Fractional calculus

3467: 2951: 4696: 5858: 3462:{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}(x-t)^{\alpha -1}\left(J^{\beta }f\right)(t)\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\int _{0}^{t}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}f(s)\,ds\,dt\\&={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}f(s)\left(\int _{s}^{x}\left(x-t\right)^{\alpha -1}\left(t-s\right)^{\beta -1}\,dt\right)\,ds\end{aligned}}} 4238: 5312: 2933: 2011:'s early papers where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and even the unified notation for differentiation and integration of arbitrary real order. Independently, the foundations of the subject were laid by 9858: 5258: 5218: 25: 4691:{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{-\alpha }}Df(t)&=\sideset {_{a}}{_{t}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau \\\sideset {_{t}}{_{b}^{-\alpha }}Df(t)&=\sideset {_{t}}{_{b}^{\alpha }}If(t)\\&={\frac {1}{\Gamma (\alpha )}}\int _{t}^{b}\left(\tau -t\right)^{\alpha -1}f(\tau )\,d\tau \end{aligned}}} 5853:{\displaystyle {\begin{aligned}\sideset {_{a}}{_{t}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{a}}{_{t}^{n-\alpha }}If(t)\\\sideset {_{t}}{_{b}^{\alpha }}Df(t)&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{-(n-\alpha )}}Df(t)\\&={\frac {d^{n}}{dt^{n}}}\sideset {_{t}}{_{b}^{n-\alpha }}If(t)\end{aligned}}} 7504: 2645: 10717: 4167: 3768: 7260: 9613: 10477: 4983: 4971: 10077: 5309:). 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. Similar to the definitions for the Riemann–Liouville integral, the derivative has upper and lower variants. 6880: 7267: 8911: 10508: 3943: 3518: 7033: 1485: 10285: 6087: 8125: 3936: 9216:

Anomalous diffusion processes in complex media can be well characterized by using fractional-order diffusion equation models. The time derivative term corresponds to long-time heavy tail decay and the spatial derivative for diffusion nonlocality. The time-space fractional diffusion governing equation

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See also Holm & Näsholm (2011) and the references therein. Such models are linked to the commonly recognized hypothesis that multiple relaxation phenomena give rise to the attenuation measured in complex media. This link is further described in Näsholm & Holm (2011b) and in the survey paper,

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Pandey and Holm gave a physical meaning to fractional differential equations by deriving them from physical principles and interpreting the fractional-order in terms of the parameters of the acoustical media, example in fluid-saturated granular unconsolidated marine sediments. Interestingly, Pandey

2928:{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&=\left(J^{\beta }\right)\left(J^{\alpha }f\right)(x)\\&=\left(J^{\alpha +\beta }f\right)(x)\\&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-t\right)^{\alpha +\beta -1}f(t)\,dt\,.\end{aligned}}} 5253:

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

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When studying the redox behavior of a substrate in solution, a voltage is applied at an electrode surface to force electron transfer between electrode and substrate. The resulting electron transfer is measured as a current. The current depends upon the concentration of substrate at the electrode

6598: 9853:{\displaystyle \nabla ^{2}u-{\dfrac {1}{c_{0}^{2}}}{\frac {\partial ^{2}u}{\partial t^{2}}}+\tau _{\sigma }^{\alpha }{\dfrac {\partial ^{\alpha }}{\partial t^{\alpha }}}\nabla ^{2}u-{\dfrac {\tau _{\epsilon }^{\beta }}{c_{0}^{2}}}{\dfrac {\partial ^{\beta +2}u}{\partial t^{\beta +2}}}=0\,.} 4775: 9904: 9175:

which relates the concentration of substrate at the electrode surface to the current. This relationship is applied in electrochemical kinetics to elucidate mechanistic behavior. For example, it has been used to study the rate of dimerization of substrates upon electrochemical reduction.

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in his 1967 paper. 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

6273: 8759: 2631: 5213:{\displaystyle \sideset {_{{\hphantom {A}}a}^{\operatorname {AB} }}{_{t}^{\alpha }}If(t)={\frac {1-\alpha }{\operatorname {AB} (\alpha )}}f(t)+{\frac {\alpha }{\operatorname {AB} (\alpha )\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau } 2006:

in 1695. Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions. Fractional calculus was introduced in one of

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The propagation of acoustical waves in complex media, such as in biological tissue, commonly implies attenuation obeying a frequency power-law. This kind of phenomenon may be described using a causal wave equation which incorporates fractional time derivatives:

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in electrical transmission line analysis circa 1890. The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.

5896: 7995: 2480: 7499:{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(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 ,} 3779: 2141: 10712:{\displaystyle i\hbar {\frac {\partial \psi ^{\alpha (\mathbf {r} )}(\mathbf {r} ,t)}{\partial t^{\alpha (\mathbf {r} )}}}=\left(-\hbar ^{2}\Delta \right)^{\frac {\beta (t)}{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t),} 6290: 4162:{\displaystyle {\begin{aligned}\left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)&={\frac {1}{\Gamma (\alpha +\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\,ds\\&=\left(J^{\alpha +\beta }f\right)(x)\end{aligned}}} 3763:{\displaystyle \left(J^{\alpha }\right)\left(J^{\beta }f\right)(x)={\frac {1}{\Gamma (\alpha )\Gamma (\beta )}}\int _{0}^{x}\left(x-s\right)^{\alpha +\beta -1}f(s)\left(\int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr\right)\,ds} 9300: 9208:. The stability and convergence in numerical simulations showed that the modified equation is more reliable in predicting the movement of pollution in deformable aquifers than equations with constant fractional and integer derivatives 9188:

is generalized by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow.

7935: 7255:{\displaystyle \sideset {_{{\hphantom {AB}}a}^{\text{ABC}}}{_{t}^{\alpha }}Df(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 ,} 6974:. 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 10472:{\displaystyle (-\hbar ^{2}\Delta )^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)={\frac {1}{(2\pi \hbar )^{3}}}\int d^{3}pe^{{\frac {i}{\hbar }}\mathbf {p} \cdot \mathbf {r} }|\mathbf {p} |^{\alpha }\varphi (\mathbf {p} ,t)\,.} 8516: 10773: 10198: 6094: 2493: 9058: 4966:{\displaystyle \sideset {_{a}}{_{t}^{-\alpha }}{\mathbf {D} }f(t)={\frac {1}{\Gamma (\alpha )}}\int _{a}^{t}\left(\log {\frac {t}{\tau }}\right)^{\alpha -1}f(\tau ){\frac {d\tau }{\tau }},\qquad t>a\,.} 4749:) would be more appropriately named the Abel–Riemann integral, on the basis of history of discovery and use, and in the same vein the integral over the entire real line be named Liouville–Weyl integral. 10072:{\displaystyle i\hbar {\frac {\partial \psi (\mathbf {r} ,t)}{\partial t}}=D_{\alpha }\left(-\hbar ^{2}\Delta \right)^{\frac {\alpha }{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t)\,.} 6286:

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

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article. See Holm & Nasholm (2013) for a paper which compares fractional wave equations which model power-law attenuation. This book on power-law attenuation also covers the topic in more detail.

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Atangana and Kilicman extended the fractional advection dispersion equation to a variable order equation. In their work, the hydrodynamic dispersion equation was generalized using the concept of a

9017: 164: 7723: 6295: 5317: 4243: 3948: 2956: 2650: 2146: 1312: 11007:"Oplösning af et Par Opgaver ved Hjelp af bestemte Integraler (Solution de quelques problèmes à l'aide d'intégrales définies, Solution of a couple of problems by means of definite integrals)" 2134: 2336: 1949: 1868: 1217: 6875:{\displaystyle \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 ,} 1119: 10502: 9894: 8906:{\displaystyle -\rho \left(\nabla ^{\alpha }\cdot {\vec {u}}\right)=\Gamma (\alpha +1)\Delta x^{1-\alpha }\rho \left(\beta _{s}+\phi \beta _{w}\right){\frac {\partial p}{\partial t}}} 6933: 9220: 10268: 1649: 12674:

Gorenflo, Rudolf; Mainardi, Francesco (2007). "Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk". In Rangarajan, G.; Ding, M. (eds.).

9898: 9201: 7976: 7858: 7847: 7663: 7545: 1480:{\displaystyle {\begin{aligned}D^{n}(f)&=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)\\&=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots ))).\end{aligned}}} 8151: 13728:

Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications

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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

8232: 7574: 6972: 12509: 7618: 7748: 6082:{\displaystyle \sideset {^{C}}{_{t}^{\alpha }}Df(t)={\frac {1}{\Gamma (n-\alpha )}}\int _{0}^{t}{\frac {f^{(n)}(\tau )}{\left(t-\tau \right)^{\alpha +1-n}}}\,d\tau .} 1793: 1721: 7782: 7028: 6675: 1574: 7001: 6648: 8405: 8120:{\displaystyle {\mathcal {F}}\left\{{\frac {\partial ^{\alpha }u}{\partial \left|x\right|^{\alpha }}}\right\}(k)=-\left|k\right|^{\alpha }{\mathcal {F}}\{u\}(k),} 4747: 14422: 13123:

Pandey, Vikash; Holm, Sverre (2016-12-01). "Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations".

10911: 10890: 10870: 8478: 8449: 8426: 8379: 8359: 8335: 8315: 8293: 7804: 7594: 3931:{\displaystyle \int _{0}^{1}\left(1-r\right)^{\alpha -1}r^{\beta -1}\,dr=B(\alpha ,\beta )={\frac {\Gamma (\alpha )\,\Gamma (\beta )}{\Gamma (\alpha +\beta )}}} 1969: 1896: 1817: 1766: 1744: 1694: 1672: 1616: 1596: 1302: 1277: 1253: 1145: 1054: 2317:{\displaystyle {\begin{aligned}\left(J^{2}f\right)(x)&=\int _{0}^{x}(Jf)(t)\,dt\\&=\int _{0}^{x}\left(\int _{0}^{t}f(s)\,ds\right)dt\,,\end{aligned}}} 12507:

Wheatcraft, Stephen W.; Meerschaert, Mark M.; Schumer, Rina; Benson, David A. (2001-01-01). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests".

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In 2013–2014 Atangana et al. described some groundwater flow problems using the concept of a derivative with fractional order. In these works, the classical

12098:

Anderson, Douglas R.; Ulness, Darin J. (2015-06-01). "Properties of the Katugampola fractional derivative with potential application in quantum mechanics".

11863:

Aslan, İsmail (2015-01-15). "An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation".

6593:{\displaystyle {\begin{aligned}\sideset {_{a}^{b}}{^{n}u}Df(t)&=\int _{a}^{b}\phi (\nu )\left\,d\nu \\&=\int _{a}^{b}\left\,d\nu \end{aligned}}} 8207: 14817: 12739: 1982:, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus. 13365:

Bhrawy, A.H.; Zaky, M.A. (2017). "An improved collocation method for multi-dimensional space–time variable-order fractional Schrödinger equations".

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and when the flux within the control volume is non-linear. In the referenced paper, the fractional conservation of mass equation for fluid flow is:

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Tenreiro Machado, J. A.; Silva, Manuel F.; Barbosa, Ramiro S.; Jesus, Isabel S.; Reis, Cecília M.; Marcos, Maria G.; Galhano, Alexandra F. (2010).

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using the framework of fractional calculus. Nutting's law was used to model the wave propagation in marine sediments using fractional derivatives.

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The kernel used in Atangana–Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all

14812: 11006: 2490:

to remove the discrete nature of the factorial function gives us a natural candidate for applications of the fractional integral operator as

10724: 10149: 8934: 260: 14415: 6621:

In a paper of 2015, M. Caputo and M. Fabrizio presented a definition of fractional derivative with a non singular kernel, for a function

8931:. Taking the Laplace transform of Fick's second law yields an ordinary second-order differential equation (here in dimensionless form): 14472: 13558:

Tenreiro Machado, J.A.; Galhano, A.M.S.F.; Trujillo, J.J. (2014). "On development of fractional calculus during the last fifty years".

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Caputo, Michele; Fabrizio, Mauro (2016-01-01). "Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels".

1975:

subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.

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Holm, S.; Näsholm, S. P. (2013). "Comparison of fractional wave equations for power law attenuation in ultrasound and elastography".

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As described by Wheatcraft and Meerschaert (2008), a fractional conservation of mass equation is needed to model fluid flow when the

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Bayın, Selçuk Ş. (5 December 2016). "Definition of the Riesz derivative and its application to space fractional quantum mechanics".

8655:{\displaystyle {\frac {x^{-\nu -\alpha +1}}{\Gamma (\alpha )}}\int _{0}^{x}\left(t-x\right)^{\alpha -1}t^{-\alpha -\nu }f(t)\,dt\,,} 1149: 14585: 14195: 13970: 1058: 4223:

whose integrals evaluate to zero). The Riemann–Liouville integral exists in two forms, upper and lower. Considering the interval

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Tenreiro Machado, J.A.; Galhano, A.M.; Trujillo, J.J. (2013). "Science metrics on fractional calculus development since 1966".

6268:{\displaystyle 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} 14629: 8361:

is an integer; this is not the case for non-integer power derivatives. In other words, a non-integer fractional derivative of

2626:{\displaystyle \left(J^{\alpha }f\right)(x)={\frac {1}{\Gamma (\alpha )}}\int _{0}^{x}\left(x-t\right)^{\alpha -1}f(t)\,dt\,.} 14523: 14065: 14028: 13999: 13959: 13913: 13876: 13849: 13820: 13771: 13736: 13715: 13696: 13677: 13658: 13417: 13349: 13096: 12709: 12074: 11976: 11485: 11368: 11303: 11263: 10953: 14296: 11077:"Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions" 14807: 12800: 526: 501: 14717: 7670: 14732: 14513: 14301: 8169: 4753: 13932: 9166:{\displaystyle {\frac {d}{dx}}C(x,t)={\frac {d^{\scriptstyle {\frac {1}{2}}}}{dt^{\scriptstyle {\frac {1}{2}}}}}C(x,t)} 1002: 565: 12900:

Näsholm, S. P.; Holm, S. (2011). "Linking multiple relaxation, power-law attenuation, and fractional wave equations".

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Podlubny, Igor; Magin, Richard L.; Trymorush, Irina (2017). "Niels Henrik Abel and the birth of fractional calculus".

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One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator

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Metzler, R.; Klafter, J. (2000). "The random walk's guide to anomalous diffusion: a fractional dynamics approach".

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Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000). "Application of a fractional advection-dispersion equation".

11655:"New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model" 10806: 9533:{\displaystyle u(t)=K_{\mathrm {p} }e(t)+K_{\mathrm {i} }D_{t}^{-\alpha }e(t)+K_{\mathrm {d} }D_{t}^{\beta }e(t)} 842: 516: 491: 173: 14863: 14462: 10505:, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena: 6613:

is a weight function and which is used to represent mathematically the presence of multiple memory formalisms.

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Fractional derivatives of a Gaussian, interpolating continuously between the function and its first derivative.

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The corresponding derivative is calculated using Lagrange's rule for differential operators. To find the

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Holm, S.; Näsholm, S. P. (2011). "A causal and fractional all-frequency wave equation for lossy media".

2475:{\displaystyle \left(J^{n}f\right)(x)={\frac {1}{(n-1)!}}\int _{0}^{x}\left(x-t\right)^{n-1}f(t)\,dt\,,} 1998:

is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to

11605:"Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model" 10228: 8708: 8202: 7264:

If the function is continuous, the Atangana–Baleanu derivative in Riemann–Liouville sense is given by:

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The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order

12056: 7943: 7814: 7626: 7512: 14843: 14762: 13194:"Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity" 9205: 7852: 2003: 1541:

for the differentiation operator, that is, an expression for some linear operator that, when applied

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Ross, B. (1975). "A brief history and exposition of the fundamental theory of fractional calculus".

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Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media

13030: 12523: 7807: 645: 205: 43: 34: 12637:; Pagnini, G. (2001). "The fundamental solution of the space-time fractional diffusion equation". 12427: 10970: 14767: 14737: 14727: 14624: 10823: 10121: 8210: 7552: 6950: 6945: 959: 751: 640: 14214: 14202: 14189: 14176: 9197:

This equation has been shown useful for modeling contaminant flow in heterogenous porous media.

4219:, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the 13804:

Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models

13327: 13278: 13025: 12518: 12422: 11654: 11287: 10919: 4207:, which is essentially what has been described above. The theory of fractional integration for 1538: 1256: 1034: 995: 924: 885: 769: 705: 629: 14304:, a Matlab and Simulink Toolbox dedicated to fractional calculus, which is freely downloadable 8711:; and the generalisation of the classical theory to higher dimensions is called the theory of 7600: 14883: 14878: 14772: 14696: 14675: 14543: 14538: 14533: 14528: 14452: 14431: 13802: 12567:"On the Generalized Mass Transport Equation to the Concept of Variable Fractional Derivative" 11461: 11360: 10100: 9369:

to use fractional orders can increase their degree of freedom. The new equation relating the

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A simple extension of the fractional derivative is the variable-order fractional derivative,

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surface. As substrate is consumed, fresh substrate diffuses to the electrode as described by

7733: 1979: 1019: 969: 635: 406: 351: 312: 218: 9295:{\displaystyle {\frac {\partial ^{\alpha }u}{\partial t^{\alpha }}}=-K(-\Delta )^{\beta }u.} 1772: 1700: 14777: 14701: 14670: 14327: 13981: 13893: 13887: 13831: 13751: 13489: 13270: 13205: 13142: 13076: 13070: 12909: 12848: 12758: 12689: 12656: 12607: 12471: 12414: 12271: 12229: 12210: 12107: 11925: 11872: 11758: 11616: 11568: 11538: 11465: 11430: 11243: 11103: 10796: 10212: 9865: 9578: 7760: 7006: 6653: 4211:(therefore including the "boundary condition" of repeating after a period) is given by the 2023: 1282: 974: 954: 880: 549: 468: 442: 356: 14357:"Relativistic wave equations with fractional derivatives and pseudodifferential operators" 14149: 14132: 12176: 11328: 6977: 6624: 8: 14787: 14680: 14574: 14276: 12374:"Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation" 9882: 8692: 8677: 8453:

Therefore, it is expected that the fractional derivative operation involves some sort of

8384: 7930:{\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},} 7753: 4726: 1991: 949: 919: 796: 650: 447: 303: 186: 181: 16:

Branch of mathematical analysis with fractional applications of derivatives and integrals

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Special Functions in Fractional Calculus and Related Fractional Differintegral Equations

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In 2016, Atangana and Baleanu suggested differential operators based on the generalized

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Kilbas, A. Anatolii Aleksandrovich; Srivastava, Hari Mohan; Trujillo, Juan J. (2006).

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For a historical review of the subject up to the beginning of the 20th century, see:

11058: 10949: 10811: 8481: 8154: 4208: 2008: 1820: 988: 822: 600: 478: 431: 288: 283: 39: 13634: 13589: 13178: 13055: 12947: 12780: 12719: 12493: 12283: 12194: 11419:"Applying fractional quantum mechanics to systems with electrical screening effects" 8506: 14893: 14873: 14392: 14378: 14347: 14335: 14053: 14016: 13987: 13947: 13901: 13864: 13837: 13808: 13759: 13620: 13610: 13575: 13567: 13536: 13528: 13505: 13497: 13456: 13405: 13374: 13337: 13308: 13288: 13223: 13213: 13150: 13084: 13035: 12980: 12925: 12917: 12886: 12864: 12856: 12819: 12809: 12766: 12733:

Colbrook, Matthew J.; Ma, Xiangcheng; Hopkins, Philip F.; Squire, Jonathan (2017).

