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

 

C++

Compiler support

Freestanding and hosted

Language

Standard library

Standard library headers

Named requirements

Feature test macros (C++20)

Language support library

Concepts library (C++20)

Metaprogramming library (C++11)

Diagnostics library

General utilities library

Strings library

Containers library

Iterators library

Ranges library (C++20)

Algorithms library

Numerics library

Localizations library

Input/output library

Filesystem library (C++17)

Regular expressions library (C++11)

Concurrency support library (C++11)

Execution support library (C++26)

Technical specifications

Symbols index

External libraries

 

Numerics library

Common mathematical functions
Mathematical special functions (C++17)
Mathematical constants (C++20)
Basic linear algebra algorithms (C++26)
Floating-point environment (C++11)
Complex numbers
Numeric array (valarray)
Pseudo-random number generation
Factor operations
gcd (C++17)	lcm (C++17)
Interpolations
midpoint (C++20)	lerp (C++20)
Saturation arithmetic
add_sat (C++26) sub_sat (C++26) saturate_cast (C++26)	mul_sat (C++26) div_sat (C++26)
Generic numeric operations
iota (C++11) ranges::iota (C++23) accumulate inner_product adjacent_difference partial_sum	reduce (C++17) transform_reduce (C++17) inclusive_scan (C++17) exclusive_scan (C++17) transform_inclusive_scan (C++17) transform_exclusive_scan (C++17)
Bit operations
has_single_bit (C++20)     bit_cast (C++20) bit_ceil (C++20) bit_floor (C++20) bit_width (C++20) rotl (C++20) rotr (C++20)	popcount (C++20) countl_zero (C++20) countl_one (C++20) countr_zero (C++20) countr_one (C++20) byteswap (C++23) endian (C++20)

 

Mathematical special functions

assoc_laguerreassoc_laguerrefassoc_laguerrel   assoc_legendreassoc_legendrefassoc_legendrel betabetafbetal comp_ellint_1comp_ellint_1fcomp_ellint_1l comp_ellint_2comp_ellint_2fcomp_ellint_2l comp_ellint_3comp_ellint_3fcomp_ellint_3l cyl_bessel_icyl_bessel_ifcyl_bessel_il

cyl_bessel_jcyl_bessel_jfcyl_bessel_jl   cyl_bessel_kcyl_bessel_kfcyl_bessel_kl cyl_neumanncyl_neumannfcyl_neumannl ellint_1ellint_1fellint_1l ellint_2ellint_2fellint_2l ellint_3ellint_3fellint_3l expintexpintfexpintl

hermitehermitefhermitel laguerrelaguerreflaguerrel legendrelegendreflegendrel riemann_zetariemann_zetafriemann_zetal sph_besselsph_besselfsph_bessell sph_legendresph_legendrefsph_legendrel sph_neumannsph_neumannfsph_neumannl

 

Defined in header <cmath>
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 \(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)∫10tx-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 <cassert>
#include <iomanip>
#include <iostream>
#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(k + 1)
                      / std::tgamma(n + 1 - k));
}
 
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';
    }
}

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

tgammatgammaftgammal (C++11)(C++11)(C++11)

gamma function (function)

External links

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