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The basic idea of "Riemann-Liouville" type fractional integration comes from the following observation:
Given any integrable function $f:{\mathbb R}\mapsto {\mathbb R}$ in one variable, we have the following Cauchy Integration Formula:
when switching the index from integer $n$ to non-integer $\alpha$ gives the ideas of the following definitions:
Definition 1: Left-Hand Riemann-Liouville Integration
where
Definition 2: Right-Hand Riemann-Liouville Integration
where
Definition 3: Riesz Potential
where
,
${\rm sign}(x)=1$ for $x>0$ ${\rm sign}(x)=0$ for $x=0$ ${\rm sign}(x)=-1$ for $x<0$ and $\Gamma(x)$ is the gamma function of $x$
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