# Beta function

The beta function, written \( \mathrm{B}(a,b) \), is a two-variable special function defined for \( \Re a>0 \) and \( \Re b>0 \) by Euler's beta integral,

\[ \mathrm{B}(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1}\,dt = \frac{\Gamma(a)\,\Gamma(b)}{\Gamma(a+b)}, \]

where \( \Gamma \) is the gamma function.<sup>[1](https://dlmf.nist.gov/5.12)</sup> It is also called the Euler integral of the first kind, or Euler's \( B \)-function.<sup>[2](https://encyclopediaofmath.org/wiki/Beta-function)</sup> The identity with the gamma function is its most used property: it converts a large class of definite integrals into quotients of gamma values.<sup>[3](https://www.cambridge.org/core/books/abs/special-functions/gamma-and-beta-functions/433931F88B43817757346B22E927B2EB)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \mathrm{B}(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}\,dt \), convergent for \( \Re a>0 \), \( \Re b>0 \)<sup>[1](https://dlmf.nist.gov/5.12)</sup> |
| Gamma relation | \( \mathrm{B}(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b) \)<sup>[1](https://dlmf.nist.gov/5.12)</sup> |
| Symmetry and recurrence | \( \mathrm{B}(p,q)=\mathrm{B}(q,p) \); \( \mathrm{B}(p,q)=\frac{q-1}{p+q-1}\mathrm{B}(p,q-1) \) for \( q>1 \)<sup>[2](https://encyclopediaofmath.org/wiki/Beta-function)</sup> |
| Reflection value | \( \mathrm{B}(p,1-p)=\pi/\sin(p\pi) \) for \( 0<p<1 \); in particular \( \mathrm{B}(1/2,1/2)=\pi \)<sup>[2](https://encyclopediaofmath.org/wiki/Beta-function)</sup><sup> • </sup><sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/05%3A_Special_Distributions/5.17%3A_The_Beta_Distribution)</sup> |
| Integer case | \( \mathrm{B}(j,k)=(j-1)!(k-1)!/(j+k-1)! \) for positive integers \( j,k \)<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/05%3A_Special_Distributions/5.17%3A_The_Beta_Distribution)</sup> |
| History | Integral computed by Euler in 1730; earlier tabulated by Pietro Mengoli (1659, 1672); named "beta function" by Jacques Binet in 1839<sup>[5](https://web.mat.upc.edu/amadeu.delshams/articles/MassaD09.pdf)</sup><sup> • </sup><sup>[6](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/functions/)</sup> |
| Statistics | The regularized incomplete beta function \( I_x(a,b) \) is the cumulative distribution function of the beta distribution<sup>[7](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betainc.html)</sup> |

## Basic properties

The integral converges when the real parts of both parameters are positive; for complex \( p \) and \( q \) this is the stated condition.<sup>[2](https://encyclopediaofmath.org/wiki/Beta-function)</sup> The function is symmetric in its two arguments, which follows from the substitution \( t \mapsto 1-t \) in the integral, and it satisfies the recurrence \( \mathrm{B}(p,q)=\frac{q-1}{p+q-1}\mathrm{B}(p,q-1) \) for \( q>1 \).<sup>[2](https://encyclopediaofmath.org/wiki/Beta-function)</sup><sup> • </sup><sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/05%3A_Special_Distributions/5.17%3A_The_Beta_Distribution)</sup> A simple special case is \( \mathrm{B}(1,b)=1/b \).<sup>[8](https://scipy.github.io/devdocs/reference/generated/scipy.special.beta.html)</sup>

For positive integers the gamma relation reduces to a factorial quotient, \( \mathrm{B}(j,k)=(j-1)!(k-1)!/(j+k-1)! \), so the beta function generalizes binomial-coefficient reciprocals to non-integer parameters.<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/05%3A_Special_Distributions/5.17%3A_The_Beta_Distribution)</sup> The reflection value \( \mathrm{B}(p,1-p)=\pi/\sin(p\pi) \) gives \( \mathrm{B}(1/2,1/2)=\pi \), the value behind Mengoli's quadrature of the circle.<sup>[2](https://encyclopediaofmath.org/wiki/Beta-function)</sup><sup> • </sup><sup>[5](https://web.mat.upc.edu/amadeu.delshams/articles/MassaD09.pdf)</sup>

