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Beta function

From Wikipedia, the free encyclopedia
(Redirected fromRegularized incomplete beta function)
Mathematical function
This article is about the Euler beta function. For other uses, seeBeta function (disambiguation).
Contour plot of the beta function

Inmathematics, thebeta function, also called theEuler integral of the first kind, is aspecial function that is closely related to thegamma function and tobinomial coefficients. It is defined by theintegral

B(z1,z2)=∫01tz1−1(1−t)z2−1dt{\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}forcomplex number inputsz1,z2{\displaystyle z_{1},z_{2}} such thatRe⁡(z1),Re⁡(z2)>0{\displaystyle \operatorname {Re} (z_{1}),\operatorname {Re} (z_{2})>0}.

The beta function was studied byLeonhard Euler andAdrien-Marie Legendre and was given its name byJacques Binet; its symbolΒ is aGreek capitalbeta.

Properties

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The beta function issymmetric, meaning thatB(z1,z2)=B(z2,z1){\displaystyle \mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{2},z_{1})} for all inputsz1{\displaystyle z_{1}} andz2{\displaystyle z_{2}}.[1]

A key property of the beta function is its close relationship to thegamma function:[1]

B(z1,z2)=Γ(z1)Γ(z2)Γ(z1+z2){\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\,\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}

A proof is given below in§ Relationship to the gamma function.

The beta function is also closely related tobinomial coefficients. Whenm (orn, by symmetry) is a positive integer, it follows from the definition of the gamma functionΓ that[1]

B(m,n)=(m−1)!(n−1)!(m+n−1)!=m+nmn/(m+nm){\displaystyle \mathrm {B} (m,n)={\frac {(m-1)!\,(n-1)!}{(m+n-1)!}}={\frac {m+n}{mn}}{\Bigg /}{\binom {m+n}{m}}}

Relationship to the gamma function

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To derive this relation, write the product of two factorials as integrals. Since they are integrals in two separate variables, we can combine them into aniterated integral:

Γ(z1)Γ(z2)=∫u=0∞ e−uuz1−1du⋅∫v=0∞ e−vvz2−1dv=∫v=0∞∫u=0∞ e−u−vuz1−1vz2−1dudv.{\displaystyle {\begin{aligned}\Gamma (z_{1})\Gamma (z_{2})&=\int _{u=0}^{\infty }\ e^{-u}u^{z_{1}-1}\,du\cdot \int _{v=0}^{\infty }\ e^{-v}v^{z_{2}-1}\,dv\\[6pt]&=\int _{v=0}^{\infty }\int _{u=0}^{\infty }\ e^{-u-v}u^{z_{1}-1}v^{z_{2}-1}\,du\,dv.\end{aligned}}}

Changing variables byu =st andv =s(1 −t), becauseu + v =s andu / (u+v) =t, we have that the limits of integrations fors are 0 to ∞ and the limits of integration fort are 0 to 1. Thus produces

Γ(z1)Γ(z2)=∫s=0∞∫t=01e−s(st)z1−1(s(1−t))z2−1sdtds=∫s=0∞e−ssz1+z2−1ds⋅∫t=01tz1−1(1−t)z2−1dt=Γ(z1+z2)⋅B(z1,z2).{\displaystyle {\begin{aligned}\Gamma (z_{1})\Gamma (z_{2})&=\int _{s=0}^{\infty }\int _{t=0}^{1}e^{-s}(st)^{z_{1}-1}(s(1-t))^{z_{2}-1}s\,dt\,ds\\[6pt]&=\int _{s=0}^{\infty }e^{-s}s^{z_{1}+z_{2}-1}\,ds\cdot \int _{t=0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt\\[1ex]&=\Gamma (z_{1}+z_{2})\cdot \mathrm {B} (z_{1},z_{2}).\end{aligned}}}

Dividing both sides byΓ(z1+z2){\displaystyle \Gamma (z_{1}+z_{2})} gives the desired result.

