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

 

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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>
	(1)
float       beta (float x, float y ); double      beta (double x, double y ); longdouble beta (longdouble x, longdouble 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)
longdouble betal(longdouble x, longdouble y );		(3)	(since C++17)
Additional overloads
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 \(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)∫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.

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

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

.

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 longdouble, then std::beta(num1, num2) has the same effect as std::beta(static_cast<longdouble>(num1),           static_cast<longdouble>(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){returnstd::lround(1/((n +1)* std::beta(n - k +1, k +1)));}   long binom_via_gamma(int n, int k){returnstd::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-checkconstlongdouble p =0.123;// a random value in [0, 1]constlongdouble q =1- p;constlongdouble π =std::numbers::pi_v<longdouble>;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

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

gamma function (function)

External links

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