NAME

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

Synopsis

Defined in header <cmath>
double beta( double x, double y );

float betaf( float x, float y ); (1) (since C++17)

long double betal( long double x, long double y );
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 \(\int_{0}^{1}{
{t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)∫1
0tx-1
(1-t)(y-1)
dt, or, equivalently, \(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)

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

External links

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