The stated identity may be seen as a particular case of the identity for theintegral of a convolution. Taking

f(u):=e−uuz1−11R+g(u):=e−uuz2−11R+,{\displaystyle {\begin{aligned}f(u)&:=e^{-u}u^{z_{1}-1}1_{\mathbb {R} _{+}}\\g(u)&:=e^{-u}u^{z_{2}-1}1_{\mathbb {R} _{+}},\end{aligned}}}

one has:

Γ(z1)Γ(z2)=∫Rf(u)du⋅∫Rg(u)du=∫R(f∗g)(u)du=B(z1,z2)Γ(z1+z2).{\displaystyle \Gamma (z_{1})\Gamma (z_{2})=\int _{\mathbb {R} }f(u)\,du\cdot \int _{\mathbb {R} }g(u)\,du=\int _{\mathbb {R} }(f*g)(u)\,du=\mathrm {B} (z_{1},z_{2})\,\Gamma (z_{1}+z_{2}).}

SeeThe Gamma Function, page 18–19[2] for a derivation of this relation.

Differentiation of the beta function

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We have

∂∂z1B(z1,z2)=B(z1,z2)(Γ′(z1)Γ(z1)−Γ′(z1+z2)Γ(z1+z2))=B(z1,z2)(ψ(z1)−ψ(z1+z2)),{\displaystyle {\frac {\partial }{\partial z_{1}}}\mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{1},z_{2})\left({\frac {\Gamma '(z_{1})}{\Gamma (z_{1})}}-{\frac {\Gamma '(z_{1}+z_{2})}{\Gamma (z_{1}+z_{2})}}\right)=\mathrm {B} (z_{1},z_{2}){\big (}\psi (z_{1})-\psi (z_{1}+z_{2}){\big )},}

∂∂zmB(z1,z2,…,zn)=B(z1,z2,…,zn)(ψ(zm)−ψ(∑k=1nzk)),1≤m≤n,{\displaystyle {\frac {\partial }{\partial z_{m}}}\mathrm {B} (z_{1},z_{2},\dots ,z_{n})=\mathrm {B} (z_{1},z_{2},\dots ,z_{n})\left(\psi (z_{m})-\psi {\left(\sum _{k=1}^{n}z_{k}\right)}\right),\quad 1\leq m\leq n,}

whereψ(z){\displaystyle \psi (z)} denotes thedigamma function.

Approximation

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Stirling's approximation gives the asymptotic formula

B(x,y)∼2πxx−1/2yy−1/2(x+y)x+y−1/2{\displaystyle \mathrm {B} (x,y)\sim {\sqrt {2\pi }}{\frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}

for largex and largey.

If on the other handx is large andy is fixed, then

B(x,y)∼Γ(y)x−y.{\displaystyle \mathrm {B} (x,y)\sim \Gamma (y)\,x^{-y}.}

Other identities and formulas

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The integral defining the beta function may be rewritten in a variety of ways, including the following:B(z1,z2)=2∫0π/2(sin⁡θ)2z1−1(cos⁡θ)2z2−1dθ,=∫0∞tz1−1(1+t)z1+z2dt,=n∫01tnz1−1(1−tn)z2−1dt,=(1−a)z2∫01(1−t)z1−1tz2−1(1−at)z1+z2dtfor any a∈R≤1,{\displaystyle {\begin{aligned}\mathrm {B} (z_{1},z_{2})&=2\int _{0}^{\pi /2}(\sin \theta )^{2z_{1}-1}(\cos \theta )^{2z_{2}-1}\,d\theta ,\\[6pt]&=\int _{0}^{\infty }{\frac {t^{z_{1}-1}}{(1+t)^{z_{1}+z_{2}}}}\,dt,\\[6pt]&=n\int _{0}^{1}t^{nz_{1}-1}(1-t^{n})^{z_{2}-1}\,dt,\\&=(1-a)^{z_{2}}\int _{0}^{1}{\frac {(1-t)^{z_{1}-1}t^{z_{2}-1}}{(1-at)^{z_{1}+z_{2}}}}dt\qquad {\text{for any }}a\in \mathbb {R} _{\leq 1},\end{aligned}}}

where in the second-to-last identityn is any positive real number. One may move from the first integral to the second one by substitutingt=tan2⁡(θ){\displaystyle t=\tan ^{2}(\theta )}.

