View source for Riemann–Liouville integral

{{Calculus|expanded=Fractional calculus}}
In [[Mathematics|mathematics]], the '''Riemann–Liouville integral''' associates with a real [[Function (mathematics)|function]] <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> another function {{math|''I''<sup>''α''</sup> ''f''}} of the same kind for each value of the parameter {{math|''α'' > 0}}. The integral is a manner of generalization of the repeated [[Antiderivative|antiderivative]] of {{mvar|f}} in the sense that for positive integer values of {{mvar|α}}, {{math|''I''<sup>''α''</sup> ''f''}} is an iterated antiderivative of {{mvar|f}} of order {{mvar|α}}. The Riemann–Liouville integral is named for [[Biography:Bernhard Riemann|Bernhard Riemann]] and [[Biography:Joseph Liouville|Joseph Liouville]], the latter of whom was the first to consider the possibility of [[Fractional calculus|fractional calculus]] in 1832.<ref name="Lizorkin">{{harvnb|Lizorkin|2001}}</ref><ref>{{Citation|last=Liouville|first=Joseph|year=1832|title=Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions| journal=Journal de l'École Polytechnique|volume=13|pages=1–69|location=Paris|url=https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f2.item.r=Joseph%20Liouville}}.</ref><ref>{{Citation|last=Liouville|first=Joseph|author-link=Joseph Liouville| year=1832| title=Mémoire sur le calcul des différentielles à indices quelconques|journal=Journal de l'École Polytechnique| volume=13| pages=71–162|location=Paris|url=https://gallica.bnf.fr/ark:/12148/bpt6k4336778/f72.image}}.</ref><ref>{{Citation |last=Riemann| first=Georg Friedrich Bernhard|author-link=Georg Friedrich Bernhard Riemann|editor=[[Biography:Heinrich Friedrich Weber|Weber, H.]]| origyear=1847|year=1896|chapter=Versuch einer allgemeinen Auffassung der integration und differentiation|title=Gesammelte Mathematische Werke|location=Leipzig|chapter-url=https://diglib.tugraz.at/download.php?id=4f6ae22892b1b&location=browse}}.</ref> The operator agrees with the '''Euler transform''', after [[Biography:Leonhard Euler|Leonhard Euler]], when applied to [[Analytic function|analytic function]]s.<ref>{{harvnb|Brychkov|Prudnikov|2001}}</ref> It was generalized to arbitrary dimensions by [[Biography:Marcel Riesz|Marcel Riesz]], who introduced the [[Riesz potential]].

==Motivation==
The Riemann-Liouville integral is motivated from Cauchy formula for repeated integration. Let f be continuous on [{{mvar|α}},x], then Cauchy repeated integration formula states that  

<math display="block">f^{(-n)}(x) = \frac{1}{(n-1)!} \int_a^x\left(x-t\right)^{n-1} f(t)\,\mathrm{d}t.</math>

Now, this formula can be generalized to any positive real number by replacing positive integer n with {{mvar|α}}, Therefore we obtain the definition of Riemann-Liouville fractional Integral by 
:<math>I^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_a^xf(t)(x-t)^{\alpha-1}\,dt</math>

==Definition==
The Riemann–Liouville integral is defined by

:<math>I^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_a^xf(t)(x-t)^{\alpha-1}\,dt</math>

where {{math|Γ}} is the [[Gamma function|gamma function]] and {{mvar|a}} is an arbitrary but fixed base point. The integral is well-defined provided {{mvar|f}} is a [[Locally integrable function|locally integrable function]], and {{mvar|α}} is a [[Complex number|complex number]] in the half-plane {{math|Re(''α'') > 0}}. The dependence on the base-point {{mvar|a}} is often suppressed, and represents a freedom in [[Constant of integration|constant of integration]]. Clearly {{math|''I''<sup>1</sup> ''f''}} is an antiderivative of {{mvar|f}} (of first order), and for positive integer values of {{mvar|α}}, {{math|''I''<sup>''α''</sup> ''f''}} is an antiderivative of order {{mvar|α}} by [[Cauchy formula for repeated integration]]. Another notation, which emphasizes the base point, is<ref>{{harvnb|Miller|Ross|1993|p=21}}</ref>

