fractional integration

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:

$D^{-n}(f)(x)=\int_{t_{n}=0}^{x}dt_{n}\ldots\int_{t_{1}=0}^{t_{2}}f(t_{1})\,dt_% {1}=\frac{1}{(n-1)!}\int_{t=0}^{x}f(t)(x-t)^{n-1}\,dt$

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

$I^{\alpha}_{L}(f)(s,t)=\frac{1}{\Gamma(\alpha)}\int_{u=s}^{t}f(u)(t-u)^{\alpha% -1}\,du=\int_{u=s}^{t}f(u)\,dg^{\alpha}_{t}(u)$

where

$g^{\alpha}_{t}(u)=\frac{t^{\alpha}-(t-u)^{\alpha}}{\Gamma(\alpha+1)}$

Definition 2: Right-Hand Riemann-Liouville Integration

$I^{\alpha}_{R}(f)(s,t)=\frac{1}{\Gamma(\alpha)}\int_{u=s}^{t}f(u)(u-s)^{\alpha% -1}\,du=\int_{u=s}^{t}f(u)\,dh^{\alpha}_{t}(u)$

where

$h^{\alpha}_{t}(u)=\frac{s^{\alpha}+(u-s)^{\alpha}}{\Gamma(\alpha+1)}$

Definition 3: Riesz Potential

$I^{\alpha}_{C}(f)(s,t;p)=\frac{1}{\Gamma(\alpha)}\int_{u=s}^{t}f(u)|u-p|^{% \alpha-1}\,du=\int_{u=s}^{t}f(u)\,dr^{\alpha}_{p}(u)$

where

$r^{\alpha}_{p}(u)=\frac{p^{\alpha}+{\rm sign}(u-p)|u-p|^{\alpha}}{\Gamma(% \alpha+1)}$

${\rm sign}(x)=1$ 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 $x$