For valuesz=z1=z2≠1{\displaystyle z=z_{1}=z_{2}\neq 1} we have:

B(z,z)=1z∫0π/21(sin⁡θz+cos⁡θz)2zdθ{\displaystyle \mathrm {B} (z,z)={\frac {1}{z}}\int _{0}^{\pi /2}{\frac {1}{\left({\sqrt[{z}]{\sin \theta }}+{\sqrt[{z}]{\cos \theta }}\right)^{2z}}}\,d\theta }

The beta function can be written as an infinite sum[3]B(x,y)=∑n=0∞(1−x)n(y+n)n!{\displaystyle \mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\frac {(1-x)_{n}}{(y+n)\,n!}}} Ifx{\displaystyle x} andy{\displaystyle y} are equal to a numberz{\displaystyle z} we get:B(z,z)=2∑n=0∞(2z+n−1)n(−1)n(z+n)n!=limx→1−2∑n=0∞(−2z)nxn(z+n)n!{\displaystyle \mathrm {B} (z,z)=2\sum _{n=0}^{\infty }{\frac {(2z+n-1)_{n}(-1)^{n}}{(z+n)n!}}=\lim _{x\to 1^{-}}2\sum _{n=0}^{\infty }{\frac {(-2z)_{n}x^{n}}{(z+n)n!}}}where(x)n{\displaystyle (x)_{n}} is therising factorial,and as an infinite productB(x,y)=x+yxy∏n=1∞(1+xyn(x+y+n))−1.{\displaystyle \mathrm {B} (x,y)={\frac {x+y}{xy}}\prod _{n=1}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1}.}

The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version ofPascal's identity

B(x,y)=B(x,y+1)+B(x+1,y){\displaystyle \mathrm {B} (x,y)=\mathrm {B} (x,y+1)+\mathrm {B} (x+1,y)}

and a simple recurrence on one coordinate:[4]

B(x+1,y)=B(x,y)⋅xx+y,B(x,y+1)=B(x,y)⋅yx+y.{\displaystyle \mathrm {B} (x+1,y)=\mathrm {B} (x,y)\cdot {\dfrac {x}{x+y}},\quad \mathrm {B} (x,y+1)=\mathrm {B} (x,y)\cdot {\dfrac {y}{x+y}}.}

The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integersm{\displaystyle m} andn{\displaystyle n},B(m+1,n+1)=∂m+nh∂am∂bn(0,0),{\displaystyle \mathrm {B} (m+1,n+1)={\frac {\partial ^{m+n}h}{\partial a^{m}\,\partial b^{n}}}(0,0),}whereh(a,b)=ea−eba−b.{\displaystyle h(a,b)={\frac {e^{a}-e^{b}}{a-b}}.}The Pascal-like identity above implies that this function is a solution to thefirst-order partial differential equationh=ha+hb.{\displaystyle h=h_{a}+h_{b}.}

Forx,y≥1{\displaystyle x,y\geq 1}, the beta function may be written in terms of aconvolution involving thetruncated power functiont↦t+x{\displaystyle t\mapsto t_{+}^{x}}:B(x,y)⋅(t↦t+x+y−1)=(t↦t+x−1)∗(t↦t+y−1){\displaystyle \mathrm {B} (x,y)\cdot \left(t\mapsto t_{+}^{x+y-1}\right)={\Big (}t\mapsto t_{+}^{x-1}{\Big )}*{\Big (}t\mapsto t_{+}^{y-1}{\Big )}}

Evaluations at particular points may simplify significantly; for example,B(1,x)=1x{\displaystyle \mathrm {B} (1,x)={\dfrac {1}{x}}}and[5]B(x,1−x)=πsin⁡(πx),x∉Z{\displaystyle \mathrm {B} (x,1-x)={\dfrac {\pi }{\sin(\pi x)}},\qquad x\not \in \mathbb {Z} }

By takingx=12{\displaystyle x={\frac {1}{2}}} in this last formula, it follows thatΓ(1/2)=π{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}.Generalizing this into a bivariate identity for a product of beta functions leads to:B(x,y)⋅B(x+y,1−y)=πxsin⁡(πy).{\displaystyle \mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\frac {\pi }{x\sin(\pi y)}}.}

Euler's integral for the beta function may be converted into an integral over thePochhammer contourC as

(1−e2πiα)(1−e2πiβ)B(α,β)=∫Ctα−1(1−t)β−1dt.{\displaystyle \left(1-e^{2\pi i\alpha }\right)\left(1-e^{2\pi i\beta }\right)\mathrm {B} (\alpha ,\beta )=\int _{C}t^{\alpha -1}(1-t)^{\beta -1}\,dt.}

This Pochhammer contour integral converges for all values ofα andβ and so gives theanalytic continuation of the beta function.

