I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Can anyone help me solve this? I really appreciate.
If $f$ is continuous on $[0, \infty)$, for $\alpha \gt 0$ and $x \ge 0$, let $$I_{\alpha}f(x) = \frac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-t)^{\alpha -1}f(t)dt$$
$I_{\alpha}f$ is called the $\alpha th$ fractional integral of $f$, here $\Gamma(x) = \int_{0}^{\infty}t^{x-1}e^{-t}dt$. Prove that $I_{\alpha + \beta}f = I_{\alpha}(I_{\beta}f)$ for all $\alpha, \beta \gt 0$.
I know the related Gamma equation: $\Gamma(x)\Gamma(y)/\Gamma(x+y) = \int_{0}^{1}t^{x-1}(1-t)^{y-1}dt$, but I still can't solve this. Thanks so much.