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Beta function

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the Euler beta function. For other uses, see <a href="Beta_function_(disambiguation)" class="mw-disambig" title="Beta function (disambiguation)">Beta function (disambiguation)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the beta function, also called the <a href="Euler_integral_(disambiguation)" class="mw-redirect" title="Euler integral (disambiguation)">Euler integral</a> of the first kind, is a <a href="Special_function" class="mw-redirect" title="Special function">special function</a> that is closely related to the <a href="Gamma_function" title="Gamma function">gamma function</a> and to <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>. It is defined by the <a href="Integral" title="Integral">integral</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}</annotation>

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</math><img src="./af2e0e1ed581918ee9225e0f011d791df29591d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.555ex; height:6.176ex;" alt="{\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}" loading="lazy"></dd></dl>

for <a href="Complex_number" title="Complex number">complex number</a> inputs

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<annotation encoding="application/x-tex">{\displaystyle z_{1},z_{2}}</annotation>

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</math><img src="./dd9e6b30dcd6b3d0ba19760e47ed97204c0a6cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.305ex; height:2.009ex;" alt="{\displaystyle z_{1},z_{2}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Re} (z_{1}),\operatorname {Re} (z_{2})>0}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Re} (z_{1}),\operatorname {Re} (z_{2})>0}</annotation>

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</math><img src="./256d2f629486f6816d92a2853892c0f88782eb1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.67ex; height:2.843ex;" alt="{\displaystyle \operatorname {Re} (z_{1}),\operatorname {Re} (z_{2})>0}" loading="lazy">.

The beta function was studied by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> and <a href="Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Adrien-Marie Legendre</a> and was given its name by <a href="Jacques_Philippe_Marie_Binet" title="Jacques Philippe Marie Binet">Jacques Binet</a>; its symbol Β is a <a href="Greek_alphabet" title="Greek alphabet">Greek</a> capital <a href="Beta_(letter)" class="mw-redirect" title="Beta (letter)">beta</a>.

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The beta function is <a href="Symmetric_function" title="Symmetric function">symmetric</a>, meaning that

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{2},z_{1})}</annotation>

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</math><img src="./6bb5c0d47ff0e4f01d036ab5e2f6f6a7c3bce01a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.618ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{2},z_{1})}" loading="lazy"> for all inputs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1}}</annotation>

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</math><img src="./c3621e468231ab352b7caa30bcf0ce9b452241a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{2}}</annotation>

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</math><img src="./5abf655fa14f7ea44ad0ca781b59ff59c5f49117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.135ex; height:2.009ex;" alt="{\displaystyle z_{2}}" loading="lazy">.<a href="#cite_note-Davis622-1">[1]</a>

A key property of the beta function is its close relationship to the <a href="Gamma_function" title="Gamma function">gamma function</a>:<a href="#cite_note-Davis622-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\,\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}</annotation>

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</math><img src="./a32ce5bb8d56c4822c5032dfd051a4fe42456c53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.876ex; height:6.509ex;" alt="{\displaystyle \mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\,\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}}" loading="lazy"></dd></dl>

A proof is given below in <a href="#Relationship_to_the_gamma_function">§ Relationship to the gamma function</a>.

The beta function is also closely related to <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>. When m (or n, by symmetry) is a positive integer, it follows from the definition of the gamma function Γ that<a href="#cite_note-Davis622-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (m,n)={\frac {(m-1)!\,(n-1)!}{(m+n-1)!}}={\frac {m+n}{mn}}{\Bigg /}{\binom {m+n}{m}}}</annotation>

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</math><img src="./3b9d3c6fe4930ec47e3783d6cfe7b8de87b71e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:51.475ex; height:7.509ex;" alt="{\displaystyle \mathrm {B} (m,n)={\frac {(m-1)!\,(n-1)!}{(m+n-1)!}}={\frac {m+n}{mn}}{\Bigg /}{\binom {m+n}{m}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Relationship_to_the_gamma_function">Relationship to the gamma function</h2></div>

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 an iterated integral:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./0602e0665bdaf2c38b4056fa812ae684ba773b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:50.152ex; height:13.509ex;" alt="{\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}}}" loading="lazy"></dd></dl>

Changing variables by u = st and v = s(1 − t), because u + v = s and u / (u+v) = t, we have that the limits of integrations for s are 0 to ∞ and the limits of integration for t are 0 to 1. Thus produces

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<annotation encoding="application/x-tex">{\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\\&=\Gamma (z_{1}+z_{2})\cdot \mathrm {B} (z_{1},z_{2}).\end{aligned}}}</annotation>

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</math><img src="./39f6cdeb03466b3835ea08e07d50371712318dd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.782ex; margin-bottom: -0.222ex; width:57.458ex; height:17.176ex;" alt="{\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\\&=\Gamma (z_{1}+z_{2})\cdot \mathrm {B} (z_{1},z_{2}).\end{aligned}}}" loading="lazy"></dd></dl>

Dividing both sides by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z_{1}+z_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (z_{1}+z_{2})}</annotation>

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</math><img src="./485ee17370747dced0059a15dd84d530fe1f1f1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.373ex; height:2.843ex;" alt="{\displaystyle \Gamma (z_{1}+z_{2})}" loading="lazy"> gives the desired result.

