std::beta,std::betaf,std::betal (3) - Linux Manuals

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

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) (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 ∫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

External links

Weisstein,_Eric_W._"Beta_Function." From MathWorld--A Wolfram Web Resource.