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Miller, Kenneth S. (1975). "The Weyl fractional calculus". In Ross, Bertram (ed.).

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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

9342:. Its applications in anomalous diffusion modeling can be found in the reference. 14853: 14782: 14207: 14169: 13726: 13460: 12206: 11354: 10943: 10104: 8712: 4980:

The Atangana–Baleanu fractional integral of a continuous function is defined as:

1972: 1233: 964: 837: 791: 786: 673: 586: 531: 14858: 14237: 13670:

An Introduction to the Fractional Calculus and Fractional Differential Equations

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Näsholm, S. P.; Holm, S. (2012). "On a Fractional Zener Elastic Wave Equation".

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where in the last step we exchanged the order of integration and pulled out the

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Závada, Petr (1998). "Operator of Fractional Derivative in the Complex Plane".

13599:"On the Selection and Meaning of Variable Order Operators for Dynamic Modeling" 13292: 13218: 13193: 12199:

Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino

11125:

For the history of the subject, see the thesis (in French): Stéphane Dugowson,

9366: 8749: 8180: 4216: 4192: 2487: 1027: 847: 655: 422: 59: 14383: 14356: 13991: 13905: 13841: 13571: 13532: 13447:

Debnath, L. (2004). "A brief historical introduction to fractional calculus".

13088: 12984: 12532: 11937: 11198: 14923: 14256: 13162: 12540: 12291: 12162: 12127: 11945: 11690: 11636: 11216: 9874: 8921: 8753: 8510: 8197: 4212: 3773: 1549:. More generally, one can look at the question of defining a linear operator 827: 591: 341: 298: 14057: 12701: 7786:

which is the exponential function, the Mittag-Leffler distribution of order

14903: 14838: 14752: 14049:

Theory and Numerical Approximations of Fractional Integrals and Derivatives

13300: 13237: 13170: 13047: 12939: 12878: 12771: 12734: 12154: 11815:

de Oliveira, Edmundo Capelas; Tenreiro Machado, José António (2014-06-10).

11681: 10082: 9351: 5247: 581: 326: 14339: 14047: 13615: 13598: 12824: 12814: 12795: 12583: 12566: 12390: 12373: 12357: 12340: 11833: 11816: 11724: 11705: 11329:"Essai sur l'étude des fonctions données par leur développement de Taylor" 11185:(2014-01-01). "Some pioneers of the applications of fractional calculus". 11050: 9055:

and then the inverse Laplace transform yields the following relationship:

14660: 14400: 14322: 14089: 13580: 13541: 13510: 13449:

International Journal of Mathematical Education in Science and Technology

13265: 12651: 12634: 12484: 12459: 12436: 11963:. Lecture Notes in Mathematics. Vol. 457. Springer. pp. 80–89. 11843: 11207: 11146:

Bertram Ross (1977). "The development of fractional calculus 1695–1900".

8715:. So there are a number of contemporary theories available, within which 7752:

It is also very well-known that, all these probability distributions are

5243: 4220: 1023: 944: 13625: 13393: 12930: 12869: 12341:"The Use of Fractional Order Derivative to Predict the Groundwater Flow" 12040: 12013: 4723:

It has been suggested that the integral on the positive real axis (i.e.

14888: 14373: 13653:. Mathematics in Science and Engineering. Vol. V. Academic Press. 13409: 13228: 11968: 10768:{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}} 10193:{\textstyle \Delta ={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}} 9878: 9586: 5257: 2016: 1546: 1031: 690: 614: 336: 331: 235: 13763: 13154: 12921: 12860: 12119: 11916:

Ma, Li; Li, Changpin (2017-05-11). "On hadamard fractional calculus".

11893: 11770: 11402: 11385: 10892:

in order to avoid confusion with other concepts identified by similar

14650: 14137: 14118: 14099: 14083: 12735:"Scaling laws of passive-scalar diffusion in the interstellar medium" 11884: 11242:(2nd ed.). New Jersey: World Scientific Publishing. p. 46. 11133:

histoire et philosophie de la généralisation de l'ordre de dérivation

9185: 2012: 1874: 619: 609: 13986:. Developments in Mathematics. Vol. 41. Switzerland: Springer. 11460:(2nd ed.). New Jersey: World Scientific Publishing Co. p. 10282:

is the 3-dimensional fractional quantum Riesz derivative defined by

9888: 14690: 13137: 12753: 12030: 11753: 11706:"High-Order Algorithms for Riesz Derivative and Their Applications" 11671: 11569:"A new Definition of Fractional Derivative without Singular Kernel" 11041: 9582: 7756:. In particular, the function Mittag-Leffler has a particular case 1222: 1124: 685: 427: 384: 73: 14246: 14020: 13341: 13020: 12975: 12684: 12506: 12224:

Kober, Hermann (1940). "On fractional integrals and derivatives".

12066: 10990: 9211: 7980:

the expectation is infinite. In addition, these distributions are

5234:

Unfortunately, the comparable process for the derivative operator

14291: 13951: 13868: 13812: 13475: 12793: 11817:"A Review of Definitions for Fractional Derivatives and Integral" 11477: 11295: 11255: 11104:"Mémoire sur le calcul des différentielles à indices quelconques" 10496: 9354: 4191:

This relationship is called the semigroup property of fractional

1619: 14163: 14010: 13480:; Mainardi, F. (2011). "Recent history of fractional calculus". 13557: 13518: 14127: 13404:. Lecture Notes in Mathematics. Vol. 457. pp. 1–36. 13689:

Fractional Integrals and Derivatives: Theory and Applications

12678:. Lecture Notes in Physics. Vol. 621. pp. 148–166. 12255:

Wheatcraft, Stephen W.; Meerschaert, Mark M. (October 2008).

10915: 4976:

Atangana–Baleanu fractional integral (AB fractional integral)

1022:

that studies the several different possibilities of defining

14108: 13482:

Communications in Nonlinear Science and Numerical Simulation

11356:

Theory And Applications of Fractional Differential Equations

9604: 2482:

leads in a straightforward way to a generalization for real

1489:

For example, one may ask for a meaningful interpretation of

14286: 14182: 12457: 12404: 11814: 11704:

Chen, YangQuan; Li, Changpin; Ding, Hengfei (22 May 2014).

10787:

is the variable-order fractional quantum Riesz derivative.

9192: 5873:

Another option for computing fractional derivatives is the

4183:

operator is applied is irrelevant and completes the proof.

1528:{\displaystyle {\sqrt {D}}=D^{\scriptstyle {\frac {1}{2}}}} 10485:

in the fractional Schrödinger equation is the Lévy index,

5265: 4203:

The classical form of fractional calculus is given by the

14266:"History, Definitions, and Applications for the Engineer" 13746:

West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003).

12796:"Some Applications of Fractional Calculus in Engineering" 6939: 5254:

necessary to be explicit about which definition is used.

12460:"The fractional-order governing equation of Lévy motion" 11352: 6616: 5238:

is significantly more complex, but it can be shown that

4198: 13708:

Fractals and Fractional Calculus in Continuum Mechanics

12732: 12143:

Progress in Fractional Differentiation and Applications

12061:. Singapore: World Scientific Pub Co Inc. p. 328. 11573:

Progress in Fractional Differentiation and Applications

11284:

Fractional Calculus and Waves in Linear Viscoelasticity

11180: 9204:. The modified equation was numerically solved via the 14208:

Journal of Fractional Calculus and Applications (JFCA)

13892:. Nonlinear Physical Science. Higher Education Press. 13251:

Laskin, N. (2002). "Fractional Schrodinger equation".

11026: 10971:"A New Approach To Generalized Fractional Derivatives" 10727: 10231: 10152: 9128: 9105: 8269: 6091:

There is the Caputo fractional derivative defined as:

1511: 13394:

Articles regarding the history of fractional calculus

12632: 12254: 10899: 10878: 10858: 10802:

Autoregressive fractionally integrated moving average

10511: 10288: 9907: 9793: 9757: 9710: 9634: 9616: 9403: 9223: 9061: 9012:{\displaystyle {\frac {d^{2}}{dx^{2}}}C(x,s)=sC(x,s)} 8937: 8762: 8519: 8466: 8457:, involving information on the function further out. 8437: 8414: 8387: 8367: 8347: 8323: 8303: 8281: 8213: 8135: 7998: 7946: 7861: 7817: 7792: 7763: 7736: 7673: 7629: 7603: 7582: 7555: 7515: 7270: 7036: 7009: 6980: 6953: 6891: 6683: 6656: 6627: 6293: 6097: 5899: 5315: 4986: 4778: 4729: 4241: 3946: 3782: 3521: 2954: 2648: 2496: 2339: 2144: 2063: 1957: 1910: 1884: 1829: 1805: 1775: 1754: 1732: 1703: 1682: 1660: 1629: 1604: 1584: 1555: 1495: 1310: 1290: 1265: 1241: 1152: 1133: 1061: 1042: 86: 14012:

Fractional Calculus – An Introduction for Physicists

13745: 13075:. Springer and Acoustical Society of America Press. 12458:

Benson, D.; Wheatcraft, S.; Meerschaert, M. (2000).

12054: 12014:"Weyl and Marchaud Derivatives: A Forgotten History" 9577:, all non-negative, denote the coefficients for the 7718:{\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })} 4756:

starts with the derivative instead of the integral.

2030: 13889:

Fractional Derivatives for Physicists and Engineers

13686: 12197:(1950–1951). "On some functional transformations". 11240:

Fractional Calculus: An Introduction for Physicists

1225:for such operators generalizing the classical one. 13705: 11810: 11808: 11648: 11646: 10978:Bulletin of Mathematical Analysis and Applications 10905: 10884: 10864: 10767: 10711: 10471: 10262: 10192: 10071: 9852: 9532: 9294: 9165: 9011: 8905: 8743: 8654: 8472: 8443: 8420: 8399: 8373: 8353: 8329: 8309: 8287: 8226: 8145: 8119: 7970: 7929: 7841: 7798: 7776: 7742: 7717: 7657: 7612: 7588: 7568: 7539: 7498: 7254: 7022: 6995: 6966: 6927: 6874: 6669: 6642: 6592: 6267: 6081: 5852: 5212: 4965: 4741: 4690: 4161: 3930: 3762: 3461: 2927: 2625: 2474: 2316: 2128: 1963: 1943: 1890: 1862: 1811: 1787: 1760: 1738: 1715: 1688: 1666: 1643: 1610: 1590: 1568: 1527: 1479: 1296: 1271: 1247: 1211: 1139: 1113: 1048: 158: 14818:List of nonlinear ordinary differential equations 13861:Basic Theory of Fractional Differential Equations 12740:Monthly Notices of the Royal Astronomical Society 12371: 11806: 11804: 11802: 11800: 11798: 11796: 11794: 11792: 11790: 11788: 11004: 10941: 9889:Fractional Schrödinger equation in quantum theory 8460:The fractional derivative of a function of order 7311: 7077: 6721: 6324: 5921: 5806: 5697: 5606: 5541: 5432: 5341: 5024: 4802: 4553: 4488: 4332: 4267: 14921: 14823:List of nonlinear partial differential equations 13668:Miller, Kenneth S.; Ross, Bertram, eds. (1993). 13125:The Journal of the Acoustical Society of America 12673: 12564: 12560: 12558: 11603:Algahtani, Obaid Jefain Julaighim (2016-08-01). 11598: 11596: 11594: 2129:{\displaystyle (Jf)(x)=\int _{0}^{x}f(t)\,dt\,.} 159:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)} 13922: 13603:International Journal of Differential Equations 12999: 11652: 11643: 9212:Time-space fractional diffusion equation models 5862: 4759: 1944:{\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}} 1863:{\displaystyle \{D^{a}\mid a\in \mathbb {R} \}} 14813:List of linear ordinary differential equations 14251:collection of books, articles, preprints, etc. 14222: 13687:Samko, S.; Kilbas, A.A.; Marichev, O. (1993). 13596: 12597: 12338: 12140: 12097: 11785: 11566: 10497:Variable-order fractional Schrödinger equation 1545:to any function, will have the same effect as 1212:{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,} 14416: 14015:(3rd ed.). Singapore: World Scientific. 13750:. Vol. 56. Springer Verlag. p. 65. 13724: 13648: 13118: 13116: 12555: 11591: 10935: 9357:in certain types of materials like polymers. 8752:is not large enough compared to the scale of 7576:is increasing on the real line, converges to 5278:th order derivative of the integral of order 996: 13925:Fractional Calculus: Theory and Applications 12902:Journal of the Acoustical Society of America 12841:Journal of the Acoustical Society of America 12193: 11865:Mathematical Methods in the Applied Sciences 11858: 11856: 11854: 11383: 11377: 11336:Journal de Mathématiques Pures et Appliquées 11145: 10817: 9345: 8732: 8691:more general than powers are studied in the 8102: 8096: 1938: 1911: 1857: 1830: 1114:{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,} 13706:Carpinteri, A.; Mainardi, F., eds. (1998). 13192:Pandey, Vikash; Holm, Sverre (2016-09-23). 13005: 12960: 12899: 12838: 11386:"Fractional paradigms in quantum chemistry" 11277: 11275: 10968: 9179: 8915: 14430: 14423: 14409: 14223:Lorenzo, Carl F.; Hartley, Tom T. (2002). 13941: 13667: 13649:Oldham, Keith B.; Spanier, Jerome (1974). 13364: 13191: 13122: 13113: 12372:Atangana, Abdon; Vermeulen, P. D. (2014). 11703: 11653:Atangana, Abdon; Baleanu, Dumitru (2016). 11390:International Journal of Quantum Chemistry 10872:is commonly used instead of the intuitive 10081:where the solution of the equation is the 8480:is nowadays often defined by means of the 8165:Classical fractional derivatives include: 6275:which has the advantage that is zero when 1003: 989: 14382: 14372: 14321: 14292:Introductory Notes on Fractional Calculus 13624: 13614: 13597:Ramirez, L.E.S.; Coimbra, C.F.M. (2010). 13579: 13540: 13509: 13331: 13282: 13264: 13227: 13217: 13136: 13029: 13019: 12974: 12929: 12868: 12823: 12813: 12770: 12752: 12683: 12650: 12582: 12522: 12483: 12426: 12389: 12356: 12039: 12029: 11892: 11851: 11842: 11832: 11752: 11723: 11680: 11670: 11602: 11567:Caputo, Michele; Fabrizio, Mauro (2015). 11550: 11401: 11206: 11098: 11071: 11040: 10989: 10969:Katugampola, Udita N. (15 October 2014). 10962: 10465: 10065: 9846: 9605:Acoustic wave equations for complex media 9350:Fractional derivatives are used to model 8665: 8648: 8641: 8496: 7863: 6579: 6567: 6431: 6230: 6069: 5222: 5203: 4959: 4677: 4456: 4100: 3892: 3846: 3753: 3741: 3448: 3436: 3275: 3268: 3112: 2917: 2910: 2635:This is in fact a well-defined operator. 2619: 2612: 2468: 2461: 2306: 2288: 2224: 2122: 2115: 1934: 1853: 1637: 1205: 1198: 1107: 119: 14254: 14244: 14235: 14170:Fractional Calculus and Applied Analysis 14164:Fractional Calculus and Applied Analysis 14008: 13972:Numerical Methods for Fractional Calcuus 13885: 13836:. Nonlinear Physical Science. Springer. 13800: 13521:Fractional Calculus and Applied Analysis 13402:Fractional Calculus and Its Applications 13062: 12954: 12639:Fractional Calculus and Applied Analysis 12565:Atangana, Abdon; Kilicman, Adem (2014). 12339:Atangana, Abdon; Bildik, Necdet (2013). 11961:Fractional Calculus and Its Applications 11455: 11416: 11410: 11326: 11281: 11272: 11187:Fractional Calculus and Applied Analysis 11029:Fractional Calculus and Applied Analysis 9193:Fractional advection dispersion equation 9036:contains a one-half power dependence on 8707:. The operators arising are examples of 5256: 3776:which satisfies the following property: 14263: 13829: 13446: 12011: 11237: 9589:terms, respectively (sometimes denoted 7855:. Their Laplace transform is given by: 7289: 7055: 6928:{\displaystyle a<0,\alpha \in (0,1]} 6702: 5266:Riemann–Liouville fractional derivative 5005: 4772:and is given by the following formula, 2638:It is straightforward to show that the 2331:Cauchy formula for repeated integration 2053:. Form the definite integral from 0 to 527:Differentiating under the integral sign 14922: 14354: 14310:Communications in Mathematical Physics 14307: 14183:Fractional Differential Calculus (FDC) 13979: 13968: 13321: 13250: 12676:Processes with Long-Range Correlations 12055:Khalili Golmankhaneh, Alireza (2022). 11958: 11520: 11135:), Thèse, Université Paris Nord (1994) 8703:also allows one to consider powers of 8671: 8505:is an integral operator introduced by 7667:Therefore, we have that, the function 6940:Atangana–Baleanu fractional derivative 2326:and this can be extended arbitrarily. 14404: 14231:. NASA John H. Glenn Research Center. 14196:Communications in Fractional Calculus 14128: 14109: 14090: 14045: 12223: 12058:Fractal Calculus and its Applications 11862: 11738: 11611:. Nonlinear Dynamics and Complexity. 8728: 8666:Riemann–Liouville fractional integral 7851:the Mittag-Leffler distributions are 6617:Caputo–Fabrizio fractional derivative 5294:is the smallest integer greater than 4199:Riemann–Liouville fractional integral 3940:Substituting back into the equation: 1232:refers to iterative application of a 14808:List of named differential equations 13858: 13399: 13068: 13008:Ultrasound in Medicine & Biology 12801:Mathematical Problems in Engineering 12571:Mathematical Problems in Engineering 12345:Mathematical Problems in Engineering 12226:The Quarterly Journal of Mathematics 11915: 11821:Mathematical Problems in Engineering 11527:is almost frequency independent. II" 8239:New fractional derivatives include: 18: 14733:Method of undetermined coefficients 14514:Dependent and independent variables 14297:Power Law & Fractional Dynamics 14257:"Les Différentielles métaphysiques" 13969:Li, Changpin; Zeng, Fanhai (2015). 12048: 11523:"Linear model of dissipation whose 10501:As a natural generalization of the 9550:are positive fractional orders and 8270:Nature of the fractional derivative 7992:The Riesz derivative is defined as 7987: 1985: 13: 14095:"Fractional Differential Equation" 13388: 13040:10.1016/j.ultrasmedbio.2013.09.033 11552:10.1111/j.1365-246x.1967.tb02303.x 10945:Handbook of Differential Equations 10747: 10737: 10728: 10613: 10564: 10521: 10305: 10263:{\textstyle D_{2}={\frac {1}{2m}}} 10211:is a scale constant with physical 10172: 10162: 10153: 9983: 9942: 9917: 9817: 9797: 9741: 9723: 9713: 9675: 9661: 9618: 9497: 9452: 9425: 9360: 9273: 9242: 9228: 8894: 8886: 8825: 8807: 8775: 8548: 8491: 8138: 8091: 8028: 8014: 8001: 7607: 6485: 6129: 5964: 5129: 4850: 4603: 4382: 4179:shows that the order in which the 4014: 3907: 3893: 3880: 3593: 3581: 3310: 3298: 3147: 3135: 3022: 2824: 2538: 68:Part of a series of articles about 14: 14946: 14225:"Initialized Fractional Calculus" 14075: 13725:Igor Podlubny (27 October 1998). 12257:"Fractional conservation of mass" 11531:Geophysical Journal International 11128:Les différentielles métaphysiques 10942:Daniel Zwillinger (12 May 2014). 10515: 10403: 10361: 10103:for the particle to have a given 9911: 2031:Computing the fractional integral 2024:fractional differential operators 1644:{\displaystyle n\in \mathbb {Z} } 58:For the associated operator, see 14630:Carathéodory's existence theorem 14238:"Fractional calculus: Resources" 13923:Daftardar-gejji, Varsha (2013). 13072:Waves with Power-Law Attenuation 12012:Ferrari, Fausto (January 2018). 11282:Mainardi, Francesco (May 2010). 11181:Valério, Duarte; Machado, José; 11108:Journal de l'École Polytechnique 11081:Journal de l'École Polytechnique 10752: 10693: 10673: 10650: 10579: 10549: 10536: 10452: 10429: 10417: 10409: 10330: 10177: 10052: 10032: 10009: 9927: 7971:{\displaystyle \alpha \in (0,1)} 7939:This directly implies that, for 7842:{\displaystyle \alpha \in (0,1)} 7658:{\displaystyle E_{\alpha }(0)=1} 7540:{\displaystyle \alpha \in (0,1]} 4806: 4787: 2022:introduced the practical use of 23: 14046:Li, Changpin; Cai, Min (2019). 13946:. Singapore: World Scientific. 13863:. Singapore: World Scientific. 13358: 13315: 13244: 13185: 12893: 12832: 12787: 12726: 12667: 12626: 12591: 12500: 12451: 12398: 12365: 12332: 12315: 12298: 12284:10.1016/j.advwatres.2008.07.004 12248: 12217: 12187: 12169: 12134: 12100:Journal of Mathematical Physics 12091: 12005: 11952: 11909: 11741:Journal of Mathematical Physics 11732: 11697: 11560: 11514: 11456:Herrmann, Richard, ed. (2014). 11449: 11384:Mostafanejad, Mohammad (2021). 11346: 11320: 11231: 11174: 10807:Initialized fractional calculus 10503:fractional Schrödinger equation 9895:fractional Schrödinger equation 8744:Fractional conservation of mass 8738: 6237: 4949: 4235:, the integrals are defined as 14361:Journal of Applied Mathematics 13691:. Taylor & Francis Books. 11609:Chaos, Solitons & Fractals 11423:Chaos, Solitons & Fractals 11138: 11119: 11092: 11065: 11020: 11014:Magazin for Naturvidenskaberne 10998: 10846: 10703: 10689: 10683: 10669: 10660: 10646: 10633: 10627: 10583: 10575: 10559: 10545: 10540: 10532: 10462: 10448: 10435: 10424: 10365: 10352: 10340: 10326: 10309: 10289: 10062: 10048: 10042: 10028: 10019: 10005: 9937: 9923: 9527: 9521: 9485: 9479: 9440: 9434: 9413: 9407: 9277: 9267: 9160: 9148: 9092: 9080: 9006: 8994: 8982: 8970: 8822: 8810: 8793: 8638: 8632: 8557: 8551: 8160: 8146:{\displaystyle {\mathcal {F}}} 8111: 8105: 8062: 8056: 7982:geometric stable distributions 7965: 7953: 7893: 7867: 7836: 7824: 7712: 7696: 7677: 7646: 7640: 7534: 7522: 7459: 7446: 7419: 7413: 7363: 7357: 7342: 7336: 7215: 7202: 7175: 7169: 7129: 7123: 7108: 7102: 6990: 6984: 6922: 6910: 6805: 6799: 6752: 6746: 6637: 6631: 6564: 6558: 6500: 6488: 6480: 6474: 6423: 6417: 6409: 6403: 6390: 6384: 6356: 6350: 6250: 6238: 6227: 6221: 6216: 6210: 6200: 6182: 6178: 6165: 6144: 6132: 6117: 6111: 6025: 6019: 6014: 6008: 5979: 5967: 5952: 5946: 5843: 5837: 5743: 5737: 5725: 5713: 5637: 5631: 5578: 5572: 5478: 5472: 5460: 5448: 5372: 5366: 5200: 5194: 5138: 5132: 5126: 5120: 5102: 5096: 5087: 5081: 5055: 5049: 4928: 4922: 4859: 4853: 4838: 4832: 4700:Where the former is valid for 4674: 4668: 4612: 4606: 4584: 4578: 4522: 4516: 4453: 4447: 4391: 4385: 4363: 4357: 4301: 4295: 4152: 4146: 4097: 4091: 4029: 4017: 3998: 3992: 3922: 3910: 3902: 3896: 3889: 3883: 3871: 3859: 3670: 3664: 3602: 3596: 3590: 3584: 3569: 3563: 3349: 3343: 3319: 3313: 3307: 3301: 3265: 3259: 3156: 3150: 3144: 3138: 3109: 3103: 3065: 3052: 3031: 3025: 3006: 3000: 2907: 2901: 2839: 2827: 2805: 2799: 2757: 2751: 2700: 2694: 2609: 2603: 2547: 2541: 2526: 2520: 2458: 2452: 2393: 2381: 2369: 2363: 2285: 2279: 2221: 2215: 2212: 2203: 2178: 2172: 2112: 2106: 2082: 2076: 2073: 2064: 1467: 1464: 1461: 1455: 1449: 1428: 1422: 1416: 1396: 1390: 1387: 1341: 1331: 1325: 1195: 1189: 1165: 1159: 1104: 1098: 1074: 1068: 153: 147: 138: 132: 116: 110: 1: 14287:Fractional Calculus Modelling 13367:Applied Numerical Mathematics 12620:10.1016/s0370-1573(00)00070-3 12378:Abstract and Applied Analysis 11711:Abstract and Applied Analysis 10929: 10834:Prabhakar fractional calculus 8701:pseudo-differential operators 2138:Repeating this process gives 1994:and mathematical analysis, a 453:Integral of inverse functions 14458:Notation for differentiation 13748:Physics of Fractal Operators 13461:10.1080/00207390410001686571 13324:Fractional Quantum Mechanics 11160:10.1016/0315-0860(77)90039-8 10829:Fractional Fourier transform 9899:fractional quantum mechanics 9897:, a fundamental equation of 9202:variational order derivative 8170:Grünwald–Letnikov derivative 7275: 7041: 6688: 6302: 5904: 5875:Caputo fractional derivative 5869:Caputo fractional derivative 5863:Caputo fractional derivative 5789: 5680: 5589: 5524: 5415: 5324: 4991: 4783: 4766:Hadamard fractional integral 4760:Hadamard fractional integral 4754:Grünwald–Letnikov derivative 4710:and the latter is valid for 4536: 4471: 4315: 4250: 1653:it coincides with the usual 7: 14554:Exact differential equation 14255:Dugowson, Stéphane (2006). 13942:Srivastava, Hari M (2014). 13927:. Narosa Publishing House. 13502:10.1016/j.cnsns.2010.05.027 13379:10.1016/j.apnum.2016.09.009 12264:Advances in Water Resources 11629:10.1016/j.chaos.2016.03.026 11443:10.1016/j.chaos.2021.111209 11016:. Kristiania (Oslo): 55–68. 10790: 9040:. Taking the derivative of 8719:can be discussed. See also 8709:singular integral operators 8265:Atangana–Baleanu derivative 8227:{\displaystyle F^{\alpha }} 7727:Mittag-Leffler distribution 7569:{\displaystyle E_{\alpha }} 6967:{\displaystyle E_{\alpha }} 876:Calculus on Euclidean space 294:Logarithmic differentiation 49:Proposed since August 2024. 