Just as the gamma function for integers describesfactorials, the beta function can define abinomial coefficient after adjusting indices:(nk)=1(n+1)B(n−k+1,k+1).{\displaystyle {\binom {n}{k}}={\frac {1}{(n+1)\,\mathrm {B} (n-k+1,\,k+1)}}.}

Moreover, for integern,Β can be factored to give a closed form interpolation function for continuous values ofk:(nk)=(−1)nn!⋅sin⁡(πk)π∏i=0n(k−i).{\displaystyle {\binom {n}{k}}=(-1)^{n}\,n!\cdot {\frac {\sin(\pi k)}{\pi \displaystyle \prod _{i=0}^{n}(k-i)}}.}

Reciprocal beta function

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Thereciprocal beta function is thefunction about the form

f(x,y)=1B(x,y){\displaystyle f(x,y)={\frac {1}{\mathrm {B} (x,y)}}}

Interestingly, their integral representations closely relate as thedefinite integral oftrigonometric functions with product of its power andmultiple-angle:[6]

∫0πsinx−1⁡θsin⁡yθ dθ=πsin⁡yπ22x−1xB(x+y+12,x−y+12)∫0πsinx−1⁡θcos⁡yθ dθ=πcos⁡yπ22x−1xB(x+y+12,x−y+12)∫0πcosx−1⁡θsin⁡yθ dθ=πcos⁡yπ22x−1xB(x+y+12,x−y+12)∫0π2cosx−1⁡θcos⁡yθ dθ=π2xxB(x+y+12,x−y+12){\displaystyle {\begin{aligned}\int _{0}^{\pi }\sin ^{x-1}\theta \sin y\theta ~d\theta &={\frac {\pi \sin {\frac {y\pi }{2}}}{2^{x-1}x\mathrm {B} {\left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}\\[1ex]\int _{0}^{\pi }\sin ^{x-1}\theta \cos y\theta ~d\theta &={\frac {\pi \cos {\frac {y\pi }{2}}}{2^{x-1}x\mathrm {B} {\left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}\\[1ex]\int _{0}^{\pi }\cos ^{x-1}\theta \sin y\theta ~d\theta &={\frac {\pi \cos {\frac {y\pi }{2}}}{2^{x-1}x\mathrm {B} {\left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}\\[1ex]\int _{0}^{\frac {\pi }{2}}\cos ^{x-1}\theta \cos y\theta ~d\theta &={\frac {\pi }{2^{x}x\mathrm {B} {\left({\frac {x+y+1}{2}},{\frac {x-y+1}{2}}\right)}}}\end{aligned}}}

Incomplete beta function

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Theincomplete beta function, a generalization of the beta function, is defined as[7][8]

B(x;a,b)=∫0xta−1(1−t)b−1dt.{\displaystyle \mathrm {B} (x;\,a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,dt.}

Forx = 1, the incomplete beta function coincides with the complete beta function. For positive integersa andb, the incomplete beta function will be a polynomial of degreea +b − 1 with rational coefficients.

By the substitutiont=sin2⁡θ{\displaystyle t=\sin ^{2}\theta } andt=11+s{\displaystyle t={\frac {1}{1+s}}}, we can show thatB(x;a,b)=2∫0arcsin⁡xsin2a−1⁡θcos2b−1⁡θdθ=∫1−xx∞sb−1(1+s)a+bds{\displaystyle {\begin{aligned}\mathrm {B} (x;\,a,b)&=2\int _{0}^{\arcsin {\sqrt {x}}}\sin ^{2a-1\!}\theta \cos ^{2b-1\!}\theta \,d\theta \\[1ex]&=\int _{\frac {1-x}{x}}^{\infty }{\frac {s^{b-1}}{(1+s)^{a+b}}}\,ds\end{aligned}}}

