View source for Riemann–Liouville integral

{{Calculus|expanded=Fractional calculus}}

In [[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]] 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 [[Bernhard Riemann]] and [[Joseph Liouville]], the latter of whom was the first to consider the possibility of [[fractional calculus]] in 1832.<ref name="Lizorkin">{{harvnb|Lizorkin|2001}}</ref><ref>{{Citation|last=Liouville|first=Joseph|author-link=Joseph Liouville|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=[[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 [[Leonhard Euler]], when applied to [[analytic function]]s.<ref>{{harvnb|Brychkov|Prudnikov|2001}}</ref> It was generalized to arbitrary dimensions by [[Marcel Riesz]], who introduced the [[Riesz potential]].

==Motivation==
The Riemann-Liouville integral is motivated from Cauchy formula for repeated integration. For a function {{mvar|f}} continuous on the interval [{{mvar|a}},{{mvar|x}}], the Cauchy formula for {{mvar|n}}-fold repeated integration states that  

<math display="block">I^n f(x) = 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 {{mvar|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^x\left(x-t\right)^{\alpha-1} f(t)\,\mathrm{d}t</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]] and {{mvar|a}} is an arbitrary but fixed base point. The integral is well-defined provided {{mvar|f}} is a [[locally integrable function]], and {{mvar|α}} is a [[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]]. 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]] 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]] of {{mvar|I}}, one can show that the limit {{math|''I''<sup>''α''</sup> ''f'' → ''f''}} holds pointwise [[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]] <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]]. One also obtains a [[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]] ({{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]] 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]]s with the [[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==
{{Unreferenced section|date=May 2020}}
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]] 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.

== See also ==
*[[Caputo fractional derivative]]

==Notes==
<references/>

==References==
{{sfn whitelist|CITEREFLizorkin2001|CITEREFBrychkovPrudnikov2001}}
*{{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 | author1-link=Marcel Riesz | 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}}

{{Bernhard Riemann}}
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[[Category:Fractional calculus]]
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