32:It has been suggested that 10: 14951: 14245:Herrmann, Richard (2018). 13807:. Imperial College Press. 13293:10.1103/PhysRevE.66.056108 13219:10.1103/PhysRevE.94.032606 11005:Niels Henrik Abel (1823). 9901:, has the following form: 8919: 8261:Caputo Fabrizio derivative 5866: 5226: 4205:Riemann–Liouville integral 3772:The inner integral is the 2046:be a function defined for 2015:in a paper from 1832. The 1228:In this context, the term 57: 14864:Józef Maria Hoene-Wroński 14844:Gottfried Wilhelm Leibniz 14831: 14800: 14710: 14643: 14635:Cauchy–Kowalevski theorem 14612: 14605: 14567: 14506: 14445: 14438: 14384:10.1155/S1110757X02110102 13992:10.1007/978-3-319-20771-1 13906:10.1007/978-3-642-33911-0 13842:10.1007/978-3-642-14003-7 13710:. Springer-Verlag Telos. 13672:. John Wiley & Sons. 13572:10.1007/s11192-013-1032-6 13533:10.2478/s13540-013-0030-y 13089:10.1007/978-3-030-14927-7 12985:10.2478/s13540-013-0003-1 12510:Transport in Porous Media 12242:10.1093/qmath/os-11.1.193 11938:10.1142/S0218348X17500335 11417:Al-Raeei, Marwan (2021). 11238:Hermann, Richard (2014). 11199:10.2478/s13540-014-0185-1 10818:Other fractional theories 10099:– the quantum mechanical 9346:Structural damping models 8430:even those far away from 8407:depends on all values of 8297:derivative of a function 8174:Sonin–Letnikov derivative 5274:th order derivative, the 2004:Gottfried Wilhelm Leibniz 1598:in such a way that, when 610:Summand limit (term test) 14758:Finite difference method 14273:University of Notre Dame 13642: 12464:Water Resources Research 12407:Water Resources Research 11521:Caputo, Michele (1967). 10839: 9180:Groundwater flow problem 8929:Fick's laws of diffusion 8916:Electrochemical analysis 8203:Erdélyi–Kober derivative 7808:exponential distribution 7613:{\displaystyle -\infty } 5227:Not to be confused with 1870:defined in this way are 289:Implicit differentiation 279:Differentiation notation 206:Inverse function theorem 35:Draft:Coimbra derivative 14738:Variation of parameters 14728:Separation of variables 14625:Peano existence theorem 14620:Picard–Lindelöf theorem 14507:Attributes of variables 14236:Podlubny, Igor (2010). 14133:"Fractional derivative" 14058:10.1137/1.9781611975888 13886:Uchaikin, V.V. (2012). 12963:Fract. Calc. Appl. Anal 12702:10.1007/3-540-44832-2_8 12533:10.1023/A:1006733002131 10824:Fractional-order system 10270:for a particle of mass 10143:depends on the system. 10122:reduced Planck constant 9383:in terms of a measured 8668:and the Weyl integral. 8513:(1940) and is given by 7743:{\displaystyle \alpha } 6946:Mittag-Leffler function 5877:. It was introduced by 752:Helmholtz decomposition 14899:Carl David Tolmé Runge 14473:Differential-algebraic 14432:Differential equations 14264:Loverro, Adam (2005). 13830:Tarasov, V.E. (2010). 13476:Tenreiro Machado, J.; 12228:. os-11 (1): 193–211. 11682:10.2298/TSCI160111018A 11288:Imperial College Press 10907: 10886: 10866: 10769: 10713: 10473: 10264: 10194: 10073: 9883:non-Newtonian rheology 9854: 9534: 9296: 9167: 9013: 8907: 8721:Erdélyi–Kober operator 8664:which generalizes the 8656: 8503:Erdélyi–Kober operator 8497:Erdélyi–Kober operator 8474: 8445: 8422: 8401: 8375: 8355: 8331: 8311: 8289: 8247:Katugampola derivative 8228: 8194:Miller–Ross derivative 8147: 8121: 7972: 7931: 7843: 7800: 7778: 7744: 7719: 7659: 7614: 7590: 7570: 7541: 7500: 7256: 7024: 6997: 6968: 6929: 6876: 6671: 6644: 6594: 6269: 6083: 5854: 5262: 5223:Fractional derivatives 5214: 4967: 4743: 4692: 4163: 3932: 3764: 3489:Changing variables to 3463: 2942:Proof of this identity 2929: 2627: 2476: 2318: 2130: 2000:Guillaume de l'Hôpital 1980:differential equations 1965: 1945: 1900:of which the original 1892: 1864: 1813: 1789: 1788:{\displaystyle n<0} 1762: 1740: 1717: 1716:{\displaystyle n>0} 1690: 1668: 1645: 1612: 1592: 1578:for every real number 1570: 1539:functional square root 1537:as an analogue of the 1529: 1481: 1298: 1273: 1249: 1213: 1141: 1115: 1050: 886:Limit of distributions 706:Directional derivative 362:Faà di Bruno's formula 160: 14884:Augustin-Louis Cauchy 14879:Joseph-Louis Lagrange 14773:Finite element method 14763:Crank–Nicolson method 14697:Numerical integration 14676:Exponential stability 14568:Relation to processes 14453:Differential operator 14355:Závada, Petr (2002). 14340:10.1007/s002200050299 14160:Specialized journals 14150:"Fractional Calculus" 14114:"Fractional calculus" 14009:Herrmann, R. (2018). 13801:Mainardi, F. (2010). 13322:Laskin, Nick (2018). 12177:"Fractional Calculus" 11429:(September): 111209. 11327:Hadamard, J. (1892). 11051:10.1515/fca-2017-0057 10908: 10887: 10867: 10770: 10714: 10474: 10265: 10195: 10101:probability amplitude 10074: 9881:and Nutting's law in 9855: 9535: 9297: 9206:Crank–Nicolson method 9168: 9014: 8908: 8657: 8488:integral transforms. 8475: 8446: 8423: 8402: 8376: 8356: 8332: 8312: 8290: 8229: 8148: 8122: 7973: 7932: 7844: 7801: 7779: 7777:{\displaystyle E_{1}} 7754:absolutely continuous 7745: 7720: 7660: 7615: 7591: 7571: 7542: 7501: 7257: 7025: 7023:{\displaystyle C^{1}} 6998: 6969: 6930: 6877: 6672: 6670:{\displaystyle C^{1}} 6645: 6595: 6270: 6084: 5855: 5260: 5215: 4968: 4744: 4693: 4164: 3933: 3765: 3464: 2930: 2628: 2477: 2319: 2131: 1996:fractional derivative 1966: 1946: 1893: 1865: 1814: 1790: 1763: 1741: 1718: 1691: 1669: 1646: 1613: 1593: 1571: 1569:{\displaystyle D^{a}} 1530: 1482: 1299: 1274: 1250: 1214: 1142: 1116: 1051: 1020:mathematical analysis 970:Mathematical analysis 881:Generalized functions 566:arithmetico-geometric 407:Leibniz integral rule 161: 14778:Finite volume method 14702:Dirac delta function 14671:Asymptotic stability 14613:Existence/uniqueness 14478:Integro-differential 12772:10.1093/mnras/stx261 12485:10.1029/2000wr900032 12437:10.1029/2000wr900031 12321:Pospíšil, L. et al. 12306:Analytical Chemistry 12155:10.18576/pfda/020101 11359:. Elsevier. p. 11148:Historia Mathematica 10948:. Elsevier Science. 10897: 10876: 10856: 10797:Acoustic attenuation 10725: 10509: 10286: 10274:), and the operator 10229: 10150: 9905: 9866:Acoustic attenuation 9614: 9401: 9221: 9059: 8935: 8760: 8517: 8464: 8435: 8412: 8385: 8365: 8345: 8321: 8301: 8279: 8211: 8177:Liouville derivative 8133: 7996: 7944: 7859: 7815: 7790: 7761: 7734: 7671: 7627: 7601: 7580: 7553: 7513: 7297: 7281: 7268: 7063: 7047: 7034: 7007: 6996:{\displaystyle f(t)} 6978: 6951: 6889: 6707: 6694: 6681: 6654: 6643:{\displaystyle f(t)} 6625: 6308: 6291: 6095: 5910: 5897: 5795: 5686: 5595: 5530: 5421: 5330: 5313: 5010: 4997: 4984: 4791: 4776: 4727: 4542: 4477: 4321: 4256: 4239: 3944: 3780: 3519: 2952: 2646: 2494: 2337: 2142: 2061: 1955: 1908: 1882: 1827: 1803: 1773: 1752: 1730: 1701: 1680: 1658: 1627: 1602: 1582: 1553: 1493: 1308: 1288: 1281:that is, repeatedly 1263: 1239: 1150: 1131: 1059: 1040: 975:Nonstandard analysis 443:Lebesgue integration 313:Rules and identities 84: 42:into this article. ( 14930:Fractional calculus 14788:Perturbation theory 14768:Runge–Kutta methods 14748:Integral transforms 14681:Rate of convergence 14577:(discrete analogue) 14332:1998CMaPh.192..261Z 13980:Umarov, S. (2015). 13898:2013fdpe.book.....U 13756:2003PhT....56l..65W 13616:10.1155/2010/846107 13494:2011CNSNS..16.1140M 13275:2002PhRvE..66e6108L 13210:2016PhRvE..94c2606P 13147:2016ASAJ..140.4225P 13081:2019wpla.book.....H 12914:2011ASAJ..130.3038N 12853:2011ASAJ..130.2195H 12815:10.1155/2010/639801 12763:2017MNRAS.467.2421C 12694:2003LNP...621..148G 12661:2007cond.mat..2419M 12612:2000PhR...339....1M 12584:10.1155/2014/542809 12476:2000WRR....36.1413B 12419:2000WRR....36.1403B 12391:10.1155/2014/381753 12358:10.1155/2013/543026 12323:Electrochimica Acta 12276:2008AdWR...31.1377W 12234:1940QJMat..11..193K 12112:2015JMP....56f3502A 12041:10.3390/math6010006 11930:2017Fract..2550033M 11924:(3): 1750033–2980. 11877:2015MMAS...38...27A 11834:10.1155/2014/238459 11763:2016JMP....57l3501B 11725:10.1155/2014/653797 11621:2016CSF....89..552A 11543:1967GeoJ...13..529C 11470:2014fcip.book.....H 11458:Fractional Calculus 11435:2021CSF...15011209A 11248:2014fcip.book.....H 11183:Kiryakova, Virginia 9788: 9773: 9708: 9653: 9517: 9475: 8717:fractional calculus 8693:functional calculus 8678:functional analysis 8672:Functional calculus 8577: 8455:boundary conditions 8400:{\displaystyle x=c} 8254:Davidson derivative 8188:Marchaud derivative 8185:Hadamard derivative 7409: 7328: 7310: 7298: 7290: 7283: 7277: 7160: 7094: 7076: 7064: 7056: 7049: 7043: 6790: 6738: 6720: 6708: 6703: 6696: 6690: 6520: 6462: 6380: 6323: 6310: 6304: 6164: 5999: 5938: 5912: 5906: 5829: 5797: 5791: 5729: 5688: 5682: 5623: 5597: 5591: 5564: 5532: 5526: 5464: 5423: 5417: 5358: 5332: 5326: 5290:is computed, where 5158: 5041: 5023: 5011: 5006: 4999: 4993: 4879: 4824: 4793: 4785: 4742:{\displaystyle a=0} 4632: 4570: 4544: 4538: 4508: 4479: 4473: 4411: 4349: 4323: 4317: 4287: 4258: 4252: 4215:. It is defined on 4049: 3797: 3692: 3622: 3371: 3339: 3191: 3176: 3051: 2859: 2642:operator satisfies 2567: 2416: 2275: 2255: 2202: 2102: 1992:applied mathematics 1823:of operator powers 1304:with itself, as in 1185: 1016:Fractional calculus 646:Cauchy condensation 448:Contour integration 174:Fundamental theorem 101: 14909:Sofya Kovalevskaya 14743:Integrating factor 14666:Lyapunov stability 14586:Stochastic partial 14130:Weisstein, Eric W. 14111:Weisstein, Eric W. 14092:Weisstein, Eric W. 13410:10.1007/BFb0067096 11969:10.1007/bfb0067098 10903: 10882: 10862: 10765: 10709: 10469: 10260: 10190: 10112:at any given time 10069: 9850: 9838: 9790: 9774: 9759: 9738: 9694: 9655: 9639: 9530: 9503: 9458: 9397:can be written as 9292: 9217:can be written as 9163: 9139: 9116: 9009: 8903: 8676:In the context of 8652: 8563: 8470: 8441: 8418: 8397: 8371: 8351: 8327: 8307: 8285: 8243:Coimbra derivative 8224: 8143: 8117: 7968: 7927: 7839: 7796: 7774: 7740: 7715: 7655: 7610: 7586: 7566: 7537: 7496: 7395: 7312: 7272: 7252: 7146: 7078: 7038: 7020: 6993: 6964: 6925: 6872: 6776: 6722: 6685: 6667: 6640: 6590: 6588: 6506: 6448: 6366: 6299: 6265: 6150: 6079: 5985: 5922: 5850: 5848: 5807: 5698: 5607: 5542: 5433: 5342: 5263: 5229:Fractal derivative 5210: 5144: 5025: 4988: 4963: 4865: 4803: 4768:was introduced by 4739: 4688: 4686: 4618: 4554: 4489: 4397: 4333: 4268: 4209:periodic functions 4159: 4157: 4035: 3928: 3783: 3760: 3678: 3608: 3459: 3457: 3357: 3325: 3177: 3162: 3037: 2925: 2923: 2845: 2623: 2553: 2472: 2402: 2314: 2312: 2261: 2241: 2188: 2126: 2088: 1961: 1941: 1888: 1860: 1809: 1785: 1758: 1736: 1713: 1686: 1664: 1641: 1608: 1588: 1566: 1525: 1522: 1477: 1475: 1448: 1441: 1386: 1379: 1294: 1269: 1245: 1209: 1171: 1137: 1111: 1046: 818:Partial derivative 747:generalized Stokes 641:Alternating series 522:Reduction formulae 511:Heaviside's method 492:tangent half-angle 479:Cylindrical shells 402:Integral transform 397:Lists of integrals 201:Mean value theorem 156: 87: 14917: 14916: 14796: 14795: 14601: 14600: 14302:The CRONE Toolbox 14067:978-1-61197-587-1 14030:978-981-3274-57-0 14001:978-3-319-20770-4 13975:. USA: CRC Press. 13961:978-981-4551-10-6 13915:978-3-642-33911-0 13878:978-981-4579-89-6 13859:Zhou, Y. (2010). 13851:978-3-642-14003-7 13822:978-1-84816-329-4 13773:978-0-387-95554-4 13764:10.1063/1.1650234 13738:978-0-08-053198-4 13717:978-3-211-82913-4 13698:978-2-88124-864-1 13679:978-0-471-58884-9 13660:978-0-12-525550-9 13419:978-3-540-07161-7 13351:978-981-322-379-0 13198:Physical Review E 13155:10.1121/1.4971289 13098:978-3-030-14926-0 13069:Holm, S. (2019). 12922:10.1121/1.3641457 12861:10.1121/1.3631626 12711:978-3-540-40129-2 12270:(10): 1377–1381. 12120:10.1063/1.4922018 12076:978-981-126-110-7 11978:978-3-540-69975-0 11771:10.1063/1.4968819 11487:978-981-4551-07-6 11403:10.1002/qua.26762 11370:978-0-444-51832-3 11361:75 (Property 2.4) 11305:978-1-84816-329-4 11265:978-981-4551-07-6 11100:Liouville, Joseph 11073:Liouville, Joseph 10955:978-1-4832-2096-3 10906:{\displaystyle I} 10885:{\displaystyle I} 10865:{\displaystyle J} 10812:Nonlocal operator 10779:and the operator 10763: 10640: 10589: 10406: 10375: 10320: 10258: 10188: 9999: 9949: 9873:and Holm derived 9837: 9789: 9737: 9689: 9654: 9312:are changed into 9256: 9143: 9137: 9114: 9075: 8965: 8901: 8796: 8733:Erdélyi 1950–1951 8561: 8473:{\displaystyle a} 8444:{\displaystyle c} 8421:{\displaystyle f} 8374:{\displaystyle f} 8354:{\displaystyle a} 8330:{\displaystyle x} 8310:{\displaystyle f} 8288:{\displaystyle a} 8251:Hilfer derivative 8181:Caputo derivative 8155:Fourier transform 8050: 7922: 7799:{\displaystyle 1} 7589:{\displaystyle 0} 7480: 7393: 7378: 7308: 7236: 7144: 7074: 6862: 6851: 6810: 6774: 6718: 6504: 6148: 6067: 5983: 5783: 5674: 5518: 5409: 5142: 5091: 4944: 4900: 4863: 4616: 4395: 4188: 4187: 4033: 3926: 3606: 3323: 3160: 3035: 2843: 2551: 2400: 2009:Niels Henrik Abel 1964:{\displaystyle n} 1891:{\displaystyle a} 1812:{\displaystyle D} 1761:{\displaystyle J} 1739:{\displaystyle n} 1689:{\displaystyle D} 1667:{\displaystyle n} 1611:{\displaystyle a} 1591:{\displaystyle a} 1520: 1501: 1411: 1409: 1346: 1344: 1297:{\displaystyle D} 1272:{\displaystyle f} 1248:{\displaystyle D} 1221:and developing a 1140:{\displaystyle J} 1093: 1049:{\displaystyle D} 1013: 1012: 893: 892: 855: 854: 823:Multiple integral 759: 758: 663: 662: 630:Direct comparison 601:Convergence tests 539: 538: 507:Partial fractions 374: 373: 284:Second derivative 56: 55: 51: 14942: 14894:Phyllis Nicolson 14874:Rudolf Lipschitz 14711:Solution methods 14686:Series solutions 14610: 14609: 14443: 14442: 14425: 14418: 14411: 14402: 14401: 14396: 14386: 14376: 14351: 14325: 14323:funct-an/9608002 14283: 14281: 14275:. Archived from 14270: 14260: 14250: 14241: 14232: 14157: 14143: 14142: 14124: 14123: 14105: 14104: 14071: 14042: 14005: 13976: 13965: 13938: 13919: 13882: 13855: 13826: 13797: 13791: 13787: 13785: 13777: 13742: 13721: 13702: 13683: 13664: 13638: 13628: 13618: 13593: 13583: 13554: 13544: 13515: 13513: 13488:(3): 1140–1153. 13472: 13443: 13437: 13433: 13431: 13423: 13383: 13382: 13362: 13356: 13355: 13335: 13319: 13313: 13312: 13286: 13268: 13266:quant-ph/0206098 13248: 13242: 13241: 13231: 13221: 13189: 13183: 13182: 13140: 13131:(6): 4225–4236. 13120: 13111: 13110: 13066: 13060: 13059: 13033: 13023: 13003: 12997: 12996: 12978: 12958: 12952: 12951: 12933: 12908:(5): 3038–3045. 12897: 12891: 12890: 12872: 12847:(4): 2195–2201. 12836: 12830: 12829: 12827: 12817: 12791: 12785: 12784: 12774: 12756: 12747:(2): 2421–2429. 12730: 12724: 12723: 12687: 12671: 12665: 12664: 12654: 12652:cond-mat/0702419 12630: 12624: 12623: 12595: 12589: 12588: 12586: 12562: 12553: 12552: 12526: 12517:(1–2): 211–240. 