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

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< cpp‎ | numeric‎ | special math

double beta( double x, double y ); float betaf( float x, float y ); long double betal( long double x, long double y ); | (1) | (since C++17)
Promoted beta( Arithmetic x, Arithmetic y ); | (2) | (since C++17)

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.

### 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 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

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

Run this code

    #include <cmath>
    #include <string>
    #include <iostream>
    #include <iomanip>
    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 (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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