Difference between revisions of "Beta-function"
From Encyclopedia of Mathematics
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| − | '' | + | ''$B$-function, Euler $B$-function, Euler integral of the first kind'' |
| − | + | {{MSC|33B15}} | |
| − | + | {{TEX|done}} | |
| − | + | $\newcommand{\Re}{\mathop{\mathrm{Re}}}$ | |
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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: | ||
| + | $$ | ||
| + | B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1. | ||
| + | $$ | ||
| + | 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]]: | ||
| + | $$ | ||
| + | B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}. | ||
| + | $$ | ||
| − | + | ====References==== | |
| − | + | * Harold Jeffreys, Bertha Jeffreys, ''Methods of Mathematical Physics'', 3<sup>rd</sup> edition, Cambridge University Press (1972) {{ZBL|0238.00004}} | |
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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
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