The stated identity may be seen as a particular case of the identity for the <a href="Convolution#Integration" title="Convolution">integral of a convolution</a>. Taking

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./e48faa26c6d3655f381159f6aad32b4c9e988625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.138ex; height:6.843ex;" alt="{\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}}}" loading="lazy"></dd></dl>

one has:

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<annotation encoding="application/x-tex">{\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}).}</annotation>

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</math><img src="./9c197a94f9749da177c3505a9d61ee9f0a72bf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:77.853ex; height:5.676ex;" alt="{\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}).}" loading="lazy"></dd></dl>

See The Gamma Function, page 18–19<a href="#cite_note-2">[2]</a> for a derivation of this relation.

<div class="mw-heading mw-heading2"><h2 id="Differentiation_of_the_beta_function">Differentiation of the beta function</h2></div>

We have

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<annotation encoding="application/x-tex">{\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 )},}</annotation>

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</math><img src="./ec759aa570b51ddd2e5d4efd1582605a1f4cb39c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:83.735ex; height:6.509ex;" alt="{\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 )},}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\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,}</annotation>

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</math><img src="./b226a05b979a5e6386271e4ffbbfefea23c8c1da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:80.15ex; height:7.509ex;" alt="{\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,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \psi (z)}</annotation>

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</math><img src="./0c02965f8dd8bfe2c0352b07c1193b8dc276c1d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle \psi (z)}" loading="lazy"> denotes the <a href="Digamma_function" title="Digamma function">digamma function</a>.

<div class="mw-heading mw-heading2"><h2 id="Approximation">Approximation</h2></div>

<a href="Stirling's_approximation" title="Stirling's approximation">Stirling's approximation</a> gives the asymptotic formula

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,y)\sim {\sqrt {2\pi }}{\frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}</annotation>

</semantics>

</math><img src="./1d3c23c2369e933ad4008b36d8d8f709507b8aa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.486ex; height:6.843ex;" alt="{\displaystyle \mathrm {B} (x,y)\sim {\sqrt {2\pi }}{\frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}" loading="lazy"></dd></dl>

for large x and large y.

If on the other hand x is large and y is fixed, then

</mrow>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,y)\sim \Gamma (y)\,x^{-y}.}</annotation>

</semantics>

</math><img src="./5f62b080b80dfab1dcf71e91383ca2253bffc03d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.181ex; height:3.009ex;" alt="{\displaystyle \mathrm {B} (x,y)\sim \Gamma (y)\,x^{-y}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Other_identities_and_formulas">Other identities and formulas</h2></div>

The integral defining the beta function may be rewritten in a variety of ways, including the following:

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<msub>

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<mtr>

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}</annotation>

</semantics>

</math><img src="./0163d50bcc0f2359f68f9ac5acc1fccbf45a262f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.667ex; margin-bottom: -0.338ex; width:65.016ex; height:29.009ex;" alt="{\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}}}" loading="lazy"></dd></dl>

where in the second-to-last identity n is any positive real number. One may move from the first integral to the second one by substituting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\tan ^{2}(\theta )}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle t=\tan ^{2}(\theta )}</annotation>

</semantics>

</math><img src="./faa02c6a80d661dc830b53dc98023338ef2ca7b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.252ex; height:3.176ex;" alt="{\displaystyle t=\tan ^{2}(\theta )}" loading="lazy">.

For values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z_{1}=z_{2}\neq 1}">

<msub>

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</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=z_{1}=z_{2}\neq 1}</annotation>

</semantics>

</math><img src="./19aa22a36bdd53294ce35409f30e3fe339a1b149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.817ex; height:2.676ex;" alt="{\displaystyle z=z_{1}=z_{2}\neq 1}" loading="lazy"> we have:

</mrow>

<mfrac>

</mfrac>

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</msubsup>

<mfrac>

<mrow>

<mroot>

<mrow>

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</mrow>

</mroot>

</mrow>

<mroot>

<mrow>

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</mrow>

</mroot>

</mrow>

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (z,z)={\frac {1}{z}}\int _{0}^{\pi /2}{\frac {1}{({\sqrt[{z}]{\sin \theta }}+{\sqrt[{z}]{\cos \theta }})^{2z}}}\,d\theta }</annotation>