## Integral representations

The DLMF lists several equivalent integrals. Besides the defining integral on \( (0,1) \), an alternative over the half-line is

\[ \int_0^{\infty} \frac{t^{a-1}\,dt}{(1+t)^{a+b}} = \mathrm{B}(a,b), \]

and a trigonometric form gives \( \int_0^{\pi/2} \sin^{2a-1}\theta\,\cos^{2b-1}\theta\,d\theta = \tfrac{1}{2}\mathrm{B}(a,b) \).<sup>[1](https://dlmf.nist.gov/5.12)</sup> Further representations include integrals involving \( \cos(bt) \), hyperbolic cosines, and contour (loop) integrals such as Pochhammer's integral, which extend the function to complex parameters.<sup>[1](https://dlmf.nist.gov/5.12)</sup>

## Analytic continuation

The gamma quotient \( \Gamma(a)\Gamma(b)/\Gamma(a+b) \) defines the beta function beyond the half-planes \( \Re a>0 \), \( \Re b>0 \): it extends \( \mathrm{B} \) to a meromorphic function of both variables, holomorphic wherever neither \( a \) nor \( b \) is a pole of \( \Gamma \).<sup>[9](https://link.springer.com/chapter/10.1007/978-3-030-54533-8_10)</sup> The gamma function itself is meromorphic with simple poles at the non-positive integers and no zeros, so the poles of \( \mathrm{B} \) come from the gamma factors in the quotient.<sup>[10](https://dlmf.nist.gov/5.2)</sup>

## Incomplete beta function and statistics

Replacing the upper limit 1 by \( x \in (0,1) \) gives the incomplete beta function \( \mathrm{B}(x;a,b)=\int_0^x u^{a-1}(1-u)^{b-1}\,du \).<sup>[4](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/05%3A_Special_Distributions/5.17%3A_The_Beta_Distribution)</sup> Scaling it by the reciprocal \( \Gamma(a+b)/(\Gamma(a)\Gamma(b)) \) gives the regularized incomplete beta function

\[ I_x(a,b) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt, \]

which is the cumulative distribution function of the beta distribution on \( [0,1] \).<sup>[7](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betainc.html)</sup> The beta distribution's density is \( f(x)=x^{a-1}(1-x)^{b-1}/\mathrm{B}(a,b) \), so the beta function appears as the normalizing constant that makes the density integrate to one.<sup>[11](https://mude.citg.tudelft.nl/book/2026/univariate_distributions/beta.html)</sup> Numerical libraries expose these pieces separately: SciPy provides `beta` for \( \mathrm{B}(a,b) \), `betainc` for the regularized incomplete function, and `betaln` for \( \ln|\mathrm{B}(a,b)| \), which avoids underflow when the parameters are large (for example, \( \beta(400,900) \) underflows to 0 in double precision while \( \mathrm{betaln}(400,900) \approx -804.3 \)).<sup>[8](https://scipy.github.io/devdocs/reference/generated/scipy.special.beta.html)</sup><sup> • </sup><sup>[12](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betaln.html)</sup>

## Generalizations

Two standard extensions appear in the DLMF. The q-beta function \( \mathrm{B}_q(a,b) \) belongs to the theory of \( q \)-gamma and \( q \)-hypergeometric functions and uses the \( q \)-[Pochhammer symbol](https://www.edgepedia.org/pochhammer-symbol) and \( q \)-differential in its integral form.<sup>[13](https://dlmf.nist.gov/5.18)</sup> The multivariate beta function \( \mathrm{B}_m(a,b) \), defined together with the multivariate gamma function \( \Gamma_m \), integrates over the space of positive-definite real symmetric \( m \times m \) matrices and appears in functions of matrix argument.<sup>[14](https://visiblecement.nist.gov/35.3)</sup> Hyperbolic and elliptic beta integrals replace the gamma function with other special functions while keeping the structure of Euler's integral.<sup>[15](https://arxiv.org/abs/math/0303178)</sup>

