Editing Fractional calculus (section)

==Fractional derivatives==
{{Distinguish|Fractal derivative}}

Unfortunately, the comparable process for the derivative operator {{mvar|D}} is significantly more complex, but it can be shown that {{mvar|D}} is neither [[commutative]] nor [[additive map|additive]] in general.<ref>{{cite book |last1=Kilbas |first1=A. Anatolii Aleksandrovich |url=https://books.google.com/books?id=LhkO83ZioQkC |title=Theory And Applications of Fractional Differential Equations |last2=Srivastava |first2=Hari Mohan |last3=Trujillo |first3=Juan J. |date=2006 |publisher=Elsevier |isbn=978-0-444-51832-3 |page=[{{google books|plainurl=yes|id=LhkO83ZioQkC|page=75}} 75 (Property 2.4)] |language=en}}</ref>

Unlike classical Newtonian derivatives, fractional derivatives can be defined in a variety of different ways that often do not all lead to the same result even for smooth functions. Some of these are defined via a fractional integral. Because of the incompatibility of definitions, it is frequently necessary to be explicit about which definition is used.
[[File:Fractionalderivative.gif|thumb|Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative]]

===Riemann–Liouville fractional derivative===
The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the {{mvar|α}}th order derivative, the {{mvar|n}}th order derivative of the integral of order {{math|(''n'' − ''α'')}} is computed, where {{mvar|n}} is the smallest integer greater than {{mvar|α}} (that is, {{math|''n'' {{=}} {{ceil|''α''}}}}). The Riemann–Liouville fractional derivative and integral has multiple applications such as in case of solutions to the equation in the case of multiple systems such as the tokamak systems, and Variable order fractional parameter.<ref name="Mostafanejad">{{Cite journal|doi = 10.1002/qua.26762|title = Fractional paradigms in quantum chemistry |year = 2021|last = Mostafanejad |first = Mohammad |journal = International Journal of Quantum Chemistry |volume = 121|issue = 20 |doi-access = free }}</ref><ref name="Al-Raeei">{{Cite journal|doi = 10.1016/j.chaos.2021.111209|title = Applying fractional quantum mechanics to systems with electrical screening effects |year = 2021|last = Al-Raeei|first = Marwan | url=https://www.sciencedirect.com/science/article/abs/pii/S0960077921005634 |journal = Chaos, Solitons & Fractals |volume = 150|issue = September|pages = 111209|bibcode = 2021CSF...15011209A }}</ref> Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants.<ref>{{cite book |editor-last=Herrmann |editor-first=Richard |date=2014 |title=Fractional Calculus: An Introduction for Physicists |edition=2nd |location=New Jersey |publisher=World Scientific Publishing Co. |page=[https://books.google.com/books?id=60S7CgAAQBAJ&pg=PA54 54]{{Verify source |date=July 2020}}|isbn=978-981-4551-07-6|doi=10.1142/8934 |bibcode=2014fcip.book.....H}}</ref>
<math display="block">\begin{align}
  \sideset{_a}{_t^\alpha}D f(t) &= \frac{d^n}{dt^n} \sideset{_a}{_t^{-(n-\alpha)}}Df(t) \\
    &= \frac{d^n}{dt^n} \sideset{_a}{_t^{n-\alpha}}I f(t) \\
  \sideset{_t}{_b^\alpha}D f(t) &= \frac{d^n}{dt^n} \sideset{_t}{_b^{-(n-\alpha)}}Df(t) \\
    &= \frac{d^n}{dt^n} \sideset{_t}{_b^{n-\alpha}}I f(t)
\end{align}</math>

===Caputo fractional derivative===
{{main|Caputo fractional derivative}}
Another option for computing fractional derivatives is the [[Caputo fractional derivative]]. It was introduced by [[Michele Caputo]] in his 1967 paper.<ref>{{cite journal |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}}.</ref> In contrast to the Riemann–Liouville fractional derivative, when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. Caputo's definition is illustrated as follows, where again {{math|1=''n'' = ⌈''α''⌉}}:
<math display="block">\sideset{^C}{_t^\alpha}D f(t)=\frac{1}{\Gamma(n-\alpha)} \int_0^t \frac{f^{(n)}(\tau)}{\left(t-\tau\right)^{\alpha+1-n}}\, d\tau.</math>

