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(References: Jeffreys & Jeffreys (1972))
 
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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b0159603.png" />-function, Euler <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b0159605.png" />-function, Euler integral of the first kind''
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''$B$-function, Euler $B$-function, Euler integral of the first kind''
  
A function of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b0159606.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b0159607.png" /> which, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b0159608.png" />, is defined by the equation
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{{MSC|33B15}}
 
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{{TEX|done}}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b0159609.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$\newcommand{\Re}{\mathop{\mathrm{Re}}}$
 
 
The values of the beta-function for various values of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596011.png" /> are connected by the following relationships:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596012.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596013.png" /></td> </tr></table>
 
  
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A function of two variables $p$ and $q$ which, for $p,\,q > 0$, is defined by the equation
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\begin{equation}
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\label{eq1}
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B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1} \rd x.
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\end{equation}
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The values of the beta-function for various values of the parameters $p$ and $q$ are connected by the following relationships:
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$$
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B(p,q)  = B(q,p),
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$$
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$$
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B(p,q) = \frac{q-1}{p+q-1}B(p,q-1), \quad q > 1.
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$$
 
The following formula is valid:
 
The following formula is valid:
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$$
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B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1.
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$$
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If $p$ and $q$ are complex, the integral \ref{eq1} converges if $\Re p > 0$ and $\Re q > 0$. The beta-function can be expressed by the [[Gamma-function|gamma-function]]:
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$$
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B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596014.png" /></td> </tr></table>
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====References====
 
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* Harold Jeffreys, Bertha Jeffreys, ''Methods of Mathematical Physics'', 3rd edition, Cambridge University Press (1972) Zbl 0238.00004
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596016.png" /> are complex, the integral (*) converges if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596018.png" />. The beta-function can be expressed by the [[Gamma-function|gamma-function]]:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596019.png" /></td> </tr></table>
 

Latest revision as of 23:17, 17 October 2014

$B$-function, Euler $B$-function, Euler integral of the first kind

2010 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL] $\newcommand{\Re}{\mathop{\mathrm{Re}}}$

A function of two variables $p$ and $q$ which, for $p,\,q > 0$, is defined by the equation \begin{equation} \label{eq1} B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1} \rd x. \end{equation} The values of the beta-function for various values of the parameters $p$ and $q$ are connected by the following relationships: $$ B(p,q) = B(q,p), $$ $$ B(p,q) = \frac{q-1}{p+q-1}B(p,q-1), \quad q > 1. $$ The following formula is valid: $$ B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1. $$ If $p$ and $q$ are complex, the integral \ref{eq1} converges if $\Re p > 0$ and $\Re q > 0$. The beta-function can be expressed by the gamma-function: $$ B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}. $$

References

  • Harold Jeffreys, Bertha Jeffreys, Methods of Mathematical Physics, 3rd edition, Cambridge University Press (1972) Zbl 0238.00004
How to Cite This Entry:
Beta-function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Beta-function&oldid=14450
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article