## History

[Leonhard Euler](https://www.edgepedia.org/leonhard-euler) calculated beta integrals for several non-integer exponents in 1730, in the work *De progressionibus transcendentibus*, while searching for a general term for the factorial function by algebraic means.<sup>[5](https://web.mat.upc.edu/amadeu.delshams/articles/MassaD09.pdf)</sup> Seventy years earlier, Pietro Mengoli (1626–1686) had computed the same integrals for natural and half-integer exponents in *Geometriae Speciosae Elementa* (1659) and *Circolo* (1672), displaying the values in triangular tables; his method yielded the quadrature of the circle, corresponding to \( \mathrm{B}(3/2,3/2)=\pi/8 \).<sup>[5](https://web.mat.upc.edu/amadeu.delshams/articles/MassaD09.pdf)</sup> [Adrien-Marie Legendre](https://www.edgepedia.org/adrien-marie-legendre) called the two Euler integrals the Eulerian integrals of the first and second kind, and the names beta function and gamma function were assigned by Jacques Binet and Gauss respectively.<sup>[16](https://mathshistory.st-andrews.ac.uk/Biographies/Euler/?level=1)</sup> Binet introduced the symbol \( B \) in 1839, in a memoir on the Eulerian definite integrals.<sup>[6](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/functions/)</sup> A historical study notes that Euler did not regard the beta integral as a true function but only as an integral; the modern function viewpoint came later.<sup>[5](https://web.mat.upc.edu/amadeu.delshams/articles/MassaD09.pdf)</sup>

## Applications

The beta function is used across mathematics, physics, and engineering, largely through the gamma relation for evaluating definite integrals.<sup>[17](https://www.sciencedirect.com/topics/mathematics/beta-function)</sup> In physics, Veneziano's 1968 observation that particle scattering amplitudes could be described by the beta function was an important early development in string theory.<sup>[18](https://dlmf.nist.gov/5.20)</sup> In statistics, the beta distribution built on \( \mathrm{B}(a,b) \) models quantities with natural lower and upper bounds, such as proportions and fractions.<sup>[11](https://mude.citg.tudelft.nl/book/2026/univariate_distributions/beta.html)</sup> In combinatorics, beta integral techniques give a systematic route to sums involving reciprocals of binomial coefficients, connecting discrete identities with hypergeometric functions.<sup>[19](https://arxiv.org/abs/2604.04566)</sup>

## References

1. [DLMF §5.12 Beta Function, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/5.12)
2. [Beta-function, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Beta-function)
3. [Andrews, Askey, Roy (1999). The Gamma and Beta Functions, Chapter 1 of *Special Functions*, Cambridge University Press](https://www.cambridge.org/core/books/abs/special-functions/gamma-and-beta-functions/433931F88B43817757346B22E927B2EB)
4. [The Beta Distribution, Statistics LibreTexts](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/05%3A_Special_Distributions/5.17%3A_The_Beta_Distribution)
5. [Massa Esteve, M. R., Delshams, A. (2009). Euler's beta integral in Pietro Mengoli's works, Archive for History of Exact Sciences 63:325–356](https://web.mat.upc.edu/amadeu.delshams/articles/MassaD09.pdf)
6. [Earliest Uses of Function Symbols, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Miller/mathsym/functions/)
7. [scipy.special.betainc, SciPy documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betainc.html)
8. [scipy.special.beta, SciPy documentation](https://scipy.github.io/devdocs/reference/generated/scipy.special.beta.html)
9. [Beals, R., Wong, R. S. C. (2020). The gamma and beta functions, in *Explorations in Complex Functions*, Springer](https://link.springer.com/chapter/10.1007/978-3-030-54533-8_10)
10. [DLMF §5.2 Definitions, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/5.2)
11. [Beta distribution, MUDE textbook, Delft University of Technology](https://mude.citg.tudelft.nl/book/2026/univariate_distributions/beta.html)
12. [scipy.special.betaln, SciPy documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.betaln.html)
13. [DLMF §5.18 q-Gamma and q-Beta Functions, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/5.18)
14. [DLMF §35.3 Multivariate Gamma and Beta Functions, NIST Digital Library of Mathematical Functions](https://visiblecement.nist.gov/35.3)
15. [Ruijsenaars, S. N. M. (2003). Hyperbolic beta integrals, arXiv:math/0303178](https://arxiv.org/abs/math/0303178)
16. [Leonhard Euler biography, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Euler/?level=1)
17. [Beta Function, an overview, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/beta-function)
18. [DLMF §5.20 Physical Applications, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/5.20)
19. [Pain, J.-C. (2026). Reciprocal binomial sums via Beta integrals, arXiv:2604.04566](https://arxiv.org/abs/2604.04566)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Number theory › Analytic number theory › Zeta and L-functions › Gamma, polygamma, and related special functions*

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