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Difference between revisions of "Euler integrals"

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The integral
 
The integral
 
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$$
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B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1}\rd x, \quad p,q > 0,
 
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$$
 
called the Euler integral of the first kind, or the [[Beta-function|beta-function]], and
 
called the Euler integral of the first kind, or the [[Beta-function|beta-function]], and
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$$
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\int_0^\infty x^{s-1}e^{-x} \rd x,
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$$
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called the Euler integral of the second kind. (The latter converges for $s>°$ and is a representation of the [[Gamma-function|gamma-function]].)
  
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These integrals were considered by L.&nbsp;Euler (1729–1731).
 
 
called the Euler integral of the second kind. (The latter converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036500/e0365003.png" /> and is a representation of the [[Gamma-function|gamma-function]].)
 
 
 
These integrals were considered by L. Euler (1729–1731).
 

Revision as of 21:17, 29 April 2012

The integral $$ B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1}\rd x, \quad p,q > 0, $$ called the Euler integral of the first kind, or the beta-function, and $$ \int_0^\infty x^{s-1}e^{-x} \rd x, $$ called the Euler integral of the second kind. (The latter converges for $s>°$ and is a representation of the gamma-function.)

These integrals were considered by L. Euler (1729–1731).

How to Cite This Entry:
Euler integrals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_integrals&oldid=25722
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article