There is the Caputo fractional derivative defined as:
<math display="block">D^\nu f(t)=\frac{1}{\Gamma(n-\nu)} \int_0^t (t-u)^{(n-\nu-1)}f^{(n)}(u)\, du \qquad (n-1)<\nu<n</math>
which has the advantage that it is zero when {{math|''f''(''t'')}} is constant and its Laplace Transform is expressed by means of the initial values of the function and its derivative. Moreover, there is the Caputo fractional derivative of distributed order defined as
<math display="block">\begin{align}
\sideset{_a^b}{^nu}Df(t) &= \int_a^b \phi(\nu)\left[D^{(\nu)}f(t)\right]\,d\nu \\
   &= \int_a^b\left[\frac{\phi(\nu)}{\Gamma(1-\nu)}\int_0^t \left(t-u\right)^{-\nu}f'(u)\,du \right]\,d\nu
\end{align}</math>

where {{math|''ϕ''(''ν'')}} is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.

===Caputo–Fabrizio fractional derivative===
In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function <math>f(t)</math> of <math>C^1</math> given by:
<math display="block">\sideset{_{\hphantom{C}a}^\text{CF}}{_t^\alpha}Df(t)=\frac{1}{1-\alpha} \int_a^t f'(\tau) \ e^\left(-\alpha\frac{t-\tau}{1-\alpha}\right) \ d\tau,</math>

where {{nowrap|<math>a < 0, \alpha \in (0,1]</math>.}}<ref>{{cite journal |last1=Caputo |first1=Michele |last2=Fabrizio |first2=Mauro |date=2015 |title=A new Definition of Fractional Derivative without Singular Kernel |url=https://www.naturalspublishing.com/ContIss.asp?IssID=255 |journal=Progress in Fractional Differentiation and Applications |volume=1 |issue=2 |pages=73–85 |access-date=7 August 2020}}</ref>

===Atangana–Baleanu fractional derivative===
In 2016, Atangana and Baleanu suggested differential operators based on the generalized [[Mittag-Leffler function]] <math> E_{\alpha}</math>. The aim was to introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in Riemann–Liouville sense and Caputo sense respectively. For a function <math>f(t)</math> of <math>C^1</math> given by <ref name=Algahtani2016/><ref name="doiserbia.nb.rs">{{cite journal |last1=Atangana |first1=Abdon |last2=Baleanu |first2=Dumitru |date=2016 |title=New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model |url=http://www.doiserbia.nb.rs/Article.aspx?ID=0354-98361600018A |journal=Thermal Science |language=en |volume=20 |issue=2 |pages=763–769 |doi=10.2298/TSCI160111018A |arxiv=1602.03408 |issn=0354-9836 |doi-access=free}}</ref>
<math display="block">\sideset{_{\hphantom{AB}a}^{\text{ABC}}}{_t^\alpha}D f(t)=\frac{\operatorname{AB}(\alpha)}{1-\alpha} \int_a^t f'(\tau)E_{\alpha}\left(-\alpha\frac{(t-\tau)^{\alpha}}{1-\alpha}\right)d\tau,</math>

If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by:
<math display="block">\sideset{_{\hphantom{AB}a}^{\text{ABC}}}{_t^\alpha}D f(t)=\frac{\operatorname{AB}(\alpha)}{1-\alpha} \frac{d}{dt}\int_a^t f(\tau)E_{\alpha}\left(-\alpha\frac{(t-\tau)^{\alpha}}{1-\alpha}\right)d\tau,</math>

