Beta

Euler's beta function $B(a, b) = \Gamma(a)\Gamma(b)/\Gamma(a+b)$, a bridge between Gamma and Binomial and the normalizing constant behind the beta distribution.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBeta function↗MathWorldBetaFunction↗WikidataQ468881↗Encyclopedia of MathematicsBeta-function↗Britannicascience/Euler-beta-function↗DLMF5.12.E1↗FungrimBetaFunction✓↗Fungrim entryWolfram LanguageBeta✓↗

Beta(a, b)the Euler beta function $B(a, b)$.

SageMathbeta(a, b)SymPybeta(a, b)mpmathbeta(a, b)Juliaenumeratio_ball((a, b) -> gamma(a) * gamma(b) * rgamma(a + b), a, b)Rustbeta(a, b)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity) -> number	compute-engine	always
(number, number, number?, number?) -> number	analyticreplaces compute-engine	always

Domain: Special functions

Details

• Defined via Gamma: $B(a, b) = \dfrac{\Gamma(a)\,\Gamma(b)}{\Gamma(a+b)}$.
• Symmetric: $B(a, b) = B(b, a)$.
• Integral form: $B(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1}\,dt$.
• Ties to Binomial: $B(k+1, n-k+1) = \dfrac{1}{(n+1)\binom{n}{k}}$.
• compute-engine evaluates Beta exactly for positive-integer a and b (an exact rational result) and detects the poles at nonpositive integers, returning ComplexInfinity.

Plot

Examples

2 more kept as data, checked against the oracles

Basic #

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In$\mathrm{B}(2, 3)$Out$\frac{1}{12}$

The unit square case: $\int_0^1 1\,dt = 1$

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In$\mathrm{B}(1, 1)$Out$1$

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In$\mathrm{B}(3, 3)$Out$\frac{1}{30}$

Threads element-wise over a list argument

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In$\mathrm{B}(\bigl\lbrack1, 2\bigr\rbrack, 2)$Out$\bigl\lbrack\frac{1}{2}, \frac{1}{6}\bigr\rbrack$

$B(5, 4) = \frac{4!\,3!}{8!} = \frac{1}{280}$

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In$\mathrm{B}(5, 4)$Out$\frac{1}{280}$

Machine-precision arguments

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In$\mathrm{B}(2.3, 3.2)$Out$0.054\,029\,791\,748\,357\,237\,815\,4$

Half-integer arguments evaluate exactly through $\Gamma(m + \tfrac12) = \frac{(2m)!}{4^m m!}\sqrt{\pi}$. Compare Binomial at half-integers

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In$\mathrm{B}(\frac{9}{2}, \frac{7}{2})$Out$\frac{5π}{2\,048}$

$B(5/2, 7/2) = \frac{3\pi}{256}$ from the half-integer gammas

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In$\mathrm{B}(\frac{5}{2}, \frac{7}{2})$Out$\frac{3π}{256}$

Inexact complex arguments evaluate numerically, through $\ln\Gamma$

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In$\mathrm{B}(2.5+i, 1-i)$Out$0.083\,107\,783\,669\,911\,63+0.142\,163\,944\,255\,870\,34i$

Three arguments: the incomplete beta $B_z(a, b) = \int_0^z t^{a-1}(1-t)^{b-1}\,dt$, here $B_{1/2}(2, 3) = 11/192$

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In$\mathrm{B}(\frac{1}{2}, 2, 3)$Out$\frac{11}{192}$

Four arguments: the generalized incomplete beta $B_{z_1}(a, b) - B_{z_0}(a, b)$, here $\int_{1/4}^{1/2} t(1-t)^2\,dt = 109/3072$

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In$\mathrm{B}(\frac{1}{4}, \frac{1}{2}, 2, 3)$Out$\frac{109}{3\,072}$

To 30 significant digits, every one correctly rounded

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In$N(\mathrm{B}(\frac{1}{3}, \frac{1}{4}), 30)$Out$6.353\,586\,485\,553\,421\,530\,528\,788\,972\,02$

$B(1/2, 1/2) = \pi$ underlies the arcsine distribution's normalizing constant

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In$\mathrm{B}(0.5, 0.5)$Out$3.141\,592\,653\,589\,793\,238\,46$

Symmetric: $B(a, b) = B(b, a)$

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In$\mathrm{B}(2, 5)=\mathrm{B}(5, 2)$Out$\mathrm{True}$

Ties to Binomial: $B(k+1, n-k+1) = \frac{1}{(n+1)\binom{n}{k}}$, here at $n=8, k=3$. See Binomial

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In$9\mathrm{B}(4, 6)\binom{8}{3}=1$Out$\mathrm{True}$

$B(a, 1) = 1/a$ for symbolic $a$

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In$\mathrm{B}(a, 1)$Out$\frac{1}{a}$

$B(2, b) = \frac{1}{b(b+1)}$; Wolfram leaves this to FunctionExpand

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In$\mathrm{B}(2, b)$Out$\frac{1}{b×(b+1)}$

$\partial_a B(a, b) = B(a, b)\,(\psi(a) - \psi(a+b))$

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In$\frac{\mathrm{d}\mathrm{B}(a, b)}{\mathrm{d}a}$Out$(\mathrm{Digamma}(a)-\mathrm{Digamma}(a+b))\mathrm{B}(a, b)$

A nonpositive-integer argument is a pole of the underlying Gamma functions, so compute-engine returns ComplexInfinity rather than leaving it unevaluated. See Gamma

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In$\mathrm{B}(-1, 2)$Out$\tilde∞$

Properties #

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In$\mathrm{B}(\frac{1}{2}, \frac{1}{2})\mathrm{B}(1, \frac{1}{2})$Out$2π$

Known value · Wolfram MathematicalFunctionData, Beta, FunctionalEquations

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In$\mathrm{B}(\frac{1}{2}, \frac{1}{2})\mathrm{B}(1, 2)$Out$\frac{π}{2}$

Known value · Wolfram MathematicalFunctionData, Beta, FunctionalEquations

Possible issues #

$B(\infty, 0) = \Gamma(\infty)\Gamma(0)/\Gamma(\infty)$ — the $\Gamma(a)$ cancels identically for any finite $a$ and stays cancelled as $a\to\infty$ (Γ smooth and nonzero out there), leaving the lone $\Gamma(0)$: a simple pole, ComplexInfinity, not a double one

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In$\mathrm{B}(∞, 0)$Out$\tilde∞$

About

Examples	24
Checked against	Wolfram Language 23 agreeSageMath 18 agree1 differ2 errorSymPy 19 agree2 differmpmath 19 agree2 differJulia 19 agree2 errorRust 14 agree2 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsBritannicaDLMFFungrimFungrim entry ×11Wolfram LanguageSageMathSymPympmath ×3Julia ×3Rust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

Incomplete beta B(z; a, b), the integral from 0 to z.

mappedmpmathpackages/oracle/src/mappings.ts

Generalized incomplete beta, the integral from z0 to z1.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

a, b > 0 only.

See also: Gamma, Binomial, BetaRegularized