Difference between revisions of "Euler integrals"
From Encyclopedia of Mathematics
(TeX) |
|||
| Line 7: | Line 7: | ||
\int_0^\infty x^{s-1}e^{-x} \rd x, | \int_0^\infty x^{s-1}e^{-x} \rd x, | ||
$$ | $$ | ||
| − | called the Euler integral of the second kind. (The latter converges for $s> | + | 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). | These integrals were considered by L. Euler (1729–1731). | ||
Revision as of 21:19, 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 $s>0$ 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=25723
Euler integrals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_integrals&oldid=25723
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article