:<math>{}_aD_x^{-\alpha}f(x) = \frac{1}{\Gamma(\alpha)}\int_a^x f(t)(x-t)^{\alpha-1}\,dt.</math>

This also makes sense if {{math|1=''a'' = −∞}}, with suitable restrictions on {{mvar|f}}.

The fundamental relations hold

:<math>\frac{d}{dx}I^{\alpha+1} f(x) = I^\alpha f(x),\quad I^\alpha(I^\beta f) = I^{\alpha+\beta}f,</math>

the latter of which is a [[Semigroup|semigroup]] property.<ref name="Lizorkin" /> These properties make possible not only the definition of fractional integration, but also of fractional differentiation, by taking enough derivatives of {{math|''I''<sup>''α''</sup> ''f''}}.

==Properties==
Fix a bounded interval {{open-open|''a'',''b''}}. The operator {{math|''I''<sup>''α''</sup>}} associates to each integrable function {{mvar|f}} on {{open-open|''a'',''b''}} the function {{math|''I''<sup>''α''</sup> ''f''}} on {{open-open|''a'',''b''}} which is also integrable by [[Fubini's theorem]]. Thus {{math|''I''<sup>''α''</sup>}} defines a linear operator on [[Lp space|{{math|''L''<sup>1</sup>(''a'',''b'')}}]]:

:<math>I^\alpha : L^1(a,b) \to L^1(a,b).</math>

Fubini's theorem also shows that this operator is [[Continuous linear operator|continuous]] with respect to the [[Banach space]] structure on {{mvar|L}}<sup>1</sup>, and that the following inequality holds:

:<math>\left \|I^\alpha f \right \|_1 \le \frac{|b-a|^{\Re(\alpha)}}{\Re(\alpha)|\Gamma(\alpha)|}\|f\|_1.</math>

Here {{math|‖ · ‖<sub>1</sub>}} denotes the [[Norm (mathematics)|norm]] on {{math|''L''<sup>1</sup>(''a'',''b'')}}.

More generally, by [[Hölder's inequality]], it follows that if {{math|''f'' ∈ ''L''<sup>''p''</sup>(''a'', ''b'')}}, then {{math|''I''<sup>''α''</sup> ''f'' ∈ ''L''<sup>''p''</sup>(''a'', ''b'')}} as well, and the analogous inequality holds

:<math>\left \|I^\alpha f \right \|_p \le \frac{|b-a|^{\Re(\alpha)/p}}{\Re(\alpha)|\Gamma(\alpha)|}\|f\|_p</math>

where {{math|‖ · ‖<sub>''p''</sub>}} is the [[Lp space|{{mvar|L}}<sup>{{mvar|p}}</sup> norm]] on the interval {{open-open|''a'',''b''}}. Thus we have a bounded linear operator {{math|''I''<sup>''α''</sup> : ''L''<sup>''p''</sup>(''a'', ''b'') → ''L''<sup>''p''</sup>(''a'', ''b'')}}. Furthermore, {{math|''I''<sup>''α''</sup> ''f'' → ''f''}} in the {{math|''L''<sup>''p''</sup>}} sense as {{math|''α'' → 0}} along the real axis. That is

:<math>\lim_{\alpha\to 0^+} \|I^\alpha f - f\|_p = 0</math>

for all {{math|''p'' ≥ 1}}. Moreover, by estimating the [[Maximal function|maximal function]] of {{mvar|I}}, one can show that the limit {{math|''I''<sup>''α''</sup> ''f'' → ''f''}} holds pointwise [[Almost everywhere|almost everywhere]].

