * 
                       std::tgamma(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_via_beta(n, k) << ' ';
            assert(binom_via_beta(n, k) == binom_via_gamma(n, k));
        }
        std::cout << '\n';
    }
 
    // A spot-check
    const long double p = 0.123; // a random value in [0, 1]
    const long double q = 1 - p;
    const long double π = std::numbers::pi_v<long double>;
    std::cout << "\n\n" << std::setprecision(19)
              << "β(p,1-p)   = " << std::beta(p, q) << '\n'
              << "π/sin(π*p) = " << π / std::sin(π * p) << '\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
 
β(p,1-p)   = 8.335989149587307836
π/sin(π*p) = 8.335989149587307834

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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