Theregularized incomplete beta function (orregularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:

Ix(a,b)=B(x;a,b)B(a,b).{\displaystyle I_{x}(a,b)={\frac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}.}

The regularized incomplete beta function is thecumulative distribution function of thebeta distribution, and is related to thecumulative distribution functionF(k;n,p){\displaystyle F(k;\,n,p)} of arandom variableX following abinomial distribution with probability of single successp and number of Bernoulli trialsn:

F(k;n,p)=Pr(X≤k)=I1−p(n−k,k+1)=1−Ip(k+1,n−k).{\displaystyle {\begin{aligned}F(k;\,n,p)&=\Pr \left(X\leq k\right)\\[1ex]&=I_{1-p}(n-k,k+1)\\[1ex]&=1-I_{p}(k+1,n-k).\end{aligned}}}

Properties

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I0(a,b)=0,I1(a,b)=1,Ix(a,1)=xa,Ix(1,b)=1−(1−x)b,Ix(a,b)=1−I1−x(b,a),Ix(a+1,b)=Ix(a,b)−xa(1−x)baB(a,b),Ix(a,b+1)=Ix(a,b)+xa(1−x)bbB(a,b),∫B(x;a,b)dx=xB(x;a,b)−B(x;a+1,b),B(x;a,b)=(−1)aB(xx−1;a,1−a−b).{\displaystyle {\begin{aligned}I_{0}(a,b)&=0,\\I_{1}(a,b)&=1,\\I_{x}(a,1)&=x^{a},\\I_{x}(1,b)&=1-(1-x)^{b},\\I_{x}(a,b)&=1-I_{1-x}(b,a),\\I_{x}(a+1,b)&=I_{x}(a,b)-{\frac {x^{a}(1-x)^{b}}{a\mathrm {B} (a,b)}},\\I_{x}(a,b+1)&=I_{x}(a,b)+{\frac {x^{a}(1-x)^{b}}{b\mathrm {B} (a,b)}},\\\int \mathrm {B} (x;a,b)\,dx&=x\mathrm {B} (x;a,b)-\mathrm {B} (x;a+1,b),\\\mathrm {B} (x;a,b)&=(-1)^{a}\mathrm {B} \left({\frac {x}{x-1}};a,1-a-b\right).\end{aligned}}}

Continued fraction expansion

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Thecontinued fraction expansion is

B(x;a,b)=xa(1−x)ba(1+d11+d21+d31+⋯),{\displaystyle \mathrm {B} (x;\,a,b)={\frac {x^{a}(1-x)^{b}}{a\left(1+{\frac {{d}_{1}}{1+{\frac {{d}_{2}}{1+{\frac {{d}_{3}}{1+\cdots }}}}}}\right)}},}

with odd and even coefficients given by

d2m+1=−(a+m)(a+b+m)x(a+2m)(a+2m+1),d2m=m(b−m)x(a+2m−1)(a+2m).{\displaystyle {\begin{aligned}{d}_{2m+1}&=-{\frac {(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}},\\[1ex]{d}_{2m}&={\frac {m(b-m)x}{(a+2m-1)(a+2m)}}.\end{aligned}}}

The4m{\displaystyle 4m} and4m+1{\displaystyle 4m+1} convergents are less thanB(x;a,b){\displaystyle \mathrm {B} (x;\,a,b)}, while the4m+2{\displaystyle 4m+2} and4m+3{\displaystyle 4m+3} convergents are greater thanB(x;a,b){\displaystyle \mathrm {B} (x;\,a,b)}.

