Euler integrals

2020 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL]

The integral $$B(p,q)={\int }_0^1{x}^{p-1}(1-x{)}^{q-1}\text{rd}x,p,q>0,$$ called the Euler integral of the first kind, or the beta-function, and $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{e}^{-x}\text{rd}x,$$ 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).