# Euler integrals

From Encyclopedia of Mathematics

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=25725

This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article