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Beta function

Table of contents: Definitions - Main formulas - Integral representations - Hypergeometric representations - Symmetry - Integer parameters - Recurrence relations

Definitions

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Symbol: BetaFunction B ⁣(a,b)\mathrm{B}\!\left(a, b\right) Beta function
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Symbol: IncompleteBeta Bx ⁣(a,b)\mathrm{B}_{x}\!\left(a, b\right) Incomplete beta function
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Symbol: IncompleteBetaRegularized Ix ⁣(a,b)I_{x}\!\left(a, b\right) Regularized incomplete beta function

Main formulas

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B ⁣(a,b)=Γ ⁣(a)Γ ⁣(b)Γ ⁣(a+b)\mathrm{B}\!\left(a, b\right) = \frac{\Gamma\!\left(a\right) \Gamma\!\left(b\right)}{\Gamma\!\left(a + b\right)}
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Ix ⁣(a,b)=Bx ⁣(a,b)B ⁣(a,b)I_{x}\!\left(a, b\right) = \frac{\mathrm{B}_{x}\!\left(a, b\right)}{\mathrm{B}\!\left(a, b\right)}
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B0 ⁣(a,b)=0\mathrm{B}_{0}\!\left(a, b\right) = 0
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B1 ⁣(a,b)=B ⁣(a,b)\mathrm{B}_{1}\!\left(a, b\right) = \mathrm{B}\!\left(a, b\right)
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I0 ⁣(a,b)=0I_{0}\!\left(a, b\right) = 0
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I1 ⁣(a,b)=1I_{1}\!\left(a, b\right) = 1

Integral representations

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B ⁣(a,b)=01ta1(1t)b1dt\mathrm{B}\!\left(a, b\right) = \int_{0}^{1} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt
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B ⁣(a,b)=20π/2sin2a1 ⁣(t)cos2b1 ⁣(t)dt\mathrm{B}\!\left(a, b\right) = 2 \int_{0}^{\pi / 2} \sin^{2 a - 1}\!\left(t\right) \cos^{2 b - 1}\!\left(t\right) \, dt
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Bx ⁣(a,b)=0xta1(1t)b1dt\mathrm{B}_{x}\!\left(a, b\right) = \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt
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Ix ⁣(a,b)=1B ⁣(a,b)0xta1(1t)b1dtI_{x}\!\left(a, b\right) = \frac{1}{\mathrm{B}\!\left(a, b\right)} \int_{0}^{x} {t}^{a - 1} {\left(1 - t\right)}^{b - 1} \, dt

Hypergeometric representations

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Bx ⁣(a,b)=xaa2F1 ⁣(a,1b,a+1,x)\mathrm{B}_{x}\!\left(a, b\right) = \frac{{x}^{a}}{a} \,{}_2F_1\!\left(a, 1 - b, a + 1, x\right)

Symmetry

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B ⁣(a,b)=B ⁣(b,a)\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(b, a\right)
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Ix ⁣(a,b)=1I1x ⁣(b,a)I_{x}\!\left(a, b\right) = 1 - I_{1 - x}\!\left(b, a\right)
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B ⁣(a,b)B ⁣(a+b,c)=B ⁣(b,c)B ⁣(a,b+c)\mathrm{B}\!\left(a, b\right) \mathrm{B}\!\left(a + b, c\right) = \mathrm{B}\!\left(b, c\right) \mathrm{B}\!\left(a, b + c\right)

Integer parameters

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B ⁣(m,n)=(m1)!(n1)!(m+n1)!\mathrm{B}\!\left(m, n\right) = \frac{\left(m - 1\right)! \left(n - 1\right)!}{\left(m + n - 1\right)!}
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B ⁣(m,n)=1m(m+n1m)\mathrm{B}\!\left(m, n\right) = \frac{1}{m {m + n - 1 \choose m}}
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B ⁣(n,b)={~,b{0,1,n1}1n(n+b1n),otherwise\mathrm{B}\!\left(n, b\right) = \begin{cases} {\tilde \infty}, & -b \in \{0, 1, \ldots n - 1\}\\\frac{1}{n {n + b - 1 \choose n}}, & \text{otherwise}\\ \end{cases}
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B ⁣(n,b)={(1)bb(nb),b{1,2,n}~,otherwise\mathrm{B}\!\left(-n, b\right) = \begin{cases} \frac{{\left(-1\right)}^{b}}{b {n \choose b}}, & b \in \{1, 2, \ldots n\}\\{\tilde \infty}, & \text{otherwise}\\ \end{cases}
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ResaB ⁣(a,b)={(nbn),nZ00,otherwise   where n=a\mathop{\operatorname{Res}}\limits_{a} \mathrm{B}\!\left(a, b\right) = \begin{cases} {n - b \choose n}, & n \in \mathbb{Z}_{\ge 0}\\0, & \text{otherwise}\\ \end{cases}\; \text{ where } n = -a

Recurrence relations

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(a+b)B ⁣(a+1,b)=aB ⁣(a,b)\left(a + b\right) \mathrm{B}\!\left(a + 1, b\right) = a \mathrm{B}\!\left(a, b\right)
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B ⁣(a,b)=B ⁣(a+1,b)+B ⁣(a,b+1)\mathrm{B}\!\left(a, b\right) = \mathrm{B}\!\left(a + 1, b\right) + \mathrm{B}\!\left(a, b + 1\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC