Problem 1349 (difficulty: 8/10)

For all continuous functions \(\displaystyle f:\R\to\R\) let \(\displaystyle I_0f=f\) and for \(\displaystyle a\ge 0\) let \(\displaystyle I_af\) be the function for which

\(\displaystyle (I_a f)(x) = \int_0^x f(t) \frac{(x-y)^{a-1}}{\Gamma(a)} \dx. \)

Prove that (a) \(\displaystyle (I_1f)(x)=\int_0^xf\); (b) \(\displaystyle I_{a+b}=I_aI_b\).

Give me another random problem!