</semantics>

</math><img src="./63de0c2556f3ef41ce9bea83ca673d6337663ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.615ex; height:7.009ex;" alt="{\displaystyle \mathrm {B} (z,z)={\frac {1}{z}}\int _{0}^{\pi /2}{\frac {1}{({\sqrt[{z}]{\sin \theta }}+{\sqrt[{z}]{\cos \theta }})^{2z}}}\,d\theta }" loading="lazy"></dd></dl>

The beta function can be written as an infinite sum<a href="#cite_note-3">[3]</a>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

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</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\frac {(1-x)_{n}}{(y+n)\,n!}}}</annotation>

</semantics>

</math><img src="./d19695d9b0c43221e84de75a51b32a5ae1d614cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.279ex; height:6.843ex;" alt="{\displaystyle \mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\frac {(1-x)_{n}}{(y+n)\,n!}}}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> are equal to a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> we get:

</mrow>

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<msup>

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<msup>

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</munder>

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</munderover>

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<mrow>

<msub>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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!}}}</annotation>

</semantics>

</math><img src="./4ff46f4ab1991e20c85c6021e909423d89be2f95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.262ex; height:6.843ex;" alt="{\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!}}}" loading="lazy"></dd>

<dd>(where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x)_{n}}</annotation>

</semantics>

</math><img src="./667ec9c60835aa2b84a8d017562b4294ad6dcaef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle (x)_{n}}" loading="lazy"> is the <a href="Falling_and_rising_factorials" title="Falling and rising factorials">rising factorial</a>)</dd></dl>

and as an infinite product

</mrow>

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<mrow>

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</munderover>

<msup>

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</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,y)={\frac {x+y}{xy}}\prod _{n=1}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1}.}</annotation>

</semantics>

</math><img src="./9a38115f521daaa975db48ba7380a9393f9db53b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.1ex; height:7.009ex;" alt="{\displaystyle \mathrm {B} (x,y)={\frac {x+y}{xy}}\prod _{n=1}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1}.}" loading="lazy"></dd></dl>

The beta function satisfies several identities analogous to corresponding identities for binomial coefficients, including a version of <a href="Pascal's_identity" class="mw-redirect" title="Pascal's identity">Pascal's identity</a>

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,y)=\mathrm {B} (x,y+1)+\mathrm {B} (x+1,y)}</annotation>

</semantics>

</math><img src="./ebb31673bda551a97237fca9ef28a920a84c7e9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.866ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} (x,y)=\mathrm {B} (x,y+1)+\mathrm {B} (x+1,y)}" loading="lazy"></dd></dl>

and a simple recurrence on one coordinate:

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<annotation encoding="application/x-tex">{\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}}.}</annotation>

</semantics>

</math><img src="./9344ca82987f970e46d6f1d601f588925bc58b2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.783ex; height:5.343ex;" alt="{\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}}.}" loading="lazy"><a href="#cite_note-4">[4]</a></dd></dl>

The positive integer values of the beta function are also the partial derivatives of a 2D function: for all nonnegative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

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</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">,

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<mfrac>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (m+1,n+1)={\frac {\partial ^{m+n}h}{\partial a^{m}\,\partial b^{n}}}(0,0),}</annotation>

</semantics>

</math><img src="./e23809a9b24d35d2395c4dcec16488f99253a5b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:33.823ex; height:5.676ex;" alt="{\displaystyle \mathrm {B} (m+1,n+1)={\frac {\partial ^{m+n}h}{\partial a^{m}\,\partial b^{n}}}(0,0),}" loading="lazy"></dd></dl>

where

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<msup>

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</msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(a,b)={\frac {e^{a}-e^{b}}{a-b}}.}</annotation>

</semantics>

</math><img src="./67e996e90a47a57e72822d9c2addef59a68b00da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.038ex; height:6.009ex;" alt="{\displaystyle h(a,b)={\frac {e^{a}-e^{b}}{a-b}}.}" loading="lazy"></dd></dl>

The Pascal-like identity above implies that this function is a solution to the <a href="First-order_partial_differential_equation" title="First-order partial differential equation">first-order partial differential equation</a>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle h=h_{a}+h_{b}.}</annotation>

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</math><img src="./39f1a94c4b662c8d7f746ac8419a24326ed49e69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.642ex; height:2.509ex;" alt="{\displaystyle h=h_{a}+h_{b}.}" loading="lazy"></dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\geq 1}</annotation>