The operator {{math|''I''<sup>''α''</sup>}} is well-defined on the set of locally integrable function on the whole real line <math>\mathbb{R}</math>. It defines a bounded transformation on any of the [[Banach space]]s of functions of [[Exponential type|exponential type]] <math>X_{\sigma} = L^1(e^{-\sigma|t|}dt),</math> consisting of locally integrable functions for which the norm

:<math>\|f\| = \int_{-\infty}^\infty |f(t)|e^{-\sigma|t|}\,dt</math>

is finite. For {{math|''f'' ∈ ''X''<sub>''σ''</sub>}}, the [[Laplace transform]] of {{math|''I''<sup>''α''</sup> ''f''}} takes the particularly simple form

:<math>(\mathcal{L}I^\alpha f)(s) = s^{-\alpha}F(s)</math>

for {{math|Re(''s'') > ''σ''}}. Here {{math|''F''(''s'')}} denotes the Laplace transform of {{mvar|f}}, and this property expresses that {{math|''I''<sup>''α''</sup>}} is a Fourier multiplier.

==Fractional derivatives==
One can define fractional-order derivatives of {{mvar|f}} as well by

:<math>\frac{d^\alpha}{dx^\alpha} f \overset{\text{def}}{=} \frac{d^{\lceil\alpha\rceil}}{dx^{\lceil\alpha\rceil}} I^{\lceil \alpha \rceil-\alpha}f</math>

where {{math|⌈ · ⌉}} denotes the [[Ceiling function|ceiling function]]. One also obtains a [[Differintegral|differintegral]] interpolating between differentiation and integration by defining

:<math>D^\alpha_x 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}</math>

An alternative fractional derivative was introduced by Caputo in 1967,<ref>{{Harvnb|Caputo|1967}}</ref> 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.<ref>{{Harvnb|Loverro|2004}}</ref> The Caputo fractional derivative with base point {{mvar|x}}, is then:

:<math>D_x^{\alpha}f(y)=\frac{1}{\Gamma(1-\alpha)}\int_x^y f'(y-u)(u-x)^{-\alpha}du.</math>

Another representation is:

:<math>{}_a\tilde{D}^\alpha_x f(x)=I^{\lceil \alpha\rceil-\alpha}\left(\frac{d^{\lceil \alpha\rceil}f}{dx^{\lceil \alpha\rceil}}\right).</math>

==Fractional derivative of a basic power function==
[[File:Half-derivative.svg|right|thumb|320px|The half derivative (purple curve) of the function {{math|1=''f''(''x'') = ''x''}} (blue curve) together with the first derivative (red curve).]]

[[File:Fractional Derivative of Basic Power Function (2014).gif|right|thumb|320px|The animation shows the derivative operator oscillating between the [[Antiderivative|antiderivative]] ({{math|1=''α'' = −1}}: {{math|1=''y'' = {{sfrac|1|2}}''x''<sup>2</sup>}}) and the derivative ({{math|1=''α'' = +1}}: {{math|1=''y'' = 1}}) of the simple power function {{math|1=''y'' = ''x''}} continuously.]]

Let us assume that {{math|''f''(''x'')}} is a [[Monomial|monomial]] of the form
:<math>f(x) = x^k\,.</math>

The first derivative is as usual
:<math>f'(x) = \frac{d}{dx}f(x) = k x^{k-1}\,.</math>

Repeating this gives the more general result that
:<math>\frac{d^a}{dx^a}x^k = \dfrac{k!}{(k-a)!}x^{k-a}\,,</math>
which, after replacing the [[Factorial|factorial]]s with the [[Gamma function|gamma function]], leads to
:<math>\frac{d^a}{dx^a}x^k = \dfrac{\Gamma(k+1)}{\Gamma(k-a+1)}x^{k-a}, \quad\ k > 0.</math>

