# evaluation of beta function using Laplace transform

The beta integral can be evaluated elegantly using the convolution theorem (http://planetmath.org/LaplaceTransform) for Laplace transforms.

 $s^{-\alpha}={\cal L}\left[{t^{\alpha-1}\over\Gamma(\alpha)}\right]=\int_{0}^{% \infty}e^{-st}{t^{\alpha-1}\over\Gamma(\alpha)}dt$

Since $s^{-q}s^{-p}=s^{-q-p}$, the convolution theorem imples that

 ${t^{q-1}\over\Gamma(q)}*{t^{p-1}\over\Gamma(p)}={t^{q+p-1}\over\Gamma(q+p)}$

Writing out the definition of convolution, this becomes

 $\int_{0}^{t}{(t-s)^{q-1}\over\Gamma(q)}{s^{p-1}\over\Gamma(p)}ds={t^{q+p-1}% \over\Gamma(p+q)}$

Setting $t=1$ and simplifying, we conclude that

 $\int_{0}^{1}x^{p-1}(1-x)^{q-1}\,dx={\Gamma(p)\Gamma(q)\over\Gamma(p+q)}$

QED

Title evaluation of beta function using Laplace transform EvaluationOfBetaFunctionUsingLaplaceTransform 2013-03-22 14:37:36 2013-03-22 14:37:36 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Derivation msc 33B15