Can someone give me the steps for proving the following integral:
$\frac{1}{\sqrt{\pi}}\int_{0}^{t}\left(t-\tau\right)^{-1/2}\tau^{-\alpha}d\tau=\frac{\Gamma\left(1-\alpha\right)}{\Gamma\left(3/2-\alpha\right)}t^{1/2-\alpha}$
Other Information:
$t>0$
$0\leq\alpha<1$
I have obtained the above solution using Mathematica, but I want to learn the steps. This is not my HOMEWORK! I am learning convolution and PDEs on my own. Thanks!
EDIT I have also obtained the above solution of the integral using Laplace transform. But I am interested in learning the algebraic steps/solution without going to Laplace transform.