The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all {{nowrap|<math>\alpha \in (0, 1]</math>,}} the function <math>E_\alpha</math> is increasing on the real line, converges to <math>0</math> in {{nowrap|<math>- \infty</math>,}} and {{nowrap|<math>E_\alpha (0) = 1</math>.}} Therefore, we have that, the function <math>x \mapsto 1-E_\alpha (-x^\alpha)</math> is the cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples is called a [[Mittag-Leffler distribution]] of order {{nowrap|<math>\alpha</math>.}} It is also very well-known that, all these probability distributions are [[absolute continuity|absolutely continuous]]. In particular, the function Mittag-Leffler has a particular case {{nowrap|<math>E_1</math>,}} which is the exponential function, the Mittag-Leffler distribution of order <math>1</math> is therefore an [[exponential distribution]]. However, for {{nowrap|<math>\alpha \in (0, 1)</math>,}} the Mittag-Leffler distributions are [[heavy-tailed distribution|heavy-tailed]]. Their Laplace transform is given by:
<math display="block">\mathbb{E} (e^{- \lambda X_\alpha}) = \frac{1}{1+\lambda^\alpha},</math>

This directly implies that, for {{nowrap|<math>\alpha \in (0, 1)</math>,}} the expectation is infinite. In addition, these distributions are [[geometric stable distribution]]s.

===Riesz derivative===
The Riesz derivative is defined as
<math display="block"> \mathcal{F} \left\{ \frac{\partial^\alpha u}{\partial \left|x\right|^\alpha} \right\}(k) = -\left|k\right|^{\alpha} \mathcal{F} \{u \}(k), </math>

where <math>\mathcal{F}</math> denotes the [[Fourier transform]].<ref>{{cite journal |last1=Chen |first1=YangQuan |last2=Li |first2=Changpin |last3=Ding |first3=Hengfei |date=22 May 2014 |title=High-Order Algorithms for Riesz Derivative and Their Applications |journal=[[Abstract and Applied Analysis]] |volume=2014 |pages=1–17 |language=en |doi=10.1155/2014/653797 |doi-access=free}}</ref><ref>{{cite journal |last=Bayın |first=Selçuk Ş. |date=5 December 2016 |title=Definition of the Riesz derivative and its application to space fractional quantum mechanics |journal=Journal of Mathematical Physics |volume=57 |issue=12 |pages=123501 |arxiv=1612.03046 |doi=10.1063/1.4968819 |bibcode=2016JMP....57l3501B |s2cid=119099201}}</ref>

=== Conformable fractional derivative ===
The conformable fractional derivative of a function <math>f</math> of order <math>\alpha</math> is given by<math display="block"> T_a(f)(t) = \lim_{\epsilon \rightarrow 0}\frac{f\left(t+\epsilon t^{1-\alpha}\right) - f(t)}{\epsilon} </math>Unlike other definitions of the fractional derivative, the conformable fractional derivative obeys the [[Product rule|product]] and [[quotient rule]] has analogs to [[Rolle's theorem]] and the [[mean value theorem]].<ref>{{Cite journal |last1=Khalil |first1=R. |last2=Al Horani |first2=M. |last3=Yousef |first3=A. |last4=Sababheh |first4=M. |date=2014-07-01 |title=A new definition of fractional derivative |url=https://www.sciencedirect.com/science/article/pii/S0377042714000065 |journal=Journal of Computational and Applied Mathematics |volume=264 |pages=65–70 |doi=10.1016/j.cam.2014.01.002 |issn=0377-0427|doi-access=free }}</ref><ref name=":0">{{Cite journal |last1=Gao |first1=Feng |last2=Chi |first2=Chunmei |date=2020 |title=Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations |journal=Journal of Function Spaces |language=en |volume=2020 |issue=1 |pages=5852414 |doi=10.1155/2020/5852414 |doi-access=free |issn=2314-8888}}</ref> However, this fractional derivative produces significantly different results compared to the Riemann-Liouville and Caputo fractional derivative. In 2020, Feng Gao and Chunmei Chi defined the improved Caputo-type conformable fractional derivative, which more closely approximates the behavior of the Caputo fractional derivative:<ref name=":0" /><math display="block"> ^C_a\widetilde{T}_a(f)(t) = \lim_{\epsilon \rightarrow 0}\left[(1-\alpha)(f(t)-f(a))+\alpha\frac{f\left(t+\epsilon (t-a)^{1-\alpha}\right) - f(t)}{\epsilon}\right] </math>where <math>a</math> and <math>t</math> are [[real numbers]] and <math>a<t</math>. They also defined the improved Riemann-Liouville-type conformable fractional derivative to similarly approximate the Riemann-Liouville fractional derivative:<ref name=":0" />

