Riemann–Liouville integral

Summary

The Riemann–Liouville integral of order $\alpha > 0$ is the integral operator $I_a^\alpha f(x) = \frac{1}{\Gamma(\alpha)} \int_a^x (x - t)^{\alpha - 1} f(t) \, dt$. For a whole number $\alpha = n$ it is the $n$-fold repeated integral, by Cauchy's formula for repeated integration. It is the basic operator of fractional calculus, and fractional derivatives are defined from it.