</semantics>

</math><img src="./d3cb123e6bc54cda97b70990aa8d40eb90e5dd4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\displaystyle x,y\geq 1}" loading="lazy">, the beta function may be written in terms of a <a href="Convolution" title="Convolution">convolution</a> involving the <a href="Truncated_power_function" title="Truncated power function">truncated power function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto t_{+}^{x}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\mapsto t_{+}^{x}}</annotation>

</semantics>

</math><img src="./b517c8ca47eb3b06796840f5e4433f47edec55c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.804ex; height:2.843ex;" alt="{\displaystyle t\mapsto t_{+}^{x}}" loading="lazy">:

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<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

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</mrow>

</mrow>

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</mrow>

</msubsup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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 )}}</annotation>

</semantics>

</math><img src="./ba01c737fe2061bdcee738faa8602bf99b6c7e79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.944ex; height:4.843ex;" alt="{\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 )}}" loading="lazy"></dd></dl>

Evaluations at particular points may simplify significantly; for example,

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (1,x)={\dfrac {1}{x}}}</annotation>

</semantics>

</math><img src="./7eb551457b8fbb13f056f212b66c52724f947067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.245ex; height:5.176ex;" alt="{\displaystyle \mathrm {B} (1,x)={\dfrac {1}{x}}}" loading="lazy"></dd></dl>

and

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<mfrac>

<mrow>

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</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,1-x)={\dfrac {\pi }{\sin(\pi x)}},\qquad x\not \in \mathbb {Z} }</annotation>

</semantics>

</math><img src="./7e4ddbc992cfdc9730aedfc6b2c25d05e648f171.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.812ex; height:5.509ex;" alt="{\displaystyle \mathrm {B} (x,1-x)={\dfrac {\pi }{\sin(\pi x)}},\qquad x\not \in \mathbb {Z} }" loading="lazy"><a href="#cite_note-5">[5]</a></dd></dl>

By taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {1}{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./2823014b08c738d0755d2397dc6d997a7e59c6c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.427ex; height:5.176ex;" alt="{\displaystyle x={\frac {1}{2}}}" loading="lazy"> in this last formula, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}">

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<msqrt>

</msqrt>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}</annotation>

</semantics>

</math><img src="./75337a521eb1fcd407b39db28c38ca6f559ab3b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.116ex; height:3.009ex;" alt="{\displaystyle \Gamma (1/2)={\sqrt {\pi }}}" loading="lazy">.

Generalizing this into a bivariate identity for a product of beta functions leads to:

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</mrow>

<mfrac>

<mrow>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\frac {\pi }{x\sin(\pi y)}}.}</annotation>

</semantics>

</math><img src="./39978dbc26acd73ac93243986407ff3580cc0900.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.076ex; height:5.509ex;" alt="{\displaystyle \mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\frac {\pi }{x\sin(\pi y)}}.}" loading="lazy"></dd></dl>

Euler's integral for the beta function may be converted into an integral over the <a href="Pochhammer_contour" title="Pochhammer contour">Pochhammer contour</a> C as

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<msub>

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<msup>

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<annotation encoding="application/x-tex">{\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.}</annotation>

</semantics>

</math><img src="./43b322c0cb3c9fb18e25483e95505ff0188a3ea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.135ex; height:5.676ex;" alt="{\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.}" loading="lazy"></dd></dl>

This Pochhammer contour integral converges for all values of α and β and so gives the <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a> of the beta function.

Just as the gamma function for integers describes <a href="Factorial" title="Factorial">factorials</a>, the beta function can define a <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> after adjusting indices:

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<annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}={\frac {1}{(n+1)\,\mathrm {B} (n-k+1,k+1)}}.}</annotation>

</semantics>

</math><img src="./d0ea419308696b9c473009a6a055b37e8525bd8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.143ex; height:6.343ex;" alt="{\displaystyle {\binom {n}{k}}={\frac {1}{(n+1)\,\mathrm {B} (n-k+1,k+1)}}.}" loading="lazy"></dd></dl>

Moreover, for integer n, Β can be factored to give a closed form interpolation function for continuous values of k:

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<annotation encoding="application/x-tex">{\displaystyle {\binom {n}{k}}=(-1)^{n}\,n!\cdot {\frac {\sin(\pi k)}{\pi \displaystyle \prod _{i=0}^{n}(k-i)}}.}</annotation>

</semantics>

</math><img src="./a8abf72879c235351347a6db76019510e489e0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:30.855ex; height:10.509ex;" alt="{\displaystyle {\binom {n}{k}}=(-1)^{n}\,n!\cdot {\frac {\sin(\pi k)}{\pi \displaystyle \prod _{i=0}^{n}(k-i)}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Reciprocal_beta_function">Reciprocal beta function</h2></div>