12504: 12498: 12497: 12487: 12470:(6): 1413–1423. 12455: 12449: 12448: 12430: 12413:(6): 1403–1412. 12402: 12396: 12395: 12393: 12369: 12363: 12362: 12360: 12336: 12330: 12319: 12313: 12302: 12296: 12295: 12261: 12252: 12246: 12245: 12221: 12215: 12214: 12191: 12185: 12184: 12173: 12167: 12166: 12138: 12132: 12131: 12095: 12089: 12088: 12052: 12046: 12045: 12043: 12033: 12009: 12003: 12002: 11996: 11992: 11990: 11982: 11956: 11950: 11949: 11913: 11907: 11906: 11896: 11885:10.1002/mma.3047 11860: 11849: 11848: 11846: 11836: 11812: 11783: 11782: 11756: 11736: 11730: 11729: 11727: 11701: 11695: 11694: 11684: 11674: 11650: 11641: 11640: 11600: 11589: 11588: 11586: 11584: 11564: 11558: 11556: 11554: 11518: 11512: 11511: 11505: 11501: 11499: 11491: 11453: 11447: 11446: 11414: 11408: 11407: 11405: 11381: 11375: 11374: 11350: 11344: 11343: 11333: 11324: 11318: 11317: 11279: 11270: 11269: 11235: 11229: 11228: 11210: 11178: 11172: 11171: 11142: 11136: 11123: 11117: 11115: 11096: 11090: 11088: 11069: 11063: 11062: 11044: 11035:(5): 1068–1075. 11024: 11018: 11017: 11011: 11002: 10996: 10995: 10993: 10975: 10966: 10960: 10959: 10939: 10923: 10914: 10912: 10910: 10909: 10904: 10891: 10889: 10888: 10883: 10871: 10869: 10868: 10863: 10850: 10786: 10777:Laplace operator 10774: 10772: 10771: 10766: 10764: 10762: 10761: 10760: 10755: 10745: 10744: 10735: 10718: 10716: 10715: 10710: 10696: 10676: 10653: 10642: 10641: 10636: 10622: 10620: 10616: 10612: 10611: 10590: 10588: 10587: 10586: 10582: 10562: 10552: 10544: 10543: 10539: 10519: 10492: 10484: 10478: 10476: 10475: 10470: 10455: 10444: 10443: 10438: 10432: 10427: 10422: 10421: 10420: 10412: 10407: 10399: 10389: 10388: 10376: 10374: 10373: 10372: 10347: 10333: 10322: 10321: 10313: 10304: 10303: 10281: 10273: 10269: 10267: 10266: 10261: 10259: 10257: 10246: 10241: 10240: 10224: 10217: 10216:= J·m·s = kg·m·s 10210: 10202:Laplace operator 10199: 10197: 10196: 10191: 10189: 10187: 10186: 10185: 10180: 10170: 10169: 10160: 10142: 10126:potential energy 10119: 10115: 10111: 10098: 10078: 10076: 10075: 10070: 10055: 10035: 10012: 10001: 10000: 9992: 9990: 9986: 9982: 9981: 9963: 9962: 9950: 9948: 9940: 9930: 9915: 9859: 9857: 9856: 9851: 9839: 9836: 9835: 9834: 9815: 9811: 9810: 9794: 9791: 9787: 9782: 9772: 9767: 9758: 9749: 9748: 9739: 9736: 9735: 9734: 9721: 9720: 9711: 9707: 9702: 9690: 9688: 9687: 9686: 9673: 9669: 9668: 9658: 9656: 9652: 9647: 9635: 9626: 9625: 9600: 9596: 9592: 9576: 9567: 9558: 9549: 9545: 9539: 9537: 9536: 9531: 9516: 9511: 9502: 9501: 9500: 9474: 9466: 9457: 9456: 9455: 9430: 9429: 9428: 9396: 9382: 9371:control variable 9341: 9326: 9311: 9307: 9301: 9299: 9298: 9293: 9285: 9284: 9257: 9255: 9254: 9253: 9240: 9236: 9235: 9225: 9172: 9170: 9169: 9164: 9144: 9142: 9141: 9140: 9138: 9130: 9118: 9117: 9115: 9107: 9099: 9076: 9074: 9063: 9054: 9039: 9035: 9018: 9016: 9015: 9010: 8966: 8964: 8963: 8962: 8949: 8948: 8939: 8912: 8910: 8909: 8904: 8902: 8900: 8892: 8884: 8882: 8878: 8877: 8876: 8861: 8860: 8843: 8842: 8803: 8799: 8798: 8797: 8789: 8783: 8782: 8725:special function 8713:Riesz potentials 8706: 8699:. The theory of 8690: 8661: 8659: 8658: 8653: 8628: 8627: 8609: 8608: 8597: 8593: 8576: 8571: 8562: 8560: 8546: 8545: 8521: 8479: 8477: 8476: 8471: 8452: 8450: 8448: 8447: 8442: 8429: 8427: 8425: 8424: 8419: 8406: 8404: 8403: 8398: 8380: 8378: 8377: 8372: 8360: 8358: 8357: 8352: 8336: 8334: 8333: 8328: 8316: 8314: 8313: 8308: 8296: 8294: 8292: 8291: 8286: 8233: 8231: 8230: 8225: 8223: 8222: 8191:Riesz derivative 8152: 8150: 8149: 8144: 8142: 8141: 8126: 8124: 8123: 8118: 8095: 8094: 8088: 8087: 8082: 8055: 8051: 8049: 8048: 8047: 8042: 8026: 8022: 8021: 8011: 8005: 8004: 7988:Riesz derivative 7979: 7977: 7975: 7974: 7969: 7936: 7934: 7933: 7928: 7923: 7921: 7920: 7919: 7900: 7892: 7891: 7890: 7889: 7866: 7850: 7848: 7846: 7845: 7840: 7806:is therefore an 7805: 7803: 7802: 7797: 7785: 7783: 7781: 7780: 7775: 7773: 7772: 7751: 7749: 7747: 7746: 7741: 7724: 7722: 7721: 7716: 7711: 7710: 7695: 7694: 7666: 7664: 7662: 7661: 7656: 7639: 7638: 7621: 7619: 7617: 7616: 7611: 7595: 7593: 7592: 7587: 7575: 7573: 7572: 7567: 7565: 7564: 7548: 7546: 7544: 7543: 7538: 7505: 7503: 7502: 7497: 7486: 7482: 7481: 7479: 7468: 7467: 7466: 7444: 7431: 7430: 7408: 7403: 7394: 7392: 7381: 7379: 7377: 7366: 7349: 7329: 7327: 7322: 7317: 7309: 7306: 7304: 7300: 7299: 7285: 7284: 7282: 7261: 7259: 7258: 7253: 7242: 7238: 7237: 7235: 7224: 7223: 7222: 7200: 7187: 7186: 7168: 7159: 7154: 7145: 7143: 7132: 7115: 7095: 7093: 7088: 7083: 7075: 7072: 7070: 7066: 7065: 7051: 7050: 7048: 7029: 7027: 7026: 7021: 7019: 7018: 7002: 7000: 6999: 6994: 6973: 6971: 6970: 6965: 6963: 6962: 6936: 6934: 6932: 6931: 6926: 6881: 6879: 6878: 6873: 6860: 6859: 6858: 6857: 6853: 6852: 6850: 6839: 6828: 6808: 6798: 6789: 6784: 6775: 6773: 6759: 6739: 6737: 6732: 6727: 6719: 6716: 6714: 6710: 6709: 6698: 6697: 6695: 6676: 6674: 6673: 6668: 6666: 6665: 6649: 6647: 6646: 6641: 6612: 6599: 6597: 6596: 6591: 6589: 6578: 6574: 6557: 6549: 6548: 6540: 6536: 6519: 6514: 6505: 6503: 6483: 6469: 6461: 6456: 6441: 6430: 6426: 6413: 6412: 6379: 6374: 6343: 6336: 6335: 6330: 6322: 6317: 6312: 6311: 6309: 6285: 6274: 6272: 6271: 6266: 6220: 6219: 6204: 6203: 6163: 6158: 6149: 6147: 6124: 6107: 6106: 6088: 6086: 6085: 6080: 6068: 6066: 6065: 6048: 6044: 6028: 6018: 6017: 6001: 5998: 5993: 5984: 5982: 5959: 5939: 5937: 5932: 5927: 5920: 5919: 5914: 5913: 5911: 5892: 5859: 5857: 5856: 5851: 5849: 5830: 5828: 5817: 5812: 5805: 5804: 5799: 5798: 5796: 5784: 5782: 5781: 5780: 5767: 5766: 5757: 5749: 5730: 5728: 5708: 5703: 5696: 5695: 5690: 5689: 5687: 5675: 5673: 5672: 5671: 5658: 5657: 5648: 5624: 5622: 5617: 5612: 5605: 5604: 5599: 5598: 5596: 5565: 5563: 5552: 5547: 5540: 5539: 5534: 5533: 5531: 5519: 5517: 5516: 5515: 5502: 5501: 5492: 5484: 5465: 5463: 5443: 5438: 5431: 5430: 5425: 5424: 5422: 5410: 5408: 5407: 5406: 5393: 5392: 5383: 5359: 5357: 5352: 5347: 5340: 5339: 5334: 5333: 5331: 5308: 5297: 5293: 5289: 5277: 5273: 5241: 5237: 5219: 5217: 5216: 5211: 5190: 5189: 5178: 5174: 5157: 5152: 5143: 5141: 5109: 5092: 5090: 5073: 5062: 5042: 5040: 5035: 5030: 5022: 5017: 5013: 5012: 5001: 5000: 4998: 4972: 4970: 4969: 4964: 4945: 4940: 4932: 4918: 4917: 4906: 4902: 4901: 4893: 4878: 4873: 4864: 4862: 4845: 4825: 4823: 4815: 4810: 4809: 4801: 4800: 4795: 4794: 4792: 4790: 4770:Jacques Hadamard 4752:By contrast the 4748: 4746: 4745: 4740: 4719: 4709: 4697: 4695: 4694: 4689: 4687: 4664: 4663: 4652: 4648: 4631: 4626: 4617: 4615: 4598: 4590: 4571: 4569: 4564: 4559: 4552: 4551: 4546: 4545: 4543: 4509: 4507: 4499: 4494: 4487: 4486: 4481: 4480: 4478: 4443: 4442: 4431: 4427: 4410: 4405: 4396: 4394: 4377: 4369: 4350: 4348: 4343: 4338: 4331: 4330: 4325: 4324: 4322: 4288: 4286: 4278: 4273: 4266: 4265: 4260: 4259: 4257: 4234: 4182: 4178: 4174: 4168: 4166: 4165: 4160: 4158: 4145: 4141: 4137: 4136: 4110: 4087: 4086: 4069: 4065: 4048: 4043: 4034: 4032: 4009: 3991: 3987: 3983: 3982: 3968: 3964: 3963: 3937: 3935: 3934: 3929: 3927: 3925: 3905: 3878: 3845: 3844: 3829: 3828: 3817: 3813: 3796: 3791: 3769: 3767: 3766: 3761: 3752: 3748: 3740: 3739: 3724: 3723: 3712: 3708: 3691: 3686: 3660: 3659: 3642: 3638: 3621: 3616: 3607: 3605: 3576: 3562: 3558: 3554: 3553: 3539: 3535: 3534: 3514: 3492: 3485: 3482:factor from the 3481: 3468: 3466: 3465: 3460: 3458: 3447: 3443: 3435: 3434: 3423: 3419: 3403: 3402: 3391: 3387: 3370: 3365: 3338: 3333: 3324: 3322: 3293: 3285: 3255: 3254: 3243: 3239: 3223: 3222: 3211: 3207: 3190: 3185: 3175: 3170: 3161: 3159: 3130: 3122: 3102: 3098: 3094: 3093: 3079: 3078: 3050: 3045: 3036: 3034: 3017: 2999: 2995: 2991: 2990: 2976: 2972: 2971: 2938: 2937: 2934: 2932: 2931: 2926: 2924: 2897: 2896: 2879: 2875: 2858: 2853: 2844: 2842: 2819: 2811: 2798: 2794: 2790: 2789: 2763: 2750: 2746: 2742: 2741: 2727: 2723: 2722: 2693: 2689: 2685: 2684: 2670: 2666: 2665: 2641: 2632: 2630: 2629: 2624: 2599: 2598: 2587: 2583: 2566: 2561: 2552: 2550: 2533: 2519: 2515: 2511: 2510: 2485: 2481: 2479: 2478: 2473: 2448: 2447: 2436: 2432: 2415: 2410: 2401: 2399: 2376: 2362: 2358: 2354: 2353: 2323: 2321: 2320: 2315: 2313: 2299: 2295: 2274: 2269: 2254: 2249: 2234: 2201: 2196: 2171: 2167: 2163: 2162: 2135: 2133: 2132: 2127: 2101: 2096: 2056: 2052: 2045: 2020:Oliver Heaviside 1986:Historical notes 1970: 1968: 1967: 1962: 1950: 1948: 1947: 1942: 1937: 1923: 1922: 1899: 1897: 1895: 1894: 1889: 1869: 1867: 1866: 1861: 1856: 1842: 1841: 1818: 1816: 1815: 1810: 1796: 1794: 1792: 1791: 1786: 1767: 1765: 1764: 1759: 1747: 1745: 1743: 1742: 1737: 1724: 1722: 1720: 1719: 1714: 1695: 1693: 1692: 1687: 1676:differentiation 1675: 1673: 1671: 1670: 1665: 1652: 1650: 1648: 1647: 1642: 1640: 1617: 1615: 1614: 1609: 1597: 1595: 1594: 1589: 1575: 1573: 1572: 1567: 1565: 1564: 1534: 1532: 1531: 1526: 1524: 1523: 1521: 1513: 1502: 1497: 1486: 1484: 1483: 1478: 1476: 1447: 1442: 1437: 1402: 1385: 1380: 1375: 1324: 1323: 1303: 1301: 1300: 1295: 1280: 1278: 1276: 1275: 1270: 1254: 1252: 1251: 1246: 1218: 1216: 1215: 1210: 1184: 1179: 1146: 1144: 1143: 1138: 1120: 1118: 1117: 1112: 1094: 1092: 1081: 1055: 1053: 1052: 1047: 1005: 998: 991: 939: 904: 870: 869: 866: 833:Surface integral 776: 775: 772: 680: 679: 676: 636:Limit comparison 556: 555: 552: 438:Riemann integral 391: 390: 387: 347:L'Hôpital's rule 304:Taylor's theorem 225: 224: 221: 165: 163: 162: 157: 109: 100: 95: 65: 64: 47: 27: 26: 19: 14950: 14949: 14945: 14944: 14943: 14941: 14940: 14939: 14935:Generalizations 14920: 14919: 14918: 14913: 14854:Jacob Bernoulli 14827: 14792: 14783:Galerkin method 14706: 14644:Solution topics 14639: 14597: 14563: 14502: 14434: 14429: 14399: 14279: 14268: 14148: 14078: 14068: 14031: 14002: 13962: 13935: 13916: 13879: 13852: 13823: 13789: 13788: 13779: 13778: 13774: 13739: 13718: 13699: 13680: 13661: 13645: 13435: 13434: 13425: 13424: 13420: 13396: 13391: 13389:Further reading 13386: 13363: 13359: 13352: 13333:10.1.1.247.5449 13320: 13316: 13284:10.1.1.252.6732 13249: 13245: 13190: 13186: 13121: 13114: 13099: 13067: 13063: 13004: 13000: 12959: 12955: 12898: 12894: 12837: 12833: 12792: 12788: 12731: 12727: 12712: 12672: 12668: 12631: 12627: 12596: 12592: 12563: 12556: 12505: 12501: 12456: 12452: 12403: 12399: 12370: 12366: 12337: 12333: 12320: 12316: 12303: 12299: 12259: 12253: 12249: 12222: 12218: 12195:Erdélyi, Arthur 12192: 12188: 12175: 12174: 12170: 12139: 12135: 12096: 12092: 12077: 12053: 12049: 12010: 12006: 11994: 11993: 11984: 11983: 11979: 11957: 11953: 11914: 11910: 11861: 11852: 11813: 11786: 11737: 11733: 11702: 11698: 11659:Thermal Science 11651: 11644: 11601: 11592: 11582: 11580: 11565: 11561: 11519: 11515: 11503: 11502: 11493: 11492: 11488: 11454: 11450: 11415: 11411: 11382: 11378: 11371: 11351: 11347: 11331: 11325: 11321: 11306: 11280: 11273: 11266: 11236: 11232: 11179: 11175: 11143: 11139: 11124: 11120: 11114:, Paris: 71–162 11097: 11093: 11070: 11066: 11025: 11021: 11009: 11003: 10999: 10973: 10967: 10963: 10956: 10940: 10936: 10932: 10927: 10926: 10898: 10895: 10894: 10893: 10877: 10874: 10873: 10857: 10854: 10853: 10851: 10847: 10842: 10820: 10793: 10780: 10756: 10751: 10750: 10746: 10740: 10736: 10734: 10726: 10723: 10722: 10692: 10672: 10649: 10623: 10621: 10607: 10603: 10599: 10595: 10594: 10578: 10571: 10567: 10563: 10548: 10535: 10528: 10524: 10520: 10518: 10510: 10507: 10506: 10499: 10486: 10482: 10451: 10439: 10434: 10433: 10428: 10423: 10416: 10408: 10398: 10397: 10393: 10384: 10380: 10368: 10364: 10351: 10346: 10329: 10312: 10308: 10299: 10295: 10287: 10284: 10283: 10275: 10271: 10250: 10245: 10236: 10232: 10230: 10227: 10226: 10219: 10215: 10209: 10205: 10181: 10176: 10175: 10171: 10165: 10161: 10159: 10151: 10148: 10147: 10129: 10117: 10113: 10107: 10105:position vector 10085: 10051: 10031: 10008: 9991: 9977: 9973: 9969: 9965: 9964: 9958: 9954: 9941: 9926: 9916: 9914: 9906: 9903: 9902: 9891: 9863:as well as the 9824: 9820: 9816: 9800: 9796: 9795: 9792: 9783: 9778: 9768: 9763: 9756: 9744: 9740: 9730: 9726: 9722: 9716: 9712: 9709: 9703: 9698: 9682: 9678: 9674: 9664: 9660: 9659: 9657: 9648: 9643: 9633: 9621: 9617: 9615: 9612: 9611: 9607: 9598: 9594: 9590: 9575: 9569: 9566: 9560: 9557: 9551: 9547: 9543: 9512: 9507: 9496: 9495: 9491: 9467: 9462: 9451: 9450: 9446: 9424: 9423: 9419: 9402: 9399: 9398: 9387: 9373: 9367:PID controllers 9363: 9361:PID controllers 9348: 9328: 9313: 9309: 9305: 9280: 9276: 9249: 9245: 9241: 9231: 9227: 9226: 9224: 9222: 9219: 9218: 9214: 9195: 9182: 9129: 9127: 9123: 9119: 9106: 9104: 9100: 9098: 9067: 9062: 9060: 9057: 9056: 9041: 9037: 9022: 9021:whose solution 8958: 8954: 8950: 8944: 8940: 8938: 8936: 8933: 8932: 8924: 8918: 8893: 8885: 8883: 8872: 8868: 8856: 8852: 8851: 8847: 8832: 8828: 8788: 8787: 8778: 8774: 8773: 8769: 8761: 8758: 8757: 8746: 8741: 8723:, important in 8704: 8697:spectral theory 8681: 8674: 8614: 8610: 8598: 8583: 8579: 8578: 8572: 8567: 8547: 8526: 8522: 8520: 8518: 8515: 8514: 8499: 8494: 8492:Generalizations 8465: 8462: 8461: 8436: 8433: 8432: 8431: 8413: 8410: 8409: 8408: 8386: 8383: 8382: 8366: 8363: 8362: 8346: 8343: 8342: 8322: 8319: 8318: 8302: 8299: 8298: 8280: 8277: 8276: 8275: 8272: 8257:Chen derivative 8218: 8214: 8212: 8209: 8208: 8198:Weyl derivative 8163: 8137: 8136: 8134: 8131: 8130: 8090: 8089: 8083: 8072: 8071: 8043: 8032: 8031: 8027: 8017: 8013: 8012: 8010: 8006: 8000: 7999: 7997: 7994: 7993: 7990: 7945: 7942: 7941: 7940: 7915: 7911: 7904: 7899: 7885: 7881: 7874: 7870: 7862: 7860: 7857: 7856: 7816: 7813: 7812: 7811: 7810:. However, for 7791: 7788: 7787: 7768: 7764: 7762: 7759: 7758: 7757: 7735: 7732: 7731: 7730: 7706: 7702: 7690: 7686: 7672: 7669: 7668: 7634: 7630: 7628: 7625: 7624: 7623: 7602: 7599: 7598: 7597: 7581: 7578: 7577: 7560: 7556: 7554: 7551: 7550: 7514: 7511: 7510: 7509: 7469: 7462: 7458: 7445: 7443: 7436: 7432: 7426: 7422: 7404: 7399: 7385: 7380: 7367: 7350: 7348: 7323: 7318: 7313: 7305: 7288: 7287: 7286: 7276: 7274: 7273: 7271: 7269: 7266: 7265: 7225: 7218: 7214: 7201: 7199: 7192: 7188: 7182: 7178: 7161: 7155: 7150: 7133: 7116: 7114: 7089: 7084: 7079: 7071: 7054: 7053: 7052: 7042: 7040: 7039: 7037: 7035: 7032: 7031: 7014: 7010: 7008: 7005: 7004: 6979: 6976: 6975: 6958: 6954: 6952: 6949: 6948: 6942: 6890: 6887: 6886: 6885: 6840: 6829: 6827: 6820: 6816: 6815: 6811: 6791: 6785: 6780: 6763: 6758: 6733: 6728: 6723: 6715: 6701: 6700: 6699: 6689: 6687: 6686: 6684: 6682: 6679: 6678: 6661: 6657: 6655: 6652: 6651: 6626: 6623: 6622: 6619: 6603: 6587: 6586: 6550: 6541: 6526: 6522: 6521: 6515: 6510: 6484: 6470: 6468: 6467: 6463: 6457: 6452: 6439: 6438: 6402: 6398: 6397: 6393: 6375: 6370: 6359: 6331: 6326: 6325: 6318: 6313: 6303: 6301: 6300: 6298: 6294: 6292: 6289: 6288: 6276: 6209: 6205: 6181: 6177: 6159: 6154: 6128: 6123: 6102: 6098: 6096: 6093: 6092: 6049: 6034: 6030: 6029: 6007: 6003: 6002: 6000: 5994: 5989: 5963: 5958: 5933: 5928: 5923: 5915: 5905: 5903: 5902: 5901: 5900: 5898: 5895: 5894: 5883: 5871: 5865: 5847: 5846: 5818: 5813: 5808: 5800: 5790: 5788: 5787: 5786: 5785: 5776: 5772: 5768: 5762: 5758: 5756: 5747: 5746: 5709: 5704: 5699: 5691: 5681: 5679: 5678: 5677: 5676: 5667: 5663: 5659: 5653: 5649: 5647: 5640: 5618: 5613: 5608: 5600: 5590: 5588: 5587: 5586: 5585: 5582: 5581: 5553: 5548: 5543: 5535: 5525: 5523: 5522: 5521: 5520: 5511: 5507: 5503: 5497: 5493: 5491: 5482: 5481: 5444: 5439: 5434: 5426: 5416: 5414: 5413: 5412: 5411: 5402: 5398: 5394: 5388: 5384: 5382: 5375: 5353: 5348: 5343: 5335: 5325: 5323: 5322: 5321: 5320: 5316: 5314: 5311: 5310: 5299: 5295: 5291: 5279: 5275: 5271: 5268: 5239: 5235: 5232: 5225: 5179: 5164: 5160: 5159: 5153: 5148: 5113: 5108: 5074: 5063: 5061: 5036: 5031: 5026: 5018: 5004: 5003: 5002: 4992: 4990: 4989: 4987: 4985: 4982: 4981: 4978: 4933: 4931: 4907: 4892: 4885: 4881: 4880: 4874: 4869: 4849: 4844: 4816: 4811: 4805: 4804: 4796: 4786: 4784: 4782: 4781: 4780: 4779: 4777: 4774: 4773: 4762: 4728: 4725: 4724: 4711: 4701: 4685: 4684: 4653: 4638: 4634: 4633: 4627: 4622: 4602: 4597: 4588: 4587: 4565: 4560: 4555: 4547: 4537: 4535: 4534: 4533: 4532: 4525: 4500: 4495: 4490: 4482: 4472: 4470: 4469: 4468: 4467: 4464: 4463: 4432: 4417: 4413: 4412: 4406: 4401: 4381: 4376: 4367: 4366: 4344: 4339: 4334: 4326: 4316: 4314: 4313: 4312: 4311: 4304: 4279: 4274: 4269: 4261: 4251: 4249: 4248: 4247: 4246: 4242: 4240: 4237: 4236: 4224: 4201: 4189: 4180: 4176: 4172: 4156: 4155: 4126: 4122: 4121: 4117: 4108: 4107: 4070: 4055: 4051: 4050: 4044: 4039: 4013: 4008: 4001: 3978: 3974: 3973: 3969: 3959: 3955: 3951: 3947: 3945: 3942: 3941: 3906: 3879: 3877: 3834: 3830: 3818: 3803: 3799: 3798: 3792: 3787: 3781: 3778: 3777: 3729: 3725: 3713: 3698: 3694: 3693: 3687: 3682: 3677: 3673: 3643: 3628: 3624: 3623: 3617: 3612: 3580: 3575: 3549: 3545: 3544: 3540: 3530: 3526: 3522: 3520: 3517: 3516: 3494: 3490: 3483: 3472: 3456: 3455: 3424: 3409: 3405: 3404: 3392: 3377: 3373: 3372: 3366: 3361: 3356: 3352: 3334: 3329: 3297: 3292: 3283: 3282: 3244: 3229: 3225: 3224: 3212: 3197: 3193: 3192: 3186: 3181: 3171: 3166: 3134: 3129: 3120: 3119: 3089: 3085: 3084: 3080: 3068: 3064: 3046: 3041: 3021: 3016: 3009: 2986: 2982: 2981: 2977: 2967: 2963: 2959: 2955: 2953: 2950: 2949: 2943: 2922: 2921: 2880: 2865: 2861: 2860: 2854: 2849: 2823: 2818: 2809: 2808: 2779: 2775: 2774: 2770: 2761: 2760: 2737: 2733: 2732: 2728: 2718: 2714: 2710: 2703: 2680: 2676: 2675: 2671: 2661: 2657: 2653: 2649: 2647: 2644: 2643: 2639: 2588: 2573: 2569: 2568: 2562: 2557: 2537: 2532: 2506: 2502: 2501: 2497: 2495: 2492: 2491: 2483: 2437: 2422: 2418: 2417: 2411: 2406: 2380: 2375: 2349: 2345: 2344: 2340: 2338: 2335: 2334: 2311: 2310: 2270: 2265: 2260: 2256: 2250: 2245: 2232: 2231: 2197: 2192: 2181: 2158: 2154: 2153: 2149: 2145: 2143: 2140: 2139: 2097: 2092: 2062: 2059: 2058: 2054: 2047: 2036: 2033: 1988: 1956: 1953: 1952: 1933: 1918: 1914: 1909: 1906: 1905: 1883: 1880: 1879: 1878: 1877:with parameter 1852: 1837: 1833: 1828: 1825: 1824: 1804: 1801: 1800: 1774: 1771: 1770: 1769: 1753: 1750: 1749: 1731: 1728: 1727: 1726: 1702: 1699: 1698: 1697: 1681: 1678: 1677: 1659: 1656: 1655: 1654: 1636: 1628: 1625: 1624: 1623: 1603: 1600: 1599: 1583: 1580: 1579: 1560: 1556: 1554: 1551: 1550: 1547:differentiation 1512: 1510: 1506: 1496: 1494: 1491: 1490: 1474: 1473: 1443: 1412: 1410: 1400: 1399: 1381: 1347: 1345: 1334: 1319: 1315: 1311: 1309: 1306: 1305: 1289: 1286: 1285: 1264: 1261: 1260: 1259: 1240: 1237: 1236: 1234:linear operator 1180: 1175: 1151: 1148: 1147: 1132: 1129: 1128: 1085: 1080: 1060: 1057: 1056: 1041: 1038: 1037: 1032:differentiation 1018:is a branch of 1009: 980: 979: 965:Integration Bee 940: 937: 930: 929: 905: 902: 895: 894: 867: 864: 857: 856: 838:Volume integral 773: 768: 761: 760: 677: 672: 665: 664: 634: 553: 548: 541: 540: 532:Risch algorithm 502:Euler's formula 388: 383: 376: 375: 357:General Leibniz 240:generalizations 222: 217: 210: 196:Rolle's theorem 191: 166: 102: 96: 91: 85: 82: 81: 63: 52: 28: 24: 17: 12: 11: 5: 14948: 14938: 14937: 14932: 14915: 14914: 14912: 14911: 14906: 14901: 14896: 14891: 14886: 14881: 14876: 14871: 14869:Ernst Lindelöf 14866: 14861: 14856: 14851: 14849:Leonhard Euler 14846: 14841: 14835: 14833: 14832:Mathematicians 14829: 14828: 14826: 14825: 14820: 14815: 14810: 14804: 14802: 14798: 14797: 14794: 14793: 14791: 14790: 14785: 14780: 14775: 14770: 14765: 14760: 14755: 14750: 14745: 14740: 14735: 14730: 14725: 14720: 14714: 14712: 14708: 14707: 14705: 14704: 14699: 14694: 14688: 14683: 14678: 14673: 14668: 14663: 14658: 14656:Phase portrait 14653: 14647: 14645: 14641: 14640: 14638: 14637: 14632: 14627: 14622: 14616: 14614: 14607: 14603: 14602: 14599: 14598: 14596: 14595: 14590: 14589: 14588: 14578: 14571: 14569: 14565: 14564: 14562: 14561: 14559:On jet bundles 14556: 14551: 14546: 14541: 14536: 14531: 14526: 14524:Nonhomogeneous 14521: 14516: 14510: 14508: 14504: 14503: 14501: 14500: 14495: 14490: 14485: 14480: 14475: 14470: 14465: 14460: 14455: 14449: 14447: 14440: 14439:Classification 14436: 14435: 14428: 14427: 14420: 14413: 14405: 14398: 14397: 14374:hep-th/0003126 14367:(4): 163–197. 14352: 14316:(2): 261–285. 14305: 14299: 14294: 14289: 14284: 14282:on 2005-10-29. 14261: 14252: 14242: 14233: 14220: 14219: 14218: 14205: 14193: 14180: 14167: 14158: 14146: 14145: 14144: 14125: 14106: 14079: 14077: 14076:External links 14074: 14073: 14072: 14066: 14043: 14029: 14006: 14000: 13977: 13966: 13960: 13939: 13934:978-8184873337 13933: 13920: 13914: 13883: 13877: 13856: 13850: 13827: 13821: 13798: 13790:|journal= 13772: 13743: 13737: 13722: 13716: 13703: 13697: 13684: 13678: 13665: 13659: 13644: 13641: 13640: 13639: 13594: 13566:(1): 577–582. 13560:Scientometrics 13555: 13527:(2): 479–500. 13516: 13473: 13455:(4): 487–501. 13444: 13436:|journal= 13418: 13395: 13392: 13390: 13387: 13385: 13384: 13357: 13350: 13314: 13243: 13184: 13112: 13097: 13061: 13031:10.1.1.765.120 13014:(4): 695–703. 12998: 12953: 12892: 12831: 12825:10400.22/13143 12786: 12725: 12710: 12666: 12645:(2): 153–192. 12633:Mainardi, F.; 12625: 12590: 12554: 12524:10.1.1.58.2062 12499: 12450: 12397: 12364: 12331: 12314: 12304:Oldham, K. B. 12297: 12247: 12216: 12186: 12168: 12133: 12090: 12075: 12047: 12004: 11977: 11951: 11908: 11850: 11784: 11747:(12): 123501. 11731: 11696: 11665:(2): 763–769. 11642: 11590: 11559: 11537:(5): 529–539. 11513: 11504:|journal= 11486: 11448: 11409: 11376: 11369: 11345: 11319: 11304: 11271: 11264: 11230: 11193:(2): 552–578. 11173: 11137: 11118: 11091: 11064: 11019: 10997: 10961: 10954: 10933: 10931: 10928: 10925: 10924: 10902: 10881: 10861: 10844: 10843: 10841: 10838: 10837: 10836: 10831: 10826: 10819: 10816: 10815: 10814: 10809: 10804: 10799: 10792: 10789: 10759: 10754: 10749: 10743: 10739: 10733: 10730: 10708: 10705: 10702: 10699: 10695: 10691: 10688: 10685: 10682: 10679: 10675: 10671: 10668: 10665: 10662: 10659: 10656: 10652: 10648: 10645: 10639: 10635: 10632: 10629: 10626: 10619: 10615: 10610: 10606: 10602: 10598: 10593: 10585: 10581: 10577: 10574: 10570: 10566: 10561: 10558: 10555: 10551: 10547: 10542: 10538: 10534: 10531: 10527: 10523: 10517: 10514: 10498: 10495: 10468: 10464: 10461: 10458: 10454: 10450: 10447: 10442: 10437: 10431: 10426: 10419: 10415: 10411: 10405: 10402: 10396: 10392: 10387: 10383: 10379: 10371: 10367: 10363: 10360: 10357: 10354: 10350: 10345: 10342: 10339: 10336: 10332: 10328: 10325: 10319: 10316: 10311: 10307: 10302: 10298: 10294: 10291: 10256: 10253: 10249: 10244: 10239: 10235: 10207: 10184: 10179: 10174: 10168: 10164: 10158: 10155: 10068: 10064: 10061: 10058: 10054: 10050: 10047: 10044: 10041: 10038: 10034: 10030: 10027: 10024: 10021: 10018: 10015: 10011: 10007: 10004: 9998: 9995: 9989: 9985: 9980: 9976: 9972: 9968: 9961: 9957: 9953: 9947: 9944: 9939: 9936: 9933: 9929: 9925: 9922: 9919: 9913: 9910: 9890: 9887: 9849: 9845: 9842: 9833: 9830: 9827: 9823: 9819: 9814: 9809: 9806: 9803: 9799: 9786: 9781: 9777: 9771: 9766: 9762: 9755: 9752: 9747: 9743: 9733: 9729: 9725: 9719: 9715: 9706: 9701: 9697: 9693: 9685: 9681: 9677: 9672: 9667: 9663: 9651: 9646: 9642: 9638: 9632: 9629: 9624: 9620: 9606: 9603: 9573: 9564: 9555: 9529: 9526: 9523: 9520: 9515: 9510: 9506: 9499: 9494: 9490: 9487: 9484: 9481: 9478: 9473: 9470: 9465: 9461: 9454: 9449: 9445: 9442: 9439: 9436: 9433: 9427: 9422: 9418: 9415: 9412: 9409: 9406: 9362: 9359: 9347: 9344: 9291: 9288: 9283: 9279: 9275: 9272: 9269: 9266: 9263: 9260: 9252: 9248: 9244: 9239: 9234: 9230: 9213: 9210: 9194: 9191: 9181: 9178: 9162: 9159: 9156: 9153: 9150: 9147: 9136: 9133: 9126: 9122: 9113: 9110: 9103: 9097: 9094: 9091: 9088: 9085: 9082: 9079: 9073: 9070: 9066: 9008: 9005: 9002: 8999: 8996: 8993: 8990: 8987: 8984: 8981: 8978: 8975: 8972: 8969: 8961: 8957: 8953: 8947: 8943: 8917: 8914: 8899: 8896: 8891: 8888: 8881: 8875: 8871: 8867: 8864: 8859: 8855: 8850: 8846: 8841: 8838: 8835: 8831: 8827: 8824: 8821: 8818: 8815: 8812: 8809: 8806: 8802: 8795: 8792: 8786: 8781: 8777: 8772: 8768: 8765: 8750:control volume 8745: 8742: 8740: 8737: 8673: 8670: 8651: 8647: 8644: 8640: 8637: 8634: 8631: 8626: 8623: 8620: 8617: 8613: 8607: 8604: 8601: 8596: 8592: 8589: 8586: 8582: 8575: 8570: 8566: 8559: 8556: 8553: 8550: 8544: 8541: 8538: 8535: 8532: 8529: 8525: 8507:Arthur Erdélyi 8498: 8495: 8493: 8490: 8469: 8440: 8417: 8396: 8393: 8390: 8370: 8350: 8339:local property 8326: 8306: 8284: 8271: 8268: 8267: 8266: 8263: 8258: 8255: 8252: 8249: 8244: 8237: 8236: 8221: 8217: 8205: 8200: 8195: 8192: 8189: 8186: 8183: 8178: 8175: 8172: 8162: 8159: 8140: 8116: 8113: 8110: 8107: 8104: 8101: 8098: 8093: 8086: 8081: 8078: 8075: 8070: 8067: 8064: 8061: 8058: 8054: 8046: 8041: 8038: 8035: 8030: 8025: 8020: 8016: 8009: 8003: 7989: 7986: 7967: 7964: 7961: 7958: 7955: 7952: 7949: 7926: 7918: 7914: 7910: 7907: 7903: 7898: 7895: 7888: 7884: 7880: 7877: 7873: 7869: 7865: 7838: 7835: 7832: 7829: 7826: 7823: 7820: 7795: 7771: 7767: 7739: 7714: 7709: 7705: 7701: 7698: 7693: 7689: 7685: 7682: 7679: 7676: 7654: 7651: 7648: 7645: 7642: 7637: 7633: 7609: 7606: 7585: 7563: 7559: 7536: 7533: 7530: 7527: 7524: 7521: 7518: 7495: 7492: 7489: 7485: 7478: 7475: 7472: 7465: 7461: 7457: 7454: 7451: 7448: 7442: 7439: 7435: 7429: 7425: 7421: 7418: 7415: 7412: 7407: 7402: 7398: 7391: 7388: 7384: 7376: 7373: 7370: 7365: 7362: 7359: 7356: 7353: 7347: 7344: 7341: 7338: 7335: 7332: 7326: 7321: 7316: 7303: 7296: 7293: 7280: 7251: 7248: 7245: 7241: 7234: 7231: 7228: 7221: 7217: 7213: 7210: 7207: 7204: 7198: 7195: 7191: 7185: 7181: 7177: 7174: 7171: 7167: 7164: 7158: 7153: 7149: 7142: 7139: 7136: 7131: 7128: 7125: 7122: 7119: 7113: 7110: 7107: 7104: 7101: 7098: 7092: 7087: 7082: 7069: 7062: 7059: 7046: 7017: 7013: 6992: 6989: 6986: 6983: 6961: 6957: 6941: 6938: 6924: 6921: 6918: 6915: 6912: 6909: 6906: 6903: 6900: 6897: 6894: 6871: 6868: 6865: 6856: 6849: 6846: 6843: 6838: 6835: 6832: 6826: 6823: 6819: 6814: 6807: 6804: 6801: 6797: 6794: 6788: 6783: 6779: 6772: 6769: 6766: 6762: 6757: 6754: 6751: 6748: 6745: 6742: 6736: 6731: 6726: 6713: 6706: 6693: 6664: 6660: 6639: 6636: 6633: 6630: 6618: 6615: 6585: 6582: 6577: 6573: 6570: 6566: 6563: 6560: 6556: 6553: 6547: 6544: 6539: 6535: 6532: 6529: 6525: 6518: 6513: 6509: 6502: 6499: 6496: 6493: 6490: 6487: 6482: 6479: 6476: 6473: 6466: 6460: 6455: 6451: 6447: 6444: 6442: 6440: 6437: 6434: 6429: 6425: 6422: 6419: 6416: 6411: 6408: 6405: 6401: 6396: 6392: 6389: 6386: 6383: 6378: 6373: 6369: 6365: 6362: 6360: 6358: 6355: 6352: 6349: 6346: 6342: 6339: 6334: 6329: 6321: 6316: 6307: 6297: 6296: 6264: 6261: 6258: 6255: 6252: 6249: 6246: 6243: 6240: 6236: 6233: 6229: 6226: 6223: 6218: 6215: 6212: 6208: 6202: 6199: 6196: 6193: 6190: 6187: 6184: 6180: 6176: 6173: 6170: 6167: 6162: 6157: 6153: 6146: 6143: 6140: 6137: 6134: 6131: 6127: 6122: 6119: 6116: 6113: 6110: 6105: 6101: 6078: 6075: 6072: 6064: 6061: 6058: 6055: 6052: 6047: 6043: 6040: 6037: 6033: 6027: 6024: 6021: 6016: 6013: 6010: 6006: 5997: 5992: 5988: 5981: 5978: 5975: 5972: 5969: 5966: 5962: 5957: 5954: 5951: 5948: 5945: 5942: 5936: 5931: 5926: 5918: 5909: 5879:Michele Caputo 5867:Main article: 5864: 5861: 5845: 5842: 5839: 5836: 5833: 5827: 5824: 5821: 5816: 5811: 5803: 5794: 5779: 5775: 5771: 5765: 5761: 5755: 5752: 5750: 5748: 5745: 5742: 5739: 5736: 5733: 5727: 5724: 5721: 5718: 5715: 5712: 5707: 5702: 5694: 5685: 5670: 5666: 5662: 5656: 5652: 5646: 5643: 5641: 5639: 5636: 5633: 5630: 5627: 5621: 5616: 5611: 5603: 5594: 5584: 5583: 5580: 5577: 5574: 5571: 5568: 5562: 5559: 5556: 5551: 5546: 5538: 5529: 5514: 5510: 5506: 5500: 5496: 5490: 5487: 5485: 5483: 5480: 5477: 5474: 5471: 5468: 5462: 5459: 5456: 5453: 5450: 5447: 5442: 5437: 5429: 5420: 5405: 5401: 5397: 5391: 5387: 5381: 5378: 5376: 5374: 5371: 5368: 5365: 5362: 5356: 5351: 5346: 5338: 5329: 5319: 5318: 5267: 5264: 5224: 5221: 5209: 5206: 5202: 5199: 5196: 5193: 5188: 5185: 5182: 5177: 5173: 5170: 5167: 5163: 5156: 5151: 5147: 5140: 5137: 5134: 5131: 5128: 5125: 5122: 5119: 5116: 5112: 5107: 5104: 5101: 5098: 5095: 5089: 5086: 5083: 5080: 5077: 5072: 5069: 5066: 5060: 5057: 5054: 5051: 5048: 5045: 5039: 5034: 5029: 5021: 5016: 5009: 4996: 4977: 4974: 4962: 4958: 4955: 4952: 4948: 4943: 4939: 4936: 4930: 4927: 4924: 4921: 4916: 4913: 4910: 4905: 4899: 4896: 4891: 4888: 4884: 4877: 4872: 4868: 4861: 4858: 4855: 4852: 4848: 4843: 4840: 4837: 4834: 4831: 4828: 4822: 4819: 4814: 4808: 4799: 4789: 4761: 4758: 4738: 4735: 4732: 4683: 4680: 4676: 4673: 4670: 4667: 4662: 4659: 4656: 4651: 4647: 4644: 4641: 4637: 4630: 4625: 4621: 4614: 4611: 4608: 4605: 4601: 4596: 4593: 4591: 4589: 4586: 4583: 4580: 4577: 4574: 4568: 4563: 4558: 4550: 4541: 4531: 4528: 4526: 4524: 4521: 4518: 4515: 4512: 4506: 4503: 4498: 4493: 4485: 4476: 4466: 4465: 4462: 4459: 4455: 4452: 4449: 4446: 4441: 4438: 4435: 4430: 4426: 4423: 4420: 4416: 4409: 4404: 4400: 4393: 4390: 4387: 4384: 4380: 4375: 4372: 4370: 4368: 4365: 4362: 4359: 4356: 4353: 4347: 4342: 4337: 4329: 4320: 4310: 4307: 4305: 4303: 4300: 4297: 4294: 4291: 4285: 4282: 4277: 4272: 4264: 4255: 4245: 4244: 4217:Fourier series 4200: 4197: 4193:differintegral 4186: 4185: 4171:Interchanging 4154: 4151: 4148: 4144: 4140: 4135: 4132: 4129: 4125: 4120: 4116: 4113: 4111: 4109: 4106: 4103: 4099: 4096: 4093: 4090: 4085: 4082: 4079: 4076: 4073: 4068: 4064: 4061: 4058: 4054: 4047: 4042: 4038: 4031: 4028: 4025: 4022: 4019: 4016: 