For {{math|1=''k'' = 1}} and {{math|1=''a'' = {{sfrac|1|2}}}}, we obtain the half-derivative of the function <math>x \mapsto x</math> as
:<math>\frac{d^\frac12}{dx^\frac12}x=\frac{\Gamma(1+1)}{\Gamma\left(1-\frac12 + 1\right)} x^{1-\frac12}=\frac{\Gamma(2)}{\Gamma\left(\frac{3}{2}\right)}x^\frac12 = \frac{1}{\frac{\sqrt{\pi}}{2}}x^\frac12.</math>

To demonstrate that this is, in fact, the "half derivative" (where {{math|1=''H''{{i_sup|2}}''f''(''x'') = ''Df''(''x'')}}), we repeat the process to get:
:<math>\dfrac{d^\frac12}{dx^\frac12} \dfrac{2x^{\frac12}}{\sqrt{\pi}}
=\frac{2}{\sqrt{\pi}}\dfrac{\Gamma(1+\frac12)}{\Gamma(\frac12-\frac12+1)}x^{\frac12-\frac12}
=\frac{2}{\sqrt{\pi}} \frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma(1)} x^0
=\frac{2 \frac{\sqrt{\pi}}{2} x^0}{\sqrt{\pi}}=1\,,</math>

(because <math display="inline">\Gamma\!\left(\frac{3}{2}\right) = \frac{\sqrt{\pi}}{2}</math> and {{math|1=Γ(1) = 1}}) which is indeed the expected result of
:<math>\left(\frac{d^\frac12}{dx^\frac12} \frac{d^\frac12}{dx^\frac12}\right)\!x = \frac{d}{dx} x = 1\,.</math>

For negative integer power {{math|''k''}}, 1/<math display="inline">\Gamma</math> is 0, so it is convenient to use the following relation:<ref>{{Citation |last1=Bologna |first1=Mauro |title=Short Introduction to Fractional Calculus |publisher=Universidad de Tarapaca, Arica, Chile |url=http://www.uta.cl/charlas/volumen19/Indice/MAUROrevision.pdf |access-date=2014-04-06 |url-status=dead |archive-url=https://web.archive.org/web/20161017232309/http://www.uta.cl/charlas/volumen19/Indice/MAUROrevision.pdf |archive-date=2016-10-17}}</ref>
:<math>\frac{d^a}{dx^a}x^{-k} = \left(-1\right)^a\dfrac{\Gamma(k+a)}{\Gamma(k)}x^{-(k+a)} \quad\text{ for } k \ge 0.</math>

This extension of the above differential operator need not be constrained only to real powers; it also applies for complex powers. For example, the {{math|(1 + ''i'')}}-th derivative of the {{math|(1 − ''i'')}}-th derivative yields the second derivative. Also setting negative values for {{mvar|a}} yields integrals.

For a general function {{math|''f''(''x'')}} and {{math|0 < ''α'' < 1}}, the complete fractional derivative is
:<math>D^\alpha f(x)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dx} \int_0^x \frac{f(t)}{\left(x-t\right)^\alpha} \, dt.</math>

For arbitrary {{mvar|α}}, since the gamma function is infinite for negative (real) integers, it is necessary to apply the fractional derivative after the integer derivative has been performed. For example,
:<math>D^\frac32 f(x) = D^\frac12 D^1 f(x)=D^\frac12 \frac d {dx} f(x).</math>

==Laplace transform==
We can also come at the question via the [[Laplace transform]]. Knowing that
:<math>\mathcal L \left\{Jf\right\}(s) = \mathcal L \left\{\int_0^t f(\tau)\,d\tau\right\}(s) = \frac1 s \bigl(\mathcal L\left\{f\right\}\bigr)(s)</math>

and
:<math>\mathcal L \left\{J^2f\right\}=\frac1s\bigl(\mathcal L \left\{Jf\right\} \bigr)(s)=\frac1{s^2}\bigl(\mathcal L\left\{f\right\}\bigr)(s)</math>

and so on, we assert
:<math>J^\alpha f=\mathcal L^{-1}\left\{s^{-\alpha}\bigl(\mathcal L\{f\}\bigr)(s)\right\}</math>.

