Namespaces
Variants
Actions

Difference between revisions of "Beta-function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(→‎References: zbl link)
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
''<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''
  
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
+
{{MSC|33B15}}
 
+
{{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>
+
$\newcommand{\Re}{\mathop{\mathrm{Re}}}$
 
 
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>
 
  
 +
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:
 +
$$
 +
B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1.
 +
$$
 +
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|gamma-function]]:
 +
$$
 +
B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.
 +
$$
  
<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>
+
====References====
 
+
* Harold Jeffreys, Bertha Jeffreys, ''Methods of Mathematical Physics'', 3<sup>rd</sup> 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 17:31, 11 November 2023

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

2020 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://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