Riemann–Liouville integral

The Riemann–Liouville integral is defined by

${\displaystyle I^{\alpha }f(x)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{x}f(t)(x-t)^{\alpha -1}\,dt}$

where Γ is the Gamma function and a is an arbitrary but fixed base point. The integral is well-defined provided ${\displaystyle f}$ is a locally integrable function, and α is a complex number in the half-plane re(α) > 0. The dependence on the base-point a is often suppressed, and represents a freedom in constant of integration. Clearly ${\displaystyle I^{1}f}$ is an antiderivative of ${\displaystyle f}$ (of first order), and for positive integer values of α, ${\displaystyle I^{\alpha }f}$ is an antiderivative of order α by Cauchy formula for repeated integration. Another notation, which emphasizes the basepoint, is[3]

${\displaystyle {}_{a}D_{x}^{-\alpha }f(x)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{x}f(t)(x-t)^{\alpha -1}\,dt.}$

This also makes sense if a = −∞, with suitable restrictions on ${\displaystyle f}$.

The fundamental relations hold

${\displaystyle {\frac {d}{dx}}I^{\alpha +1}f(x)=I^{\alpha }f(x),\quad I^{\alpha }(I^{\beta }f)=I^{\alpha +\beta }f,}$

the latter of which is a semigroup property.[1] These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of ${\displaystyle I^{\alpha }f}$.

Fix a bounded interval (a,b). The operator I^α associates to each integrable function ${\displaystyle f}$ on (a,b) the function ${\displaystyle I^{\alpha }f}$ on (a,b) which is also integrable by Fubini's theorem. Thus ${\displaystyle I^{\alpha }}$ defines a linear operator on L¹(a,b):

${\displaystyle I^{\alpha }:L^{1}(a,b)\to L^{1}(a,b).}$

Fubini's theorem also shows that this operator is continuous with respect to the Banach space structure on L¹, and that the following inequality holds:

${\displaystyle \left\|I^{\alpha }f\right\|_{1}\leq {\frac {|b-a|^{\Re (\alpha )}}{\Re (\alpha )|\Gamma (\alpha )|}}\|f\|_{1}.}$

Here ${\displaystyle \|\cdot \|_{1}}$ denotes the norm on L¹(a,b).

More generally, by Hölder's inequality, it follows that if ${\displaystyle f\in L^{p}(a,b),}$ then ${\displaystyle I^{\alpha }f\in L^{p}(a,b)}$ as well, and the analogous inequality holds

${\displaystyle \left\|I^{\alpha }f\right\|_{p}\leq {\frac {|b-a|^{\frac {\Re (\alpha )}{p}}}{\Re (\alpha )|\Gamma (\alpha )|}}\|f\|_{p}}$

where ${\displaystyle \|\cdot \|_{p}}$ is the L^p norm on the interval (a,b). Thus we have a bounded linear operator ${\displaystyle I^{\alpha }:L^{p}(a,b)\to L^{p}(a,b).}$ Furthermore, ${\displaystyle I^{\alpha }f\to f}$ in the L^p sense as α → 0 along the real axis. That is

${\displaystyle {\underset {\alpha >0}{\lim _{\alpha \to 0}}}\|I^{\alpha }f-f\|_{p}=0}$

for all p ≥ 1. Moreover, by estimating the maximal function of I, one can show that the limit ${\displaystyle I^{\alpha }f\to f}$ holds pointwise almost everywhere.

The operator ${\displaystyle I^{\alpha }}$ is well-defined on the set of locally integrable function on the whole real line ${\displaystyle \mathbb {R} .}$ It defines a bounded transformation on any of the Banach spaces of functions of exponential type ${\displaystyle X_{\sigma }=L^{1}(e^{-\sigma |t|}dt),}$ consisting of locally integrable functions for which the norm

${\displaystyle \|f\|=\int _{-\infty }^{\infty }|f(t)|e^{-\sigma |t|}\,dt}$

is finite. For ${\displaystyle f\in X_{\sigma },}$ the Laplace transform of ${\displaystyle I^{\alpha }f}$ takes the particularly simple form

${\displaystyle ({\mathcal {L}}I^{\alpha }f)(s)=s^{-\alpha }F(s)}$

for re(s) > σ. Here F(s) denotes the Laplace transform of ${\displaystyle f}$, and this property expresses that ${\displaystyle I^{\alpha }}$ is a Fourier multiplier.

One can define fractional-order derivatives of ${\displaystyle f}$ as well by

${\displaystyle {\frac {d^{\alpha }}{dx^{\alpha }}}f{\overset {\text{def}}{=}}{\frac {d^{\lceil \alpha \rceil }}{dx^{\lceil \alpha \rceil }}}I^{\lceil \alpha \rceil -\alpha }f}$

where ${\displaystyle \lceil \cdot \rceil }$ denotes the ceiling function. One also obtains a differintegral interpolating between differentiation and integration by defining

${\displaystyle D_{x}^{\alpha }f(x)={\begin{cases}{\frac {d^{\lceil \alpha \rceil }}{dx^{\lceil \alpha \rceil }}}I^{\lceil \alpha \rceil -\alpha }f(x)&\alpha >0\\f(x)&\alpha =0\\I^{-\alpha }f(x)&\alpha <0.\end{cases}}}$

An alternative fractional derivative was introduced by Caputo in 1967, and produces a derivative that has different properties: it produces zero from constant functions and, more importantly, the initial value terms of the Laplace Transform are expressed by means of the values of that function and of its derivative of integer order rather than the derivatives of fractional order as in the Riemann–Liouville derivative. The Caputo fractional derivative with base point ${\displaystyle x}$, is then:

${\displaystyle D_{x}^{\alpha }f(y)={\frac {1}{\Gamma (1-\alpha )}}\int _{x}^{y}f'(y-u)(u-x)^{-\alpha }du.}$

Another representation is:

${\displaystyle {}_{a}{\tilde {D}}_{x}^{\alpha }f(x)=I^{\lceil \alpha \rceil -\alpha }\left({\frac {d^{\lceil \alpha \rceil }f}{dx^{\lceil \alpha \rceil }}}\right).}$

• Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Euler transformation", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.
• Hille, Einar; Phillips, Ralph S. (1974), Functional analysis and semi-groups, Providence, R.I.: American Mathematical Society, MR 0423094.
• Lizorkin, P.I. (2001) [1994], "Fractional integration and differentiation", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.
• Miller, Kenneth S.; Ross, Bertram (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, ISBN 0-471-58884-9.
• Riesz, Marcel (1949), "L'intégrale de Riemann-Liouville et le problème de Cauchy", Acta Mathematica, 81 (1): 1–223, doi:10.1007/BF02395016, ISSN 0001-5962, MR 0030102.

• Alan Beardon (2000). "Fractional calculus II". University of Cambridge.
• Alan Beardon (2000). "Fractional calculus III". University of Cambridge.