The reciprocal beta function is the <a href="Special_function" class="mw-redirect" title="Special function">function</a> about the form

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)={\frac {1}{\mathrm {B} (x,y)}}}</annotation>

</semantics>

</math><img src="./e087664a0ac50c0627a0aa267da4c9cf09b5c0ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.516ex; height:6.009ex;" alt="{\displaystyle f(x,y)={\frac {1}{\mathrm {B} (x,y)}}}" loading="lazy"></dd></dl>

Interestingly, their integral representations closely relate as the <a href="Definite_integral" class="mw-redirect" title="Definite integral">definite integral</a> of <a href="Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> with product of its power and <a href="List_of_trigonometric_identities#Multiple-angle_formulae" title="List of trigonometric identities">multiple-angle</a>:<a href="#cite_note-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle \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)}}}</annotation>

</semantics>

</math><img src="./fcf70d62ede9e7403db9a59e867a70583823903f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:47.881ex; height:9.009ex;" alt="{\displaystyle \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)}}}" loading="lazy"></dd></dl>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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)}}}</annotation>

</semantics>

</math><img src="./6c1a1868d0d1eaaca1a56a028a85056180b64c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:48.137ex; height:9.009ex;" alt="{\displaystyle \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)}}}" loading="lazy"></dd></dl>

</mrow>

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</msubsup>

<msup>

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<mrow>

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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)}}}</annotation>

</semantics>

</math><img src="./7d108b279a269e70e4c463075e29868717ade2ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:48.137ex; height:9.009ex;" alt="{\displaystyle \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)}}}" loading="lazy"></dd></dl>

</mrow>

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</msubsup>

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<mfrac>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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)}}}</annotation>

</semantics>

</math><img src="./73934f2cee04d1ea2775dec082e7bc4085cb688f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:46.95ex; height:8.676ex;" alt="{\displaystyle \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)}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Incomplete_beta_function">Incomplete beta function</h2></div>

The incomplete beta function, a generalization of the beta function, is defined as<a href="#cite_note-7">[7]</a><a href="#cite_note-paris-ibf-8">[8]</a>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x;\,a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,dt.}</annotation>

</semantics>

</math><img src="./38c3c049ff3febe90def7ee2e86198f1c80e7774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.798ex; height:5.843ex;" alt="{\displaystyle \mathrm {B} (x;\,a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,dt.}" loading="lazy"></dd></dl>

For x = 1, the incomplete beta function coincides with the complete beta function. For positive integers a and b, the incomplete beta function will be a polynomial of degree a + b - 1 with rational coefficients.

By the substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\sin ^{2}\theta }">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle t=\sin ^{2}\theta }</annotation>

</semantics>

</math><img src="./818e889314d0876aee0b0aaf2650e24d5cadb220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.325ex; height:2.676ex;" alt="{\displaystyle t=\sin ^{2}\theta }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {1}{1+s}}}">

<mfrac>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle t={\frac {1}{1+s}}}</annotation>

</semantics>

</math><img src="./3da5ad3aa89cc9b43655495238f3016b110ad9c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.868ex; height:5.343ex;" alt="{\displaystyle t={\frac {1}{1+s}}}" loading="lazy">, we can show that

</mrow>

</mrow>

<mi>arcsin</mi>

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x;\,a,b)=2\int _{0}^{\arcsin {\sqrt {x}}}\sin ^{2a-1\!}\theta \cos ^{2b-1\!}\theta \,\mathrm {d} \theta =\int _{\frac {1-x}{x}}^{\infty }{\frac {s^{b-1}}{(1+s)^{a+b}}}\,\mathrm {d} s}</annotation>

</semantics>

</math><img src="./b84eb7476aa420e9cfb3dfa3775271b08e4f1f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:65.209ex; height:7.009ex;" alt="{\displaystyle \mathrm {B} (x;\,a,b)=2\int _{0}^{\arcsin {\sqrt {x}}}\sin ^{2a-1\!}\theta \cos ^{2b-1\!}\theta \,\mathrm {d} \theta =\int _{\frac {1-x}{x}}^{\infty }{\frac {s^{b-1}}{(1+s)^{a+b}}}\,\mathrm {d} s}" loading="lazy"></dd></dl>

The regularized incomplete beta function (or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function:

<msub>

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<annotation encoding="application/x-tex">{\displaystyle I_{x}(a,b)={\frac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}.}</annotation>

</semantics>

</math><img src="./df598957dcc02bd38b07e8e12949e930ad1722c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.314ex; height:6.509ex;" alt="{\displaystyle I_{x}(a,b)={\frac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}.}" loading="lazy"></dd></dl>