For example,
:<math>J^\alpha(t^k) = \mathcal L^{-1}\left\{\frac{\Gamma(k+1)}{s^{\alpha+k+1}}\right\} = \frac{\Gamma(k+1)}{\Gamma(\alpha+k+1)} t^{\alpha+k} </math>

as expected. Indeed, given the [[Convolution|convolution]] rule
:<math>\mathcal L\{f*g\}=\bigl(\mathcal L\{f\}\bigr)\bigl(\mathcal L\{g\}\bigr)</math>

and shorthanding {{math|1=''p''(''x'') = ''x''<sup>''α'' − 1</sup>}} for clarity, we find that
:<math>\begin{align}
\left(J^\alpha f\right)(t) &= \frac{1}{\Gamma(\alpha)}\mathcal L^{-1}\left\{\bigl(\mathcal L\{p\}\bigr)\bigl(\mathcal L\{f\}\bigr)\right\}\\
&=\frac{1}{\Gamma(\alpha)}(p*f)\\
&=\frac{1}{\Gamma(\alpha)}\int_0^t p(t-\tau)f(\tau)\,d\tau\\
&=\frac{1}{\Gamma(\alpha)}\int_0^t\left(t-\tau\right)^{\alpha-1}f(\tau)\,d\tau\\
\end{align}</math>

which is what Cauchy gave us above.

Laplace transforms "work" on relatively few functions, but they ''are'' often useful for solving fractional differential equations.

==Notes==
<references/>

==References==
*{{springer|title=Euler transformation|id=e/e036620|first1=Yu.A.|last1=Brychkov|first2=A.P.|last2=Prudnikov}}.
*{{Citation|last=Caputo|first=Michele|title=Linear model of dissipation whose ''Q'' is almost frequency independent. II|journal=Geophysical Journal International|year=1967|volume=13|issue=5|pages=529–539|doi=10.1111/j.1365-246x.1967.tb02303.x|bibcode=1967GeoJ...13..529C|doi-access=free}}.
*{{Citation|last1=Hille| first1=Einar | last2=Phillips | first2=Ralph S. | title=Functional analysis and semi-groups | publisher=American Mathematical Society | location=Providence, R.I. | mr=0423094 | year=1974}}.
*{{springer|title=Fractional integration and differentiation|id=f/f041230|first=P.I.|last=Lizorkin}}.
*{{Citation|last=Loverro|first=Adam|date=2004-05-08|title=Fractional Calculus: History, Definitions and Applications for the Engineer|publisher=University of Notre Dame|location=Notre Dame, IN|url=http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf|archive-url=https://web.archive.org/web/20051029113800/http://www.nd.edu/~msen/Teaching/UnderRes/FracCalc.pdf|archive-date=2005-10-29}}
*{{citation|title=An Introduction to the Fractional Calculus and Fractional Differential Equations|first1=Kenneth S.| last1=Miller| first2=Bertram| last2=Ross|publisher=John Wiley & Sons|year=1993|isbn=0-471-58884-9}}.
*{{Citation | doi=10.1007/BF02395016 | last1=Riesz | first1=Marcel |  title=L'intégrale de Riemann-Liouville et le problème de Cauchy | mr=0030102 | year=1949 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=81 | issue=1 | pages=1–223| doi-access=free }}.

==External links==
*{{cite web|author=Alan Beardon|url=http://nrich.maths.org/public/viewer.php?obj_id=1369|title=Fractional calculus II|publisher=University of Cambridge|year=2000}}
*{{cite web|author=Alan Beardon|url=http://nrich.maths.org/public/viewer.php?obj_id=1371|title=Fractional calculus III|publisher=University of Cambridge|year=2000}}

{{DEFAULTSORT:Riemann-Liouville integral}}
[[Category:Fractional calculus]]
[[Category:Integral transforms]]

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