fractional integration

The basic idea of ”Riemann-Liouville” type fractional integration comes from the following observation:

Given any integrable function $f:ℝ\mapsto ℝ$ in one variable, we have the following Cauchy Integration Formula:

$$D^{-n}(f)(x)={\int }_{t_n=0}^x𝑑t_n\mathrm{\dots }{\int }_{t_1=0}^{t_2}f(t_1)𝑑t_1=\frac{1}{(n-1)!}{\int }_{t=0}^{x}f(t){(x-t)}^{n-1}𝑑t$$

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

$$I_L^{\alpha }(f)(s,t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_{u=s}^{t}f(u){(t-u)}^{\alpha -1}𝑑u={\int }_{u=s}^{t}f(u)𝑑g_t^{\alpha }(u)$$

where

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

Definition 2: Right-Hand Riemann-Liouville Integration

$$I_R^{\alpha }(f)(s,t)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_{u=s}^{t}f(u){(u-s)}^{\alpha -1}𝑑u={\int }_{u=s}^{t}f(u)𝑑h_t^{\alpha }(u)$$

where

$$h_t^{\alpha }(u)=\frac{s^{\alpha }+{(u-s)}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}$$

Definition 3: Riesz Potential

$$I_C^{\alpha }(f)(s,t;p)=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_{u=s}^{t}f(u){|u-p|}^{\alpha -1}𝑑u={\int }_{u=s}^{t}f(u)𝑑r_p^{\alpha }(u)$$

where

$$r_p^{\alpha }(u)=\frac{p^{\alpha }+\mathrm{sign}(u-p){|u-p|}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}$$

,

$\mathrm{sign}(x)=1$ for $x>0$, $\mathrm{sign}(x)=0$ for $x=0$, $\mathrm{sign}(x)=-1$ for

and $\mathrm{\Gamma }(x)$ is the gamma function of $x$