Difference between revisions of "Euler integrals"
From Encyclopedia of Mathematics
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The integral | 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|beta-function]], and | called the Euler integral of the first kind, or the [[Beta-function|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 $s>0$ and is a representation of the [[Gamma-function|gamma-function]].) | ||
| − | + | These integrals were considered by L. Euler (1729–1731). | |
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| − | These integrals were considered by L. Euler (1729–1731). | ||
Latest revision as of 21:21, 29 April 2012
2020 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL]
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 $s>0$ 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=16378
Euler integrals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_integrals&oldid=16378
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article