The regularized incomplete beta function is the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> of the <a href="Beta_distribution" title="Beta distribution">beta distribution</a>, and is related to the <a href="Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(k;\,n,p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(k;\,n,p)}</annotation>

</semantics>

</math><img src="./cb58ba7dfb22dacfb3749c67e265ddcc3b1e1965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.78ex; height:2.843ex;" alt="{\displaystyle F(k;\,n,p)}" loading="lazy"> of a <a href="Random_variable" title="Random variable">random variable</a> X following a <a href="Binomial_distribution" title="Binomial distribution">binomial distribution</a> with probability of single success p and number of Bernoulli trials n:

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle F(k;\,n,p)=\Pr \left(X\leq k\right)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).}</annotation>

</semantics>

</math><img src="./5bf39362b381c66180be5f29be84559f2133ced0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:67.978ex; height:3.009ex;" alt="{\displaystyle F(k;\,n,p)=\Pr \left(X\leq k\right)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

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<annotation encoding="application/x-tex">{\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)\mathrm {d} x&=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}}}</annotation>

</semantics>

</math><img src="./26700f5b349fcc3ce690228b9b50ebad993bcfcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -19.948ex; margin-bottom: -0.223ex; width:49.039ex; height:41.509ex;" alt="{\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)\mathrm {d} x&=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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Continued_fraction_expansion">Continued fraction expansion</h3></div>

The <a href="Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">continued fraction</a> expansion

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<annotation encoding="application/x-tex">{\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+}}{\frac {{d}_{4}}{1+}}\cdots \right)}}}</annotation>

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</math><img src="./5c2aac9f458e6bd866831fd53bf173fe0e2d0562.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:36.654ex; height:8.509ex;" alt="{\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+}}{\frac {{d}_{4}}{1+}}\cdots \right)}}}" loading="lazy"></dd></dl>

with odd and even coefficients respectively

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<annotation encoding="application/x-tex">{\displaystyle {d}_{2m+1}=-{\frac {(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}}}</annotation>

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</math><img src="./a87bc7412d3c360aa1940ef72982698e62cec67f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.717ex; height:6.509ex;" alt="{\displaystyle {d}_{2m+1}=-{\frac {(a+m)(a+b+m)x}{(a+2m)(a+2m+1)}}}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {d}_{2m}={\frac {m(b-m)x}{(a+2m-1)(a+2m)}}}</annotation>

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</math><img src="./b9493abc5eae69456a0838c1ed7c20746f7713fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.808ex; height:6.509ex;" alt="{\displaystyle {d}_{2m}={\frac {m(b-m)x}{(a+2m-1)(a+2m)}}}" loading="lazy"></dd></dl>

converges rapidly when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is not close to 1. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4m}">

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<annotation encoding="application/x-tex">{\displaystyle 4m}</annotation>

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</math><img src="./691b73762d9780148a19d46434ac5d3db74c925b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle 4m}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4m+1}">

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<annotation encoding="application/x-tex">{\displaystyle 4m+1}</annotation>

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</math><img src="./979236eb049f13fa685f313ec9105ff33e8ebe61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 4m+1}" loading="lazy"> convergents are less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} (x;\,a,b)}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x;\,a,b)}</annotation>

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</math><img src="./fad485bb9b1df855d5a74746a6c2fbc1863055fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.467ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} (x;\,a,b)}" loading="lazy">, while the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4m+2}">

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<annotation encoding="application/x-tex">{\displaystyle 4m+2}</annotation>

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</math><img src="./a597b2013f1db484204ecdf38c04ca5f6a46abb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 4m+2}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4m+3}">

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<annotation encoding="application/x-tex">{\displaystyle 4m+3}</annotation>

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</math><img src="./d54266aa63291aaf75df2c36f0cf6162866581cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.206ex; height:2.343ex;" alt="{\displaystyle 4m+3}" loading="lazy"> convergents are greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} (x;\,a,b)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x;\,a,b)}</annotation>

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For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>{\frac {a+1}{a+b+2}}}">

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<annotation encoding="application/x-tex">{\displaystyle x>{\frac {a+1}{a+b+2}}}</annotation>

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</math><img src="./b94900d1e8fc397b8951de84c0dbc6e5bd5433d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.335ex; height:5.509ex;" alt="{\displaystyle x>{\frac {a+1}{a+b+2}}}" loading="lazy">, the function may be evaluated more efficiently using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} (x;\,a,b)=\mathrm {B} (a,b)-\mathrm {B} (1-x;\,b,a)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x;\,a,b)=\mathrm {B} (a,b)-\mathrm {B} (1-x;\,b,a)}</annotation>