<math display="block"> ^{RL}_a\widetilde{T}_a(f)(t) = \lim_{\epsilon \rightarrow 0}\left[(1-\alpha)f(t)+\alpha\frac{f\left(t+\epsilon (t-a)^{1-\alpha}\right) - f(t)}{\epsilon}\right] </math>where <math>a</math> and <math>t</math> are [[real numbers]] and <math>a<t</math>. Both improved conformable fractional derivatives have analogs to Rolle's theorem and the [[interior extremum theorem]].<ref>{{Cite journal |last=Hasanah |first=Dahliatul |date=2022-10-31 |title=On continuity properties of the improved conformable fractional derivatives |url=http://mail.fourier.or.id/index.php/FOURIER/article/view/176 |journal=Jurnal Fourier |language=en |volume=11 |issue=2 |pages=88–96 |doi=10.14421/fourier.2022.112.88-96 |issn=2541-5239|doi-access=free }}</ref>

=== Other types ===
Classical fractional derivatives include:
* [[Grünwald–Letnikov derivative]]<ref name=deOliveira2014>{{cite journal |last1=de Oliveira |first1=Edmundo Capelas |last2=Tenreiro Machado |first2=José António |date=2014-06-10 |title=A Review of Definitions for Fractional Derivatives and Integral |journal=Mathematical Problems in Engineering |volume=2014 |pages=1–6 |language=en |doi=10.1155/2014/238459 |doi-access=free|hdl=10400.22/5497 |hdl-access=free }}</ref><ref name=Aslan2015>{{cite journal |last=Aslan |first=İsmail |date=2015-01-15 |title=An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation |journal=Mathematical Methods in the Applied Sciences |language=en |volume=38 |issue=1 |pages=27–36 |doi=10.1002/mma.3047 |bibcode=2015MMAS...38...27A |hdl=11147/5562 |s2cid=120881978 |hdl-access=free}}</ref>
* Sonin–Letnikov derivative<ref name=Aslan2015/>
* Liouville derivative<ref name=deOliveira2014/>
* [[Differintegral|Caputo derivative]]<ref name=deOliveira2014/>
* Hadamard derivative<ref name=deOliveira2014/><ref>{{cite journal |last1=Ma |first1=Li |last2=Li |first2=Changpin |date=2017-05-11 |title=On hadamard fractional calculus |journal=Fractals |volume=25 |issue=3 |pages=1750033–2980 |doi=10.1142/S0218348X17500335 |bibcode=2017Fract..2550033M |issn=0218-348X}}</ref>
* Marchaud derivative<ref name=deOliveira2014/>
* Riesz derivative<ref name=Aslan2015/>
* Miller–Ross derivative<ref name=deOliveira2014/>
* [[Weyl integral|Weyl derivative]]<ref>{{cite book |last=Miller |first=Kenneth S.  |chapter=The Weyl fractional calculus |date=1975 |title=Fractional Calculus and Its Applications: Proceedings of the International Conference Held at the University of New Haven, June 1974 |pages=80–89 |editor-last=Ross |editor-first=Bertram |series=Lecture Notes in Mathematics |volume=457 |publisher=Springer |language=en |doi=10.1007/bfb0067098 |isbn=978-3-540-69975-0}}</ref><ref>{{cite journal |last=Ferrari |first=Fausto |date=January 2018 |title=Weyl and Marchaud Derivatives: A Forgotten History |journal=Mathematics |language=en |volume=6 |issue=1 |pages=6 |doi=10.3390/math6010006 |doi-access=free|arxiv=1711.08070 }}</ref><ref name=deOliveira2014/>
* [[Erdelyi–Kober operator|Erdélyi–Kober derivative]]<ref name=deOliveira2014/>
* [[Fractal calculus|<math>F^{\alpha}</math>-derivative]]<ref name="Ali">{{cite book |last= Khalili Golmankhaneh|first= Alireza |date=2022 |title=Fractal Calculus and its Applications |url=https://worldscientific.com/worldscibooks/10.1142/12988#t=aboutBook|location=Singapore |publisher= World Scientific Pub Co Inc|page=328 |doi= 10.1142/12988 |isbn=978-981-126-110-7 |s2cid= 248575991 }}</ref>
New fractional derivatives include:
* Coimbra derivative<ref name=deOliveira2014/>
* [[Katugampola fractional operators|Katugampola derivative]]<ref>{{cite journal |last1=Anderson |first1=Douglas R. |last2=Ulness |first2=Darin J. |date=2015-06-01 |title=Properties of the Katugampola fractional derivative with potential application in quantum mechanics |journal=Journal of Mathematical Physics |volume=56 |issue=6 |pages=063502 |doi=10.1063/1.4922018 |bibcode=2015JMP....56f3502A |issn=0022-2488}}</ref>
* Hilfer derivative<ref name=deOliveira2014/>
* Davidson derivative<ref name=deOliveira2014/>
* Chen derivative<ref name=deOliveira2014/>
* [[Caputo Fabrizio derivative]]<ref name=Algahtani2016>{{cite journal |last=Algahtani |first=Obaid Jefain Julaighim |date=2016-08-01 |title=Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077916301059 |journal=Chaos, Solitons & Fractals |series=Nonlinear Dynamics and Complexity |language=en |volume=89 |pages=552–559 |doi=10.1016/j.chaos.2016.03.026 |bibcode=2016CSF....89..552A |issn=0960-0779}}</ref><ref>{{cite journal |last1=Caputo |first1=Michele |last2=Fabrizio |first2=Mauro |date=2016-01-01 |title=Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels |journal=Progress in Fractional Differentiation and Applications |volume=2 |issue=1 |pages=1–11 |doi=10.18576/pfda/020101 |issn=2356-9336}}</ref>
* Atangana–Baleanu derivative<ref name=Algahtani2016/><ref name="doiserbia.nb.rs"/>

