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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'' | + | ''$B$-function, Euler $B$-function, Euler integral of the first kind'' |
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− | 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
| + | [[TEX|done]] |
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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/b0159609.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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− | 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:
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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/b01596012.png" /></td> </tr></table>
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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/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 |
| + | \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: | | The following formula is valid: |
− | | + | $$ |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596014.png" /></td> </tr></table>
| + | B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1. |
− | | + | $$ |
− | 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]]: | + | 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]]: |
− | | + | $$ |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015960/b01596019.png" /></td> </tr></table>
| + | B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}. |
| + | $$ |
Revision as of 15:52, 26 April 2012
$B$-function, Euler $B$-function, Euler integral of the first kind
done
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:
$$
B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.
$$
How to Cite This Entry:
Beta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Beta-function&oldid=25504
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article