4012: 4007: 4004: 4002: 4000: 3997: 3994: 3990: 3986: 3981: 3977: 3972: 3967: 3962: 3958: 3954: 3950: 3949: 3924: 3921: 3918: 3915: 3912: 3909: 3904: 3901: 3898: 3895: 3891: 3888: 3885: 3882: 3876: 3873: 3870: 3867: 3864: 3861: 3858: 3855: 3852: 3849: 3843: 3840: 3837: 3833: 3827: 3824: 3821: 3816: 3812: 3809: 3806: 3802: 3795: 3790: 3786: 3759: 3756: 3751: 3747: 3744: 3738: 3735: 3732: 3728: 3722: 3719: 3716: 3711: 3707: 3704: 3701: 3697: 3690: 3685: 3681: 3676: 3672: 3669: 3666: 3663: 3658: 3655: 3652: 3649: 3646: 3641: 3637: 3634: 3631: 3627: 3620: 3615: 3611: 3604: 3601: 3598: 3595: 3592: 3589: 3586: 3583: 3579: 3574: 3571: 3568: 3565: 3561: 3557: 3552: 3548: 3543: 3538: 3533: 3529: 3525: 3454: 3451: 3446: 3442: 3439: 3433: 3430: 3427: 3422: 3418: 3415: 3412: 3408: 3401: 3398: 3395: 3390: 3386: 3383: 3380: 3376: 3369: 3364: 3360: 3355: 3351: 3348: 3345: 3342: 3337: 3332: 3328: 3321: 3318: 3315: 3312: 3309: 3306: 3303: 3300: 3296: 3291: 3288: 3286: 3284: 3281: 3278: 3274: 3271: 3267: 3264: 3261: 3258: 3253: 3250: 3247: 3242: 3238: 3235: 3232: 3228: 3221: 3218: 3215: 3210: 3206: 3203: 3200: 3196: 3189: 3184: 3180: 3174: 3169: 3165: 3158: 3155: 3152: 3149: 3146: 3143: 3140: 3137: 3133: 3128: 3125: 3123: 3121: 3118: 3115: 3111: 3108: 3105: 3101: 3097: 3092: 3088: 3083: 3077: 3074: 3071: 3067: 3063: 3060: 3057: 3054: 3049: 3044: 3040: 3033: 3030: 3027: 3024: 3020: 3015: 3012: 3010: 3008: 3005: 3002: 2998: 2994: 2989: 2985: 2980: 2975: 2970: 2966: 2962: 2958: 2957: 2945: 2944: 2941: 2936: 2920: 2916: 2913: 2909: 2906: 2903: 2900: 2895: 2892: 2889: 2886: 2883: 2878: 2874: 2871: 2868: 2864: 2857: 2852: 2848: 2841: 2838: 2835: 2832: 2829: 2826: 2822: 2817: 2814: 2812: 2810: 2807: 2804: 2801: 2797: 2793: 2788: 2785: 2782: 2778: 2773: 2769: 2766: 2764: 2762: 2759: 2756: 2753: 2749: 2745: 2740: 2736: 2731: 2726: 2721: 2717: 2713: 2709: 2706: 2704: 2702: 2699: 2696: 2692: 2688: 2683: 2679: 2674: 2669: 2664: 2660: 2656: 2652: 2651: 2622: 2618: 2615: 2611: 2608: 2605: 2602: 2597: 2594: 2591: 2586: 2582: 2579: 2576: 2572: 2565: 2560: 2556: 2549: 2546: 2543: 2540: 2536: 2531: 2528: 2525: 2522: 2518: 2514: 2509: 2505: 2500: 2488:gamma function 2471: 2467: 2464: 2460: 2457: 2454: 2451: 2446: 2443: 2440: 2435: 2431: 2428: 2425: 2421: 2414: 2409: 2405: 2398: 2395: 2392: 2389: 2386: 2383: 2379: 2374: 2371: 2368: 2365: 2361: 2357: 2352: 2348: 2343: 2309: 2305: 2302: 2298: 2294: 2291: 2287: 2284: 2281: 2278: 2273: 2268: 2264: 2259: 2253: 2248: 2244: 2240: 2237: 2235: 2233: 2230: 2227: 2223: 2220: 2217: 2214: 2211: 2208: 2205: 2200: 2195: 2191: 2187: 2184: 2182: 2180: 2177: 2174: 2170: 2166: 2161: 2157: 2152: 2148: 2147: 2125: 2121: 2118: 2114: 2111: 2108: 2105: 2100: 2095: 2091: 2087: 2084: 2081: 2078: 2075: 2072: 2069: 2066: 2032: 2029: 1987: 1984: 1960: 1940: 1936: 1932: 1929: 1926: 1921: 1917: 1913: 1887: 1859: 1855: 1851: 1848: 1845: 1840: 1836: 1832: 1808: 1784: 1781: 1778: 1757: 1735: 1712: 1709: 1706: 1685: 1663: 1639: 1635: 1632: 1607: 1587: 1563: 1559: 1544: 1519: 1516: 1509: 1505: 1500: 1472: 1469: 1466: 1463: 1460: 1457: 1454: 1451: 1446: 1440: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1415: 1408: 1405: 1403: 1401: 1398: 1395: 1392: 1389: 1384: 1378: 1374: 1371: 1368: 1365: 1362: 1359: 1356: 1353: 1350: 1343: 1340: 1337: 1335: 1333: 1330: 1327: 1322: 1318: 1314: 1313: 1293: 1268: 1244: 1208: 1204: 1201: 1197: 1194: 1191: 1188: 1183: 1178: 1174: 1170: 1167: 1164: 1161: 1158: 1155: 1136: 1110: 1106: 1103: 1100: 1097: 1091: 1088: 1084: 1079: 1076: 1073: 1070: 1067: 1064: 1045: 1030:powers of the 1028:complex number 1011: 1010: 1008: 1007: 1000: 993: 985: 982: 981: 978: 977: 972: 967: 962: 960:List of topics 957: 952: 947: 941: 936: 935: 932: 931: 928: 927: 922: 917: 912: 906: 901: 900: 897: 896: 891: 890: 889: 888: 883: 878: 868: 863: 862: 859: 858: 853: 852: 851: 850: 845: 840: 835: 830: 825: 820: 812: 811: 807: 806: 805: 804: 799: 794: 789: 781: 780: 774: 767: 766: 763: 762: 757: 756: 755: 754: 749: 744: 739: 734: 729: 721: 720: 716: 715: 714: 713: 708: 703: 698: 693: 688: 678: 671: 670: 667: 666: 661: 660: 659: 658: 653: 648: 643: 638: 632: 627: 622: 617: 612: 604: 603: 597: 596: 595: 594: 589: 584: 579: 574: 569: 554: 547: 546: 543: 542: 537: 536: 535: 534: 529: 524: 519: 517:Changing order 514: 504: 499: 481: 476: 471: 463: 462: 461:Integration by 458: 457: 456: 455: 450: 445: 440: 435: 425: 423:Antiderivative 417: 416: 412: 411: 410: 409: 404: 399: 389: 382: 381: 378: 377: 372: 371: 370: 369: 364: 359: 354: 349: 344: 339: 334: 329: 324: 316: 315: 309: 308: 307: 306: 301: 296: 291: 286: 281: 273: 272: 268: 267: 266: 265: 264: 263: 258: 253: 243: 230: 229: 223: 216: 215: 212: 211: 209: 208: 203: 198: 192: 190: 189: 184: 178: 177: 176: 168: 167: 155: 152: 149: 146: 143: 140: 137: 134: 131: 128: 125: 122: 118: 115: 112: 108: 105: 99: 94: 90: 80: 77: 76: 70: 69: 60:differintegral 54: 53: 31: 29: 22: 15: 9: 6: 4: 3: 2: 14947: 14936: 14933: 14931: 14928: 14927: 14925: 14910: 14907: 14905: 14902: 14900: 14897: 14895: 14892: 14890: 14887: 14885: 14882: 14880: 14877: 14875: 14872: 14870: 14867: 14865: 14862: 14860: 14857: 14855: 14852: 14850: 14847: 14845: 14842: 14840: 14837: 14836: 14834: 14830: 14824: 14821: 14819: 14816: 14814: 14811: 14809: 14806: 14805: 14803: 14799: 14789: 14786: 14784: 14781: 14779: 14776: 14774: 14771: 14769: 14766: 14764: 14761: 14759: 14756: 14754: 14751: 14749: 14746: 14744: 14741: 14739: 14736: 14734: 14731: 14729: 14726: 14724: 14721: 14719: 14716: 14715: 14713: 14709: 14703: 14700: 14698: 14695: 14692: 14689: 14687: 14684: 14682: 14679: 14677: 14674: 14672: 14669: 14667: 14664: 14662: 14659: 14657: 14654: 14652: 14649: 14648: 14646: 14642: 14636: 14633: 14631: 14628: 14626: 14623: 14621: 14618: 14617: 14615: 14611: 14608: 14604: 14594: 14591: 14587: 14584: 14583: 14582: 14579: 14576: 14573: 14572: 14570: 14566: 14560: 14557: 14555: 14552: 14550: 14547: 14545: 14542: 14540: 14537: 14535: 14532: 14530: 14527: 14525: 14522: 14520: 14517: 14515: 14512: 14511: 14509: 14505: 14499: 14496: 14494: 14491: 14489: 14486: 14484: 14481: 14479: 14476: 14474: 14471: 14469: 14466: 14464: 14461: 14459: 14456: 14454: 14451: 14450: 14448: 14444: 14441: 14437: 14433: 14426: 14421: 14419: 14414: 14412: 14407: 14406: 14403: 14394: 14390: 14385: 14380: 14375: 14370: 14366: 14362: 14358: 14353: 14349: 14345: 14341: 14337: 14333: 14329: 14324: 14319: 14315: 14311: 14306: 14303: 14300: 14298: 14295: 14293: 14290: 14288: 14285: 14278: 14274: 14267: 14262: 14258: 14253: 14248: 14243: 14239: 14234: 14230: 14226: 14221: 14216: 14212: 14209: 14206: 14204: 14200: 14197: 14194: 14191: 14187: 14184: 14181: 14178: 14174: 14171: 14168: 14165: 14162: 14161: 14159: 14155: 14154:MathPages.com 14151: 14147: 14140: 14139: 14134: 14131: 14126: 14121: 14120: 14115: 14112: 14107: 14102: 14101: 14096: 14093: 14088: 14087: 14085: 14081: 14080: 14069: 14063: 14059: 14055: 14051: 14050: 14044: 14040: 14036: 14032: 14026: 14022: 14021:10.1142/11107 14018: 14014: 14013: 14007: 14003: 13997: 13993: 13989: 13985: 13984: 13978: 13974: 13973: 13967: 13963: 13957: 13953: 13949: 13945: 13940: 13936: 13930: 13926: 13921: 13917: 13911: 13907: 13903: 13899: 13895: 13891: 13890: 13884: 13880: 13874: 13870: 13866: 13862: 13857: 13853: 13847: 13843: 13839: 13835: 13834: 13828: 13824: 13818: 13814: 13810: 13806: 13805: 13799: 13795: 13783: 13775: 13769: 13765: 13761: 13757: 13753: 13749: 13744: 13740: 13734: 13730: 13729: 13723: 13719: 13713: 13709: 13704: 13700: 13694: 13690: 13685: 13681: 13675: 13671: 13666: 13662: 13656: 13652: 13647: 13646: 13636: 13632: 13627: 13622: 13617: 13612: 13609:(1): 846107. 13608: 13604: 13600: 13595: 13591: 13587: 13582: 13581:10400.22/3769 13577: 13573: 13569: 13565: 13561: 13556: 13552: 13548: 13543: 13542:10400.22/3773 13538: 13534: 13530: 13526: 13522: 13517: 13512: 13511:10400.22/4149 13507: 13503: 13499: 13495: 13491: 13487: 13483: 13479: 13478:Kiryakova, V. 13474: 13470: 13466: 13462: 13458: 13454: 13450: 13445: 13441: 13429: 13421: 13415: 13411: 13407: 13403: 13398: 13397: 13380: 13376: 13372: 13368: 13361: 13353: 13347: 13343: 13342:10.1142/10541 13339: 13334: 13329: 13325: 13318: 13310: 13306: 13302: 13298: 13294: 13290: 13285: 13280: 13276: 13272: 13267: 13262: 13259:(5): 056108. 13258: 13254: 13247: 13239: 13235: 13230: 13225: 13220: 13215: 13211: 13207: 13204:(3): 032606. 13203: 13199: 13195: 13188: 13180: 13176: 13172: 13168: 13164: 13160: 13156: 13152: 13148: 13144: 13139: 13134: 13130: 13126: 13119: 13117: 13108: 13104: 13100: 13094: 13090: 13086: 13082: 13078: 13074: 13073: 13065: 13057: 13053: 13049: 13045: 13041: 13037: 13032: 13027: 13022: 13017: 13013: 13009: 13002: 12994: 12990: 12986: 12982: 12977: 12972: 12968: 12964: 12957: 12949: 12945: 12941: 12937: 12932: 12927: 12923: 12919: 12915: 12911: 12907: 12903: 12896: 12888: 12884: 12880: 12876: 12871: 12866: 12862: 12858: 12854: 12850: 12846: 12842: 12835: 12826: 12821: 12816: 12811: 12807: 12803: 12802: 12797: 12790: 12782: 12778: 12773: 12768: 12764: 12760: 12755: 12750: 12746: 12742: 12741: 12736: 12729: 12721: 12717: 12713: 12707: 12703: 12699: 12695: 12691: 12686: 12681: 12677: 12670: 12662: 12658: 12653: 12648: 12644: 12640: 12636: 12629: 12621: 12617: 12613: 12609: 12605: 12601: 12594: 12585: 12580: 12576: 12572: 12568: 12561: 12559: 12550: 12546: 12542: 12538: 12534: 12530: 12525: 12520: 12516: 12512: 12511: 12503: 12495: 12491: 12486: 12481: 12477: 12473: 12469: 12465: 12461: 12454: 12446: 12442: 12438: 12434: 12429: 12428:10.1.1.1.4838 12424: 12420: 12416: 12412: 12408: 12401: 12392: 12387: 12383: 12379: 12375: 12368: 12359: 12354: 12350: 12346: 12342: 12335: 12328: 12324: 12318: 12311: 12307: 12301: 12293: 12289: 12285: 12281: 12277: 12273: 12269: 12265: 12258: 12251: 12243: 12239: 12235: 12231: 12227: 12220: 12212: 12208: 12204: 12200: 12196: 12190: 12182: 12181:MathPages.com 12178: 12172: 12164: 12160: 12156: 12152: 12148: 12144: 12137: 12129: 12125: 12121: 12117: 12113: 12109: 12106:(6): 063502. 12105: 12101: 12094: 12086: 12082: 12078: 12072: 12068: 12067:10.1142/12988 12064: 12060: 12059: 12051: 12042: 12037: 12032: 12027: 12023: 12019: 12015: 12008: 12000: 11988: 11980: 11974: 11970: 11966: 11962: 11955: 11947: 11943: 11939: 11935: 11931: 11927: 11923: 11919: 11912: 11904: 11900: 11895: 11890: 11886: 11882: 11878: 11874: 11870: 11866: 11859: 11857: 11855: 11845: 11844:10400.22/5497 11840: 11835: 11830: 11826: 11822: 11818: 11811: 11809: 11807: 11805: 11803: 11801: 11799: 11797: 11795: 11793: 11791: 11789: 11780: 11776: 11772: 11768: 11764: 11760: 11755: 11750: 11746: 11742: 11735: 11726: 11721: 11717: 11713: 11712: 11707: 11700: 11692: 11688: 11683: 11678: 11673: 11668: 11664: 11660: 11656: 11649: 11647: 11638: 11634: 11630: 11626: 11622: 11618: 11614: 11610: 11606: 11599: 11597: 11595: 11578: 11574: 11570: 11563: 11553: 11548: 11544: 11540: 11536: 11532: 11528: 11526: 11517: 11509: 11497: 11489: 11483: 11479: 11475: 11471: 11467: 11463: 11459: 11452: 11444: 11440: 11436: 11432: 11428: 11424: 11420: 11413: 11404: 11399: 11395: 11391: 11387: 11380: 11372: 11366: 11362: 11358: 11357: 11349: 11342:(8): 101–186. 11341: 11337: 11330: 11323: 11315: 11311: 11307: 11301: 11297: 11293: 11289: 11285: 11278: 11276: 11267: 11261: 11257: 11253: 11249: 11245: 11241: 11234: 11226: 11222: 11218: 11214: 11209: 11208:10400.22/5491 11204: 11200: 11196: 11192: 11188: 11184: 11177: 11169: 11165: 11161: 11157: 11153: 11149: 11141: 11134: 11130: 11129: 11122: 11113: 11109: 11105: 11101: 11095: 11087:, Paris: 1–69 11086: 11082: 11078: 11074: 11068: 11060: 11056: 11052: 11048: 11043: 11038: 11034: 11030: 11023: 11015: 11008: 11001: 10992: 10987: 10983: 10979: 10972: 10965: 10957: 10951: 10947: 10946: 10938: 10934: 10921: 10917: 10900: 10879: 10859: 10849: 10845: 10835: 10832: 10830: 10827: 10825: 10822: 10821: 10813: 10810: 10808: 10805: 10803: 10800: 10798: 10795: 10794: 10788: 10784: 10778: 10757: 10741: 10731: 10719: 10706: 10700: 10697: 10686: 10680: 10677: 10666: 10663: 10657: 10654: 10643: 10637: 10630: 10624: 10617: 10608: 10604: 10600: 10596: 10591: 10572: 10568: 10556: 10553: 10529: 10525: 10512: 10504: 10494: 10490: 10479: 10466: 10459: 10456: 10445: 10440: 10413: 10400: 10394: 10390: 10385: 10381: 10377: 10369: 10358: 10355: 10348: 10343: 10337: 10334: 10323: 10317: 10314: 10300: 10296: 10292: 10279: 10254: 10251: 10247: 10242: 10237: 10233: 10222: 10214: 10203: 10182: 10166: 10156: 10144: 10140: 10136: 10132: 10127: 10123: 10110: 10106: 10102: 10096: 10092: 10088: 10084: 10079: 10066: 10059: 10056: 10045: 10039: 10036: 10025: 10022: 10016: 10013: 10002: 9996: 9993: 9987: 9978: 9974: 9970: 9966: 9959: 9955: 9951: 9945: 9934: 9931: 9920: 9908: 9900: 9896: 9886: 9884: 9880: 9876: 9875:Lomnitz's law 9870: 9868: 9867: 9860: 9847: 9843: 9840: 9831: 9828: 9825: 9821: 9812: 9807: 9804: 9801: 9784: 9779: 9775: 9769: 9764: 9760: 9753: 9750: 9745: 9731: 9727: 9717: 9704: 9699: 9695: 9691: 9683: 9679: 9670: 9665: 9649: 9644: 9640: 9636: 9630: 9627: 9622: 9602: 9588: 9584: 9580: 9572: 9563: 9554: 9540: 9524: 9518: 9513: 9508: 9504: 9492: 9488: 9482: 9476: 9471: 9468: 9463: 9459: 9447: 9443: 9437: 9431: 9420: 9416: 9410: 9404: 9394: 9390: 9386: 9380: 9376: 9372: 9368: 9365:Generalizing 9358: 9356: 9353: 9343: 9339: 9335: 9331: 9324: 9320: 9316: 9302: 9289: 9286: 9281: 9270: 9264: 9261: 9258: 9250: 9246: 9237: 9232: 9209: 9207: 9203: 9198: 9190: 9187: 9177: 9173: 9157: 9154: 9151: 9145: 9134: 9131: 9124: 9120: 9111: 9108: 9101: 9095: 9089: 9086: 9083: 9077: 9071: 9068: 9064: 9052: 9048: 9044: 9033: 9029: 9025: 9019: 9003: 9000: 8997: 8991: 8988: 8985: 8979: 8976: 8973: 8967: 8959: 8955: 8951: 8945: 8941: 8930: 8923: 8922:Neopolarogram 8913: 8897: 8889: 8879: 8873: 8869: 8865: 8862: 8857: 8853: 8848: 8844: 8839: 8836: 8833: 8829: 8819: 8816: 8813: 8804: 8800: 8790: 8784: 8779: 8770: 8766: 8763: 8755: 8754:heterogeneity 8751: 8736: 8734: 8730: 8726: 8722: 8718: 8714: 8710: 8702: 8698: 8694: 8688: 8684: 8679: 8669: 8667: 8662: 8649: 8645: 8642: 8635: 8629: 8624: 8621: 8618: 8615: 8611: 8605: 8602: 8599: 8594: 8590: 8587: 8584: 8580: 8573: 8568: 8564: 8554: 8542: 8539: 8536: 8533: 8530: 8527: 8523: 8512: 8511:Hermann Kober 8508: 8504: 8489: 8487: 8483: 8467: 8458: 8456: 8438: 8415: 8394: 8391: 8388: 8368: 8348: 8340: 8324: 8304: 8282: 8264: 8262: 8259: 8256: 8253: 8250: 8248: 8245: 8242: 8241: 8240: 8235: 8219: 8215: 8206: 8204: 8201: 8199: 8196: 8193: 8190: 8187: 8184: 8182: 8179: 8176: 8173: 8171: 8168: 8167: 8166: 8158: 8156: 8127: 8114: 8108: 8099: 8084: 8079: 8076: 8073: 8068: 8065: 8059: 8052: 8044: 8039: 8036: 8033: 8023: 