Fractional integration and differentiation

An extension of the operations of integration and differentiation to the case of fractional powers. Let $f$ be integrable on the interval $[a,b]$, let $I_1^af(x)$ be the integral of $f$ along $[a,x]$, while $I_\alpha^af(x)$ is the integral of $I_{\alpha-1}^af(x)$ along $[a,b]$, $\alpha=2,3,\dots$. One then has

$$I_\alpha^af(x)=\frac{1}{\Gamma(\alpha)}\int\limits_a^x(x-t)^{\alpha-1}f(t)\,dt,\quad a\leq x\leq b,\label{1}\tag{1}$$

where $\Gamma(\alpha)=(\alpha-1)!$ is the gamma-function. The right-hand side makes sense for every $\alpha>0$. The relation $\eqref{1}$ defines the fractional integral (or the Riemann–Liouville integral) of order $\alpha$ of $f$ with starting point $a$. The operator $I_z^a$ was studied by B. Riemann (1847) for complex values of the parameter $z$. The operator $I_\alpha^a$ is linear and has the semi-group property:

$$I_\alpha^a[I_\beta^af(x)]=I_{\alpha+\beta}^af(x).$$

The operation inverse to fractional integration is known as fractional differentiation: If $I_\alpha f=F$, then $f$ is the fractional derivative of order $\alpha$ of $F$. If $0<\alpha<1$, Marchaut's formula applies:

$$f(x)=\frac{\alpha}{\Gamma(1-\alpha)}\int\limits_0^\infty\left\lbrace\frac{F(x)-F(x-t)}{t^{1+\alpha}}\right\rbrace dt.$$

The concept of fractional integration and differentiation was first introduced by J. Liouville (1832); he studied, in particular, the operator $I_\alpha^{-\infty}=I_\alpha$, $\alpha>0$:

$$I_\alpha f=\frac{1}{\Gamma(\alpha)}\int\limits_{-\infty}^x\frac{f(t)}{(x-t)^{1-\alpha}}dt$$

(subject to appropriate restrictions on $f$; cf. [1], which also contains estimates of the operator $I_\alpha$ in $L_p$).

The following definition (H. Weyl, 1917) is convenient for an integrable $2\pi$-periodic function $f$ with zero average value over the period. If

$$f(x)\sim\sum_{|n|>0}c^ne^{inx}=\sum'c_ne^{inx},$$

then the Weyl integral $f_\alpha$ of order $\alpha>0$ of $f$ is defined by the formula

$$f_\alpha(x)\sim\sum'\frac{c_ne^{inx}}{(in)^\alpha};\label{2}\tag{2}$$

and the derivative $f^\beta$ of order $\beta>0$ is defined by the equation

$$f^\beta(x)=\frac{d^n}{dx^n}f_{n-\beta}(x),$$

where $n$ is the smallest integer larger than $\beta$ (it should be noted that $f_\alpha(x)$ coincides with $I_\alpha f(x)$).

These definitions were further developed in the framework of the theory of generalized functions. For periodic generalized functions

$$f\sim\sum'c_ne^{inx}$$

the operation of fractional integration $I_\alpha f=f_\alpha$ is realized according to formula $\eqref{2}$ for all real $\alpha$ (if $\alpha$ is negative, $I_\alpha f$ coincides with the partial derivative of order $\alpha$) and has the semi-group property with respect to the parameter $\alpha$.

In an $n$-dimensional space $X$ the analogue of the operator of fractional integration is the Riesz potential (or the integral of potential type)

$$R_\alpha f(x)=\pi^{\alpha-n/2}\frac{\Gamma((n-\alpha)/2)}{\Gamma(\alpha/2)}\int\limits_X\frac{f(t)}{|x-t|^{n-\alpha}}dt.$$

The operation inverse to $R_\alpha$ is said to be the Riesz derivative of order $\alpha$.

References

[1]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[2]	A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
[3]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[4]	M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)