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</math><img src="./c2e599b8dde9ff4b994490dceba7092abbabb7ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.592ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} (x;\,a,b)=\mathrm {B} (a,b)-\mathrm {B} (1-x;\,b,a)}" loading="lazy">.<a href="#cite_note-paris-ibf-8">[8]</a>

<div class="mw-heading mw-heading2"><h2 id="Multivariate_beta_function">Multivariate beta function</h2></div>

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

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<annotation encoding="application/x-tex">{\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})}}.}</annotation>

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</math><img src="./b5c482e4c1a13c9d26d9a30f514eadcd35150d05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.301ex; height:6.509ex;" alt="{\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})}}.}" loading="lazy"></dd></dl>

This multivariate beta function is used in the definition of the <a href="Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a>. Its relationship to the beta function is analogous to the relationship between <a href="Multinomial_coefficient" class="mw-redirect" title="Multinomial coefficient">multinomial coefficients</a> and binomial coefficients. For example, it satisfies a similar version of Pascal's identity:

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<annotation encoding="application/x-tex">{\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).}</annotation>

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</math><img src="./0ed15f59e9251da04d575243d0e0ed08f623e8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:92.69ex; height:2.843ex;" alt="{\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).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

The beta function is useful in computing and representing the <a href="Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> for <a href="Regge_trajectories" class="mw-redirect" title="Regge trajectories">Regge trajectories</a>. Furthermore, it was the first known <a href="S_matrix" class="mw-redirect" title="S matrix">scattering amplitude</a> in <a href="String_theory" title="String theory">string theory</a>, first <a href="Veneziano_amplitude" title="Veneziano amplitude">conjectured</a> by <a href="Gabriele_Veneziano" title="Gabriele Veneziano">Gabriele Veneziano</a>. It also occurs in the theory of the <a href="Preferential_attachment" title="Preferential attachment">preferential attachment</a> process, a type of stochastic <a href="Urn_problem" title="Urn problem">urn process</a>. The beta function is also important in statistics, e.g. for the <a href="Beta_distribution" title="Beta distribution">beta distribution</a> and <a href="Beta_prime_distribution" title="Beta prime distribution">beta prime distribution</a>. As briefly alluded to previously, the beta function is closely tied with the <a href="Gamma_function" title="Gamma function">gamma function</a> and plays an important role in <a href="Calculus" title="Calculus">calculus</a>.

<div class="mw-heading mw-heading2"><h2 id="Software_implementation">Software implementation</h2></div>

Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in <a href="Spreadsheet" title="Spreadsheet">spreadsheet</a> or <a href="Computer_algebra_system" title="Computer algebra system">computer algebra systems</a>.

In <a href="Microsoft_Excel" title="Microsoft Excel">Microsoft Excel</a>, for example, the complete beta function can be computed with the <code><a href="Gamma_function#Log-gamma_function" title="Gamma function">GammaLn</a></code> function (or <code>special.gammaln</code> in <a href="Python_(programming_language)" title="Python (programming language)">Python's</a> <a href="SciPy" title="SciPy">SciPy</a> package):

<dl><dd><code>Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))</code></dd></dl>

This result follows from the properties <a href="#Properties">listed above</a>.

The incomplete beta function cannot be directly computed using such relations and other methods must be used. In <a rel="nofollow" class="external text" href="https://www.gnu.org/software/gsl/doc/html/specfunc.html#incomplete-beta-function">GNU Octave</a>, it is computed using a <a href="Continued_fraction" title="Continued fraction">continued fraction</a> expansion.

The incomplete beta function has existing implementation in common languages. For instance, <code>betainc</code> (incomplete beta function) in <a href="MATLAB" title="MATLAB">MATLAB</a> and <a href="GNU_Octave" title="GNU Octave">GNU Octave</a>, <code>pbeta</code> (probability of beta distribution) in <a href="R_(programming_language)" title="R (programming language)">R</a> and <code>betainc</code> in <a href="SymPy" title="SymPy">SymPy</a>. In <a href="SciPy" title="SciPy">SciPy</a>, <code>special.betainc</code> computes the <a href="Beta_distribution#Cumulative_distribution_function" title="Beta distribution">regularized incomplete beta function</a>—which is, in fact, the cumulative beta distribution. To get the actual incomplete beta function, one can multiply the result of <code>special.betainc</code> by the result returned by the corresponding <code>beta</code> function. In <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a>, <code>Beta[x, a, b]</code> and <code>BetaRegularized[x, a, b]</code> give <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {B} (x;\,a,b)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (x;\,a,b)}</annotation>

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</math><img src="./fad485bb9b1df855d5a74746a6c2fbc1863055fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.467ex; height:2.843ex;" alt="{\displaystyle \mathrm {B} (x;\,a,b)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{x}(a,b)}">

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<annotation encoding="application/x-tex">{\displaystyle I_{x}(a,b)}</annotation>

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</math><img src="./a3378ac827df061820ca7bf563b61262d03d66bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.266ex; height:2.843ex;" alt="{\displaystyle I_{x}(a,b)}" loading="lazy">, respectively.

