alternate integral representation of beta function

By making the change of variable $x^{p}=y$, we see that

 $\int_{0}^{1}x^{p-1}(1-x)^{q-1}\,dx={1\over p}\int_{0}^{1}(1-y^{1\over p})^{q-1% }\,dy.$

Hence, we have

 $\int_{0}^{1}(1-y^{1\over p})^{q-1}\,dy=p{\Gamma(p)\Gamma(q)\over\Gamma(p+q)}.$
Title alternate integral representation of beta function AlternateIntegralRepresentationOfBetaFunction 2013-03-22 17:10:08 2013-03-22 17:10:08 rspuzio (6075) rspuzio (6075) 4 rspuzio (6075) Result msc 33B15