====Coimbra derivative====
The '''Coimbra derivative''' is used for physical modeling:<ref> C. F. M. Coimbra (2003) "Mechanics with Variable Order Differential Equations," Annalen der Physik (12), No. 11-12, pp. 692-703.</ref> A number of applications in both mechanics and optics can be found in the works by Coimbra and collaborators,<ref>L. E. S. Ramirez, and C. F. M. Coimbra (2007) "A Variable Order Constitutive Relation for Viscoelasticity"– Annalen der Physik (16) 7-8, pp. 543-552.</ref><ref>H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, and C. F. M. Coimbra (2008) "Variable Order Modeling of Diffusive-Convective Effects on the Oscillatory Flow Past a Sphere" – Journal of Vibration and Control, (14) 9-10, pp. 1569-1672.</ref><ref>G. Diaz, and C. F. M. Coimbra (2009) "Nonlinear Dynamics and Control of a Variable Order Oscillator with Application to the van der Pol Equation" – Nonlinear Dynamics, 56, pp. 145—157.</ref><ref>L. E. S. Ramirez, and C. F. M. Coimbra (2010) "On the Selection and Meaning of Variable Order Operators for Dynamic Modeling"– International Journal of Differential Equations Vol. 2010, Article ID 846107.</ref><ref> L. E. S. Ramirez and C. F. M. Coimbra (2011) "On the Variable Order Dynamics of the Nonlinear Wake Caused by a Sedimenting Particle," Physica D (240) 13, pp. 1111-1118.</ref><ref>E. A. Lim, M. H. Kobayashi and C. F. M. Coimbra (2014) "Fractional Dynamics of Tethered Particles in Oscillatory Stokes Flows," Journal of Fluid Mechanics (746) pp. 606-625.</ref><ref>J. Orosco and C. F. M. Coimbra (2016) "On the Control and Stability of Variable Order Mechanical Systems" Nonlinear Dynamics, (86:1), pp. 695–710.</ref> as well as additional applications to physical problems and numerical implementations studied in a number of works by other authors<ref>E. C. de Oliveira, J. A. Tenreiro Machado (2014), "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459.</ref><ref>S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh (2012) "Numerical techniques for the variable order time fractional diffusion equation" Applied Mathematics and Computation
Volume 218, Issue 22, pp. 10861-10870.</ref><ref>H. Zhang and S. Shen, "The Numerical Simulation of Space-Time Variable Fractional Order Diffusion Equation," Numer. Math. Theor. Meth. Appl. Vol. 6, No. 4, pp. 571-585.</ref><ref>H. Zhang, F. Liu, M. S. Phanikumar, and M. M. Meerschaert (2013) "A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model," Computers & Mathematics with Applications, 66, issue 5, pp. 693–701.</ref>