It converges rapidly forx<(a+1)/(a+b+2){\displaystyle x<(a+1)/(a+b+2)}. Forx>(a+1)/(a+b+2){\displaystyle x>(a+1)/(a+b+2)} or1−x<(b+1)/(a+b+2){\displaystyle 1-x<(b+1)/(a+b+2)}, the function may be evaluated more efficiently through the relationB(x;a,b)=B(a,b)−B(1−x;b,a){\displaystyle \mathrm {B} (x;\,a,b)=\mathrm {B} (a,b)-\mathrm {B} (1-x;\,b,a)}.[8]

Multivariate beta function

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The beta function can be extended to a function with more than two arguments:

B(α1,α2,…αn)=Γ(α1)Γ(α2)⋯Γ(αn)Γ(α1+α2+⋯+αn).{\displaystyle \mathrm {B} (\alpha _{1},\alpha _{2},\ldots \alpha _{n})={\frac {\Gamma (\alpha _{1})\,\Gamma (\alpha _{2})\cdots \Gamma (\alpha _{n})}{\Gamma (\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n})}}.}

This multivariate beta function is used in the definition of theDirichlet distribution. Its relationship to the beta function is analogous to the relationship betweenmultinomial coefficients and binomial coefficients. For example, it satisfies a similar version of Pascal's identity:

B(α1,α2,…αn)=B(α1+1,α2,…αn)+B(α1,α2+1,…αn)+⋯+B(α1,α2,…αn+1).{\displaystyle \mathrm {B} (\alpha _{1},\alpha _{2},\ldots \alpha _{n})=\mathrm {B} (\alpha _{1}+1,\alpha _{2},\ldots \alpha _{n})+\mathrm {B} (\alpha _{1},\alpha _{2}+1,\ldots \alpha _{n})+\cdots +\mathrm {B} (\alpha _{1},\alpha _{2},\ldots \alpha _{n}+1).}

Applications

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The beta function is useful in computing and representing thescattering amplitude forRegge trajectories. Furthermore, it was the first knownscattering amplitude instring theory, firstconjectured byGabriele Veneziano. It also occurs in the theory of thepreferential attachment process, a type of stochasticurn process. The beta function is also important in statistics, e.g. for thebeta distribution andbeta prime distribution. As briefly alluded to previously, the beta function is closely tied with thegamma function and plays an important role incalculus.

Software implementation

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Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included inspreadsheet orcomputer algebra systems.

InMicrosoft Excel, for example, the complete beta function can be computed with theGammaLn function (orspecial.gammaln inPython'sSciPy package):

Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))

This result follows from the propertieslisted above.

The incomplete beta function cannot be directly computed using such relations and other methods must be used. InGNU Octave, it is computed using acontinued fraction expansion.

The incomplete beta function has existing implementation in common languages. For instance,betainc (incomplete beta function) inMATLAB andGNU Octave,pbeta (probability of beta distribution) inR andbetainc inSymPy. InSciPy,special.betainc computes theregularized incomplete beta function—which is, in fact, the cumulative beta distribution. To get the actual incomplete beta function, one can multiply the result ofspecial.betainc by the result returned by the correspondingbeta function. InMathematica,Beta[x, a, b] andBetaRegularized[x, a, b] giveB(x;a,b){\displaystyle \mathrm {B} (x;\,a,b)} andIx(a,b){\displaystyle I_{x}(a,b)}, respectively.

See also

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This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(November 2010) (Learn how and when to remove this message)

References

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  1. ^abcDavis, Philip J. (1972), "6. Gamma function and related functions", inAbramowitz, Milton;Stegun, Irene A. (eds.),Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York:Dover Publications, p. 258,ISBN 978-0-486-61272-0. Specifically, see 6.2 Beta Function.
  2. ^Artin, Emil,The Gamma Function(PDF), pp. 18–19, archived fromthe original(PDF) on 2016-11-12, retrieved2016-11-11
  3. ^Beta function : Series representations (Formula 06.18.06.0007)
  4. ^Mäklin, Tommi (2022),Probabilistic Methods for High-Resolution Metagenomics(PDF), Series of publications A / Department of Computer Science, University of Helsinki, Helsinki: Unigrafia, p. 27,ISBN 978-951-51-8695-9,ISSN 2814-4031
  5. ^"Euler's Reflection Formula - ProofWiki",proofwiki.org, retrieved2020-09-02
  6. ^Paris, R. B. (2010),"Beta Function", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.
  7. ^Zelen, M.; Severo, N. C. (1972), "26. Probability functions", inAbramowitz, Milton;Stegun, Irene A. (eds.),Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York:Dover Publications, pp. 944,ISBN 978-0-486-61272-0
  8. ^abParis, R. B. (2010),"Incomplete beta functions", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.

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