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| Title of object: |
beta function |
| Canonical Name: |
BetaFunction |
| Type: |
Definition |
| Created on: |
2003-02-09 03:44:12 |
| Modified on: |
2005-03-04 15:31:32 |
| Classification: |
msc:33B15 |
Revision comment (for changes between this and next version):
| Changes for correction #7422 ('typo'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
The beta function is defined as:
$$ B(p,q) = \int_0^1 x^{p-1} (1-x)^{q-1} dx $$
for any $p,q > 0$
The beta fuction has the property:
$$B(p,q) = \frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$$
where $\Gamma$ is the gamma function
Also,
$$B(p,q) = B(q,p)$$ and
$$B(\frac{1}{2},\frac{1}{2}) = \pi$$
The function was discovered by L. Euler (1730) and the name was given by
J. Binet. |
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