Reference request: Calculating fractional integrals

Question

I am trying to calculate/solve integrals of the type $f(x) = \int_0^x (x-t)^{\alpha-1} u(t) dt$ for a given $u(t)$ and $\alpha > 0$. Doing so by hand is pretty tedious, as you can imagine. This easier to solve if I choose $\alpha$ to be a nice fraction. But I struggle with getting a formula for arbitrary $\alpha$.

As I am pretty sure that these types of integrals are well studied since Abel first looked at them, is there a handbook, guide, a collection of known integrals and their solution or something similar that would help me solve those? My Google-foo has failed me on the quest to locate anything (probably been looking for the wrong terms).

Answer 1

Fractional Calculus is tangentially related to what I study. It's not my forte, but I've found "Fractional Integrals and Derivatives" by Samko, Kilbas, and Marichev a comprehensive and widely used reference.

If you need some better google search terms, the integral/problem you are interested in goes by the "Riemann-Liouville Fractional Integral" or "Abel Integral Equation."

Answer 2

As far as I know, one of the most extended table of fractional integrals can be found in the book :

H.Bateman, "Tables of Integral Transforms", Chapter XIII : Fractional Integrals, pp.181-212, McGraw-Hill Edit. 1954. I don't know if an open version of the book exists on the web.

Especially, the fractional integrals of the sinusoidal functions are useful in the field of complex impedance analysis : https://fr.scribd.com/doc/71923015/The-Phasance-Concept .