<ul><li><a href="Beta_distribution" title="Beta distribution">Beta distribution</a> and <a href="Beta_prime_distribution" title="Beta prime distribution">Beta prime distribution</a>, two probability distributions related to the beta function</li>

<li><a href="Jacobi_sum" title="Jacobi sum">Jacobi sum</a>, the analogue of the beta function over <a href="Finite_field" title="Finite field">finite fields</a>.</li>

<li><a href="N%C3%B8rlund%E2%80%93Rice_integral" title="Nørlund–Rice integral">Nørlund–Rice integral</a></li>

<li><a href="Yule%E2%80%93Simon_distribution" title="Yule–Simon distribution">Yule–Simon distribution</a></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFDavis1972" class="citation cs2">Davis, Philip J. (1972), "6. Gamma function and related functions", in <a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene A.</a> (eds.), <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofmathe000abra/page/258/mode/2up?view=theater">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>, New York: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, p. 258, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi></cite>. Specifically, see 6.2 Beta Function.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFArtin" class="citation cs2">Artin, Emil, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161112081854/http://www.plouffe.fr/simon/math/Artin%20E.%20The%20Gamma%20Function%20(1931)(23s).pdf">The Gamma Function</a> (PDF), pp. 18–19, archived from <a rel="nofollow" class="external text" href="http://www.plouffe.fr/simon/math/Artin%20E.%20The%20Gamma%20Function%20(1931)(23s).pdf">the original</a> (PDF) on 2016-11-12, retrieved 2016-11-11</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="https://functions.wolfram.com/GammaBetaErf/Beta/06/03/0001/">Beta function : Series representations (Formula 06.18.06.0007)</a></cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFMäklin2022" class="citation cs2">Mäklin, Tommi (2022), <a rel="nofollow" class="external text" href="https://helda.helsinki.fi/bitstream/handle/10138/349862/M%C3%A4klin_Tommi_dissertation_2022.pdf">Probabilistic Methods for High-Resolution Metagenomics</a> (PDF), Series of publications A / Department of Computer Science, University of Helsinki, Helsinki: Unigrafia, p. 27, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-951-51-8695-9</bdi>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2814-4031">2814-4031</a></cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Euler%27s_Reflection_Formula">"Euler's Reflection Formula - ProofWiki"</a>, proofwiki.org, retrieved 2020-09-02</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFParis2010" class="citation cs2">Paris, R. B. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/5.12">"Beta Function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFZelenSevero1972" class="citation cs2">Zelen, M.; Severo, N. C. (1972), "26. Probability functions", in <a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene A.</a> (eds.), <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>, New York: <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/handbookofmathe000abra/page/944">944</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi></cite>

</li>

<li id="cite_note-paris-ibf-8">^ <a href="#cite_ref-paris-ibf_8-0">a</a> <a href="#cite_ref-paris-ibf_8-1">b</a> <cite id="CITEREFParis2010" class="citation cs2">Paris, R. B. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/8.17">"Incomplete beta functions"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.

</li>

</ol></div></div>

<ul><li><cite id="CITEREFAskeyRoy2010" class="citation cs2"><a href="Richard_Askey" title="Richard Askey">Askey, R. A.</a>; Roy, R. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/5.12">"Beta function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, W. H.; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20211027043154/http://apps.nrbook.com/empanel/index.html?pg=256">"Section 6.1 Gamma Function, Beta Function, Factorials"</a>, Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=256">the original</a> on 2021-10-27, retrieved 2011-08-09</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Beta-function">"Beta-function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="https://planetmath.org/evaluationofbetafunctionusinglaplacetransform">Evaluation of beta function using Laplace transform</a> at <a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</li>

<li>Arbitrarily accurate values can be obtained from:

<ul><li><a rel="nofollow" class="external text" href="http://functions.wolfram.com">The Wolfram functions site</a>: <a rel="nofollow" class="external text" href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized">Evaluate Beta Regularized incomplete beta</a></li>

<li>danielsoper.com: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070120151547/http://www.danielsoper.com/statcalc/calc36.aspx">Incomplete beta function calculator</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070120151557/http://www.danielsoper.com/statcalc/calc37.aspx">Regularized incomplete beta function calculator</a></li></ul></li></ul>

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