What does a Fractional Integral Look Like?
What is a Fractional Integral? It was shown a previous post how to define and calculate a function’s $\alpha$-th fractional integral, where $\alpha$ is a non-negative real number. Using this technique one can also define a function’s $\alpha$-th primitive. Now one will be able to see what such a primitive actually looks like through three examples. Recall that the $\alpha$-th primitive of $f_0$ is defined as $$ f_{\alpha}(x) := \frac{1}{\Gamma(\alpha)} \int_0^{x} (x - s)^{\alpha-1} f_0(s) ds. $$ ...