Euler integrals

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