 For <math>q(t) < 1 </math>
 <math display="block"> \begin{align}
^{\mathbb{C}}_{ a}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[1-q(t)]}  \int_{0^+}^t (t-\tau)^{-q(t)}\frac{d\,f(\tau)}{d\tau}d\tau\,+\,\frac{(f(0^+)-f(0^-))\,t^{-q(t)}}{\Gamma(1-q(t))},
\end{align}</math>
where the lower limit <math>a</math> can be taken as either <math>0^-</math> or <math>-\infty</math> as long as <math>f(t)</math> is identically zero from or <math>-\infty</math> to <math>0^-</math>. Note that this operator returns the correct fractional derivatives for all values of <math>t</math> and can be applied to either the dependent function itself <math> f(t)</math> with a variable order of the form <math>q(f(t))</math> or to the independent variable with a variable order of the form <math>q(t)</math>.<math>^{[1]}</math>  

The Coimbra derivative can be generalized to any order,<ref> C. F. M. Coimbra "Methods of using generalized order
differentiation and integration of input variables to forecast
trends," U.S. Patent Application 13,641,083 (2013). </ref> leading to the Coimbra Generalized Order Differintegration Operator (GODO)<ref>J. Orosco and C. F. M. Coimbra (2018) "Variable-order Modeling of Nonlocal Emergence in Many-body Systems: Application to Radiative Dispersion," Physical Review E (98), 032208.</ref>
 For <math>q(t) < m </math>
 <math display="block"> \begin{align}
^{\mathbb{\quad C}}_{\,\,-\infty}\mathbb{D}^{q(t)} f(t)=\frac{1}{\Gamma[
m-q(t)]}  \int_{0^+}^t (t-\tau)^{m-1-q(t)}\frac{d^m f(\tau)}{d\tau^m}d\tau\,+\,\sum^{m-1}_{n = 0} \frac{(\frac{d^n f(t)}{dt^n }|_{0^+}-\frac{d^n f(t)}{dt^n}|_{0^-})\,t^{n -q(t)}}{\Gamma[n+1-q(t)]},
\end{align}</math>
where <math>m</math> is an integer larger than the larger value of <math>q(t)</math> for all values of <math>t</math>. Note that the second (summation) term on the right side of the definition above can be expressed as 

 <math display="block"> \begin{align}
\frac{1}{\Gamma[m-q(t)]}\sum^{m-1}_{n = 0} \{[\frac{d^n\!f(t)}{dt^n}|_{0^+}-\frac{d^n\!f(t)}{dt^n }|_{0^-}]\,t^{n -q(t)} \prod^{m-1}_{j=n+1} [j- q(t)]\}
\end{align}</math>
so to keep the denominator on the positive branch of the Gamma (<math>\Gamma</math>) function and for ease of numerical calculation.

=== Nature of the fractional derivative ===
The {{nowrap|<math>a</math>-th}} derivative of a function <math>f</math> at a point <math>x</math> is a ''local property'' only when <math>a</math> is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of <math>f</math> at <math>x=c</math> depends on all values of {{nowrap|<math>f</math>,}} even those far away from {{nowrap|<math>c</math>.}} Therefore, it is expected that the fractional derivative operation involves some sort of [[boundary condition]]s, involving information on the function further out.<ref>{{MathPages|id=home/kmath616/kmath616.htm|title=Fractional Calculus}}</ref>

The fractional derivative of a function of order <math>a</math> is nowadays often defined by means of the [[Fourier transform|Fourier]] or [[Mellin transform|Mellin]] integral transforms.{{Citation needed|date=November 2022|reason=Examination of recent papers does not mention this}}