8018: 8007: 7985: 7983: 7962: 7959: 7956: 7950: 7947: 7937: 7924: 7916: 7912: 7908: 7905: 7901: 7896: 7886: 7882: 7878: 7875: 7871: 7854: 7833: 7830: 7827: 7821: 7818: 7809: 7793: 7769: 7765: 7755: 7737: 7728: 7707: 7703: 7699: 7691: 7687: 7683: 7680: 7674: 7652: 7649: 7643: 7635: 7631: 7604: 7583: 7561: 7557: 7549:the function 7531: 7528: 7525: 7519: 7516: 7506: 7493: 7490: 7487: 7483: 7476: 7473: 7470: 7463: 7455: 7452: 7449: 7440: 7437: 7433: 7427: 7423: 7416: 7410: 7405: 7400: 7396: 7389: 7386: 7382: 7374: 7371: 7368: 7360: 7354: 7351: 7345: 7339: 7333: 7330: 7324: 7319: 7314: 7301: 7294: 7291: 7278: 7262: 7249: 7246: 7243: 7239: 7232: 7229: 7226: 7219: 7211: 7208: 7205: 7196: 7193: 7189: 7183: 7179: 7172: 7165: 7162: 7156: 7151: 7147: 7140: 7137: 7134: 7126: 7120: 7117: 7111: 7105: 7099: 7096: 7090: 7085: 7080: 7067: 7060: 7057: 7044: 7015: 7011: 6987: 6981: 6959: 6955: 6947: 6937: 6919: 6916: 6913: 6907: 6904: 6901: 6898: 6895: 6892: 6882: 6869: 6866: 6863: 6854: 6847: 6844: 6841: 6836: 6833: 6830: 6824: 6821: 6817: 6812: 6802: 6795: 6792: 6786: 6781: 6777: 6770: 6767: 6764: 6760: 6755: 6749: 6743: 6740: 6734: 6729: 6724: 6711: 6704: 6691: 6662: 6658: 6634: 6628: 6614: 6610: 6606: 6600: 6583: 6580: 6575: 6571: 6568: 6561: 6554: 6551: 6545: 6542: 6537: 6533: 6530: 6527: 6523: 6516: 6511: 6507: 6497: 6494: 6491: 6477: 6471: 6464: 6458: 6453: 6449: 6445: 6443: 6435: 6432: 6427: 6420: 6414: 6406: 6399: 6394: 6387: 6381: 6376: 6371: 6367: 6363: 6361: 6353: 6347: 6344: 6340: 6337: 6332: 6327: 6319: 6314: 6305: 6283: 6279: 6262: 6259: 6256: 6253: 6247: 6244: 6241: 6234: 6231: 6224: 6213: 6206: 6197: 6194: 6191: 6188: 6185: 6174: 6171: 6168: 6160: 6155: 6151: 6141: 6138: 6135: 6125: 6120: 6114: 6108: 6103: 6099: 6089: 6076: 6073: 6070: 6062: 6059: 6056: 6053: 6050: 6045: 6041: 6038: 6035: 6031: 6022: 6011: 6004: 5995: 5990: 5986: 5976: 5973: 5970: 5960: 5955: 5949: 5943: 5940: 5934: 5929: 5924: 5916: 5907: 5890: 5886: 5880: 5876: 5870: 5860: 5840: 5834: 5831: 5825: 5822: 5819: 5814: 5809: 5801: 5792: 5777: 5773: 5769: 5763: 5759: 5753: 5751: 5740: 5734: 5731: 5722: 5719: 5716: 5710: 5705: 5700: 5692: 5683: 5668: 5664: 5660: 5654: 5650: 5644: 5642: 5634: 5628: 5625: 5619: 5614: 5609: 5601: 5592: 5575: 5569: 5566: 5560: 5557: 5554: 5549: 5544: 5536: 5527: 5512: 5508: 5504: 5498: 5494: 5488: 5486: 5475: 5469: 5466: 5457: 5454: 5451: 5445: 5440: 5435: 5427: 5418: 5403: 5399: 5395: 5389: 5385: 5379: 5377: 5369: 5363: 5360: 5354: 5349: 5344: 5336: 5327: 5306: 5302: 5287: 5283: 5259: 5255: 5251: 5249: 5245: 5230: 5220: 5207: 5204: 5197: 5191: 5186: 5183: 5180: 5175: 5171: 5168: 5165: 5161: 5154: 5149: 5145: 5135: 5123: 5117: 5114: 5110: 5105: 5099: 5093: 5084: 5078: 5075: 5070: 5067: 5064: 5058: 5052: 5046: 5043: 5037: 5032: 5027: 5019: 5014: 5007: 4994: 4973: 4960: 4956: 4953: 4950: 4946: 4941: 4937: 4934: 4925: 4919: 4914: 4911: 4908: 4903: 4897: 4894: 4889: 4886: 4882: 4875: 4870: 4866: 4856: 4846: 4841: 4835: 4829: 4826: 4820: 4817: 4812: 4797: 4771: 4767: 4757: 4755: 4750: 4736: 4733: 4730: 4721: 4718: 4714: 4708: 4704: 4698: 4681: 4678: 4671: 4665: 4660: 4657: 4654: 4649: 4645: 4642: 4639: 4635: 4628: 4623: 4619: 4609: 4599: 4594: 4592: 4581: 4575: 4572: 4566: 4561: 4556: 4548: 4539: 4529: 4527: 4519: 4513: 4510: 4504: 4501: 4496: 4491: 4483: 4474: 4460: 4457: 4450: 4444: 4439: 4436: 4433: 4428: 4424: 4421: 4418: 4414: 4407: 4402: 4398: 4388: 4378: 4373: 4371: 4360: 4354: 4351: 4345: 4340: 4335: 4327: 4318: 4308: 4306: 4298: 4292: 4289: 4283: 4280: 4275: 4270: 4262: 4253: 4232: 4228: 4222: 4218: 4214: 4213:Weyl integral 4210: 4206: 4196: 4194: 4184: 4169: 4149: 4142: 4138: 4133: 4130: 4127: 4123: 4118: 4114: 4112: 4104: 4101: 4094: 4088: 4083: 4080: 4077: 4074: 4071: 4066: 4062: 4059: 4056: 4052: 4045: 4040: 4036: 4026: 4023: 4020: 4010: 4005: 4003: 3995: 3988: 3984: 3979: 3975: 3970: 3965: 3960: 3956: 3952: 3938: 3919: 3916: 3913: 3899: 3886: 3874: 3868: 3865: 3862: 3856: 3853: 3850: 3847: 3841: 3838: 3835: 3831: 3825: 3822: 3819: 3814: 3810: 3807: 3804: 3800: 3793: 3788: 3784: 3775: 3774:beta function 3770: 3757: 3754: 3749: 3745: 3742: 3736: 3733: 3730: 3726: 3720: 3717: 3714: 3709: 3705: 3702: 3699: 3695: 3688: 3683: 3679: 3674: 3667: 3661: 3656: 3653: 3650: 3647: 3644: 3639: 3635: 3632: 3629: 3625: 3618: 3613: 3609: 3599: 3587: 3577: 3572: 3566: 3559: 3555: 3550: 3546: 3541: 3536: 3531: 3527: 3523: 3513: 3509: 3505: 3501: 3497: 3487: 3486:integration. 3479: 3475: 3469: 3452: 3449: 3444: 3440: 3437: 3431: 3428: 3425: 3420: 3416: 3413: 3410: 3406: 3399: 3396: 3393: 3388: 3384: 3381: 3378: 3374: 3367: 3362: 3358: 3353: 3346: 3340: 3335: 3330: 3326: 3316: 3304: 3294: 3289: 3287: 3279: 3276: 3272: 3269: 3262: 3256: 3251: 3248: 3245: 3240: 3236: 3233: 3230: 3226: 3219: 3216: 3213: 3208: 3204: 3201: 3198: 3194: 3187: 3182: 3178: 3172: 3167: 3163: 3153: 3141: 3131: 3126: 3124: 3116: 3113: 3106: 3099: 3095: 3090: 3086: 3081: 3075: 3072: 3069: 3061: 3058: 3055: 3047: 3042: 3038: 3028: 3018: 3013: 3011: 3003: 2996: 2992: 2987: 2983: 2978: 2973: 2968: 2964: 2960: 2947: 2946: 2940: 2939: 2935: 2918: 2914: 2911: 2904: 2898: 2893: 2890: 2887: 2884: 2881: 2876: 2872: 2869: 2866: 2862: 2855: 2850: 2846: 2836: 2833: 2830: 2820: 2815: 2813: 2802: 2795: 2791: 2786: 2783: 2780: 2776: 2771: 2767: 2765: 2754: 2747: 2743: 2738: 2734: 2729: 2724: 2719: 2715: 2711: 2707: 2705: 2697: 2690: 2686: 2681: 2677: 2672: 2667: 2662: 2658: 2654: 2636: 2633: 2620: 2616: 2613: 2606: 2600: 2595: 2592: 2589: 2584: 2580: 2577: 2574: 2570: 2563: 2558: 2554: 2544: 2534: 2529: 2523: 2516: 2512: 2507: 2503: 2498: 2489: 2469: 2465: 2462: 2455: 2449: 2444: 2441: 2438: 2433: 2429: 2426: 2423: 2419: 2412: 2407: 2403: 2396: 2390: 2387: 2384: 2377: 2372: 2366: 2359: 2355: 2350: 2346: 2341: 2332: 2327: 2324: 2307: 2303: 2300: 2296: 2292: 2289: 2282: 2276: 2271: 2266: 2262: 2257: 2251: 2246: 2242: 2238: 2236: 2228: 2225: 2218: 2209: 2206: 2198: 2193: 2189: 2185: 2183: 2175: 2168: 2164: 2159: 2155: 2150: 2136: 2123: 2119: 2116: 2109: 2103: 2098: 2093: 2089: 2085: 2079: 2070: 2067: 2050: 2043: 2039: 2028: 2025: 2021: 2018: 2014: 2010: 2005: 2001: 1997: 1993: 1983: 1981: 1976: 1974: 1958: 1930: 1927: 1924: 1919: 1915: 1904:semigroup of 1903: 1885: 1876: 1873: 1849: 1846: 1843: 1838: 1834: 1822: 1806: 1797: 1782: 1779: 1776: 1755: 1733: 1725:and with the 1710: 1707: 1704: 1683: 1661: 1633: 1630: 1621: 1605: 1585: 1576: 1561: 1557: 1548: 1542: 1540: 1535: 1517: 1514: 1507: 1503: 1498: 1487: 1470: 1458: 1452: 1444: 1438: 1434: 1431: 1425: 1419: 1413: 1406: 1404: 1393: 1382: 1376: 1372: 1369: 1366: 1363: 1360: 1357: 1354: 1351: 1348: 1338: 1336: 1328: 1320: 1316: 1291: 1284: 1266: 1258: 1242: 1235: 1231: 1226: 1224: 1219: 1206: 1202: 1199: 1192: 1186: 1181: 1176: 1172: 1168: 1162: 1156: 1153: 1134: 1126: 1121: 1108: 1101: 1095: 1089: 1086: 1082: 1077: 1071: 1065: 1062: 1043: 1036: 1033: 1029: 1025: 1021: 1017: 1006: 1001: 999: 994: 992: 987: 986: 984: 983: 976: 973: 971: 968: 966: 963: 961: 958: 956: 953: 951: 948: 946: 943: 942: 934: 933: 926: 923: 921: 918: 916: 913: 911: 908: 907: 899: 898: 887: 884: 882: 879: 877: 874: 873: 872: 871: 861: 860: 849: 846: 844: 841: 839: 836: 834: 831: 829: 828:Line integral 826: 824: 821: 819: 816: 815: 814: 813: 809: 808: 803: 800: 798: 795: 793: 790: 788: 785: 784: 783: 782: 778: 777: 771: 770:Multivariable 765: 764: 753: 750: 748: 745: 743: 740: 738: 735: 733: 730: 728: 725: 724: 723: 722: 718: 717: 712: 709: 707: 704: 702: 699: 697: 694: 692: 689: 687: 684: 683: 682: 681: 675: 669: 668: 657: 654: 652: 649: 647: 644: 642: 639: 637: 633: 631: 628: 626: 623: 621: 618: 616: 613: 611: 608: 607: 606: 605: 602: 599: 598: 593: 590: 588: 585: 583: 580: 578: 575: 573: 570: 567: 563: 560: 559: 558: 557: 551: 545: 544: 533: 530: 528: 525: 523: 520: 518: 515: 512: 508: 505: 503: 500: 497: 493: 489: 488:trigonometric 485: 482: 480: 477: 475: 472: 470: 467: 466: 465: 464: 460: 459: 454: 451: 449: 446: 444: 441: 439: 436: 433: 429: 426: 424: 421: 420: 419: 418: 414: 413: 408: 405: 403: 400: 398: 395: 394: 393: 392: 386: 380: 379: 368: 365: 363: 360: 358: 355: 353: 350: 348: 345: 343: 340: 338: 335: 333: 330: 328: 325: 323: 320: 319: 318: 317: 314: 311: 310: 305: 302: 300: 299:Related rates 297: 295: 292: 290: 287: 285: 282: 280: 277: 276: 275: 274: 270: 269: 262: 259: 257: 256:of a function 254: 252: 251:infinitesimal 249: 248: 247: 244: 241: 237: 234: 233: 232: 231: 227: 226: 220: 214: 213: 207: 204: 202: 199: 197: 194: 193: 188: 185: 183: 180: 179: 175: 172: 171: 170: 169: 150: 144: 141: 135: 129: 126: 123: 120: 113: 106: 103: 97: 92: 88: 79: 78: 75: 72: 71: 67: 66: 61: 50: 45: 41: 37: 36: 30: 21: 20: 14904:Martin Kutta 14859:Émile Picard 14839:Isaac Newton 14753:Euler method 14723:Substitution 14364: 14360: 14313: 14309: 14277:the original 14259:(in French). 14247:"GigaHedron" 14228: 14153: 14136: 14117: 14098: 14048: 14011: 13982: 13971: 13952:10.1142/8936 13943: 13924: 13888: 13869:10.1142/9069 13860: 13832: 13813:10.1142/p614 13803: 13747: 13731:. Elsevier. 13727: 13707: 13688: 13669: 13650: 13626:10.1155/6314 13606: 13602: 13563: 13559: 13524: 13520: 13485: 13481: 13452: 13448: 13401: 13370: 13366: 13360: 13323: 13317: 13256: 13253:Phys. Rev. E 13252: 13246: 13201: 13197: 13187: 13128: 13124: 13071: 13064: 13011: 13007: 13001: 12966: 12962: 12956: 12931:10852/103312 12905: 12901: 12895: 12870:10852/103311 12844: 12840: 12834: 12805: 12799: 12789: 12744: 12738: 12728: 12675: 12669: 12642: 12638: 12628: 12603: 12599: 12593: 12574: 12570: 12514: 12508: 12502: 12467: 12463: 12453: 12410: 12406: 12400: 12381: 12377: 12367: 12348: 12344: 12334: 12326: 12322: 12317: 12309: 12305: 12300: 12267: 12263: 12250: 12225: 12219: 12202: 12198: 12189: 12180: 12171: 12146: 12142: 12136: 12103: 12099: 12093: 12057: 12050: 12021: 12017: 12007: 11960: 11954: 11921: 11917: 11911: 11871:(1): 27–36. 11868: 11864: 11824: 11820: 11744: 11740: 11734: 11715: 11709: 11699: 11662: 11658: 11612: 11608: 11581:. Retrieved 11576: 11572: 11562: 11534: 11530: 11524: 11516: 11478:10.1142/8934 11457: 11451: 11426: 11422: 11412: 11393: 11389: 11379: 11355: 11348: 11339: 11335: 11322: 11296:10.1142/p614 11283: 11256:10.1142/8934 11239: 11233: 11190: 11186: 11176: 11151: 11147: 11140: 11132: 11127: 11121: 11111: 11107: 11094: 11084: 11080: 11067: 11032: 11028: 11022: 11013: 11000: 10981: 10977: 10964: 10944: 10937: 10848: 10782: 10720: 10500: 10488: 10480: 10277: 10220: 10145: 10138: 10134: 10130: 10108: 10094: 10090: 10086: 10083:wavefunction 10080: 9892: 9871: 9864: 9861: 9608: 9579:proportional 9570: 9561: 9552: 9541: 9392: 9388: 9384: 9378: 9374: 9370: 9364: 9352:viscoelastic 9349: 9337: 9333: 9329: 9322: 9318: 9314: 9303: 9215: 9199: 9196: 9183: 9174: 9050: 9046: 9042: 9031: 9027: 9023: 9020: 8925: 8747: 8739:Applications 8716: 8686: 8682: 8680:, functions 8675: 8663: 8509:(1940). and 8500: 8459: 8338: 8273: 8238: 8164: 8153:denotes the 8128: 7991: 7938: 7853:heavy-tailed 7507: 7263: 6943: 6883: 6620: 6608: 6604: 6601: 6281: 6277: 6090: 5888: 5884: 5872: 5304: 5300: 5285: 5281: 5269: 5252: 5250:in general. 5233: 4979: 4765: 4763: 4751: 4722: 4716: 4712: 4706: 4702: 4699: 4230: 4226: 4202: 4195:operators. 4190: 4170: 3939: 3771: 3511: 3507: 3503: 3499: 3495: 3488: 3477: 3473: 3470: 2948: 2637: 2634: 2486:: using the 2328: 2325: 2137: 2057:. Call this 2048: 2041: 2037: 2034: 1995: 1989: 1977: 1951:for integer 1901: 1871: 1819:is that the 1798: 1577: 1536: 1488: 1229: 1227: 1220: 1122: 1015: 1014: 909: 484:Substitution 246:Differential 219:Differential 48: 33: 14661:Phase space 14519:Homogeneous 14229:Tech Briefs 13373:: 197–218. 13229:10852/53091 12606:(1): 1–77. 12205:: 217–234. 12149:(1): 1–11. 12018:Mathematics 11995:|work= 11615:: 552–559. 10984:(4): 1–15. 10852:The symbol 9385:error value 8317:at a point 8234:-derivative 8161:Other types 5244:commutative 5242:is neither 4221:unit circle 3493:defined by 1978:Fractional 1973:denumerable 1125:integration 1123:and of the 1024:real number 945:Precalculus 938:Miscellanea 903:Specialized 810:Definitions 577:Alternating 415:Definitions 228:Definitions 14924:Categories 14889:John Crank 14718:Inspection 14581:Stochastic 14575:Difference 14549:Autonomous 14493:Non-linear 14483:Fractional 14446:Operations 13138:1612.05557 12754:1610.06590 12635:Luchko, Y. 12031:1711.08070 11894:11147/5562 11754:1612.03046 11672:1602.03408 11579:(2): 73–85 11042:1802.05441 10930:References 10920:identities 10918:, such as 10481:The index 9879:seismology 9587:derivative 8920:See also: 8729:Kober 1940 8341:only when 7030:given by 6677:given by: 5298:(that is, 2017:autodidact 1875:semigroups 1872:continuous 1026:powers or 925:Variations 920:Stochastic 910:Fractional 779:Formalisms 742:Divergence 711:Identities 691:Divergence 236:Derivative 187:Continuity 14693:solutions 14651:Wronskian 14606:Solutions 14534:Decoupled 14498:Holonomic 14215:2090-5858 14203:2218-3892 14190:1847-9677 14177:1314-2224 14166:1998–2014 14138:MathWorld 14119:MathWorld 14100:MathWorld 14084:MathWorld 14039:242068092 13792:ignored ( 13782:cite book 13551:122487513 13469:122198977 13438:ignored ( 13428:cite book 13328:CiteSeerX 13279:CiteSeerX 13163:0001-4966 13107:145880744 13026:CiteSeerX 13021:1306.6507 12993:120348311 12976:1212.4024 12969:: 26–50. 12685:0709.3990 12600:Phys. 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10664:+ 10661:) 10658:t 10655:, 10651:r 10647:( 10638:2 10634:) 10631:t 10628:( 10618:) 10609:2 10597:( 10592:= 10584:) 10580:r 10576:( 10569:t 10560:) 10557:t 10554:, 10550:r 10546:( 10541:) 10537:r 10533:( 10513:i 10489:α 10483:α 10467:. 10463:) 10460:t 10457:, 10453:p 10449:( 10436:| 10430:p 10425:| 10418:r 10410:p 10401:i 10395:e 10391:p 10386:3 10382:d 10370:3 10366:) 10356:2 10353:( 10349:1 10344:= 10341:) 10338:t 10335:, 10331:r 10327:( 10318:2 10310:) 10301:2 10290:( 10278:ħ 10272:m 10255:m 10252:2 10248:1 10243:= 10238:2 10234:D 10221:α 10208:α 10206:D 10183:2 10178:r 10167:2 10157:= 10141:) 10139:t 10135:r 10133:( 10131:V 10118:ħ 10114:t 10109:r 10097:) 10095:t 10091:r 10089:( 10087:ψ 10067:. 10063:) 10060:t 10057:, 10053:r 10049:( 10043:) 10040:t 10037:, 10033:r 10029:( 10026:V 10023:+ 10020:) 10017:t 10014:, 10010:r 10006:( 9997:2 9988:) 9979:2 9967:( 9956:D 9952:= 9946:t 9938:) 9935:t 9932:, 9928:r 9924:( 9909:i 9848:. 9844:0 9841:= 9832:2 9829:+ 9822:t 9813:u 9808:2 9805:+ 9785:2 9780:0 9776:c 9751:u 9746:2 9728:t 9692:+ 9684:2 9680:t 9671:u 9666:2 9650:2 9645:0 9641:c 9637:1 9628:u 9623:2 9599:D 9595:I 9591:P 9574:d 9571:K 9565:i 9562:K 9556:p 9553:K 9548:β 9544:α 9528:) 9525:t 9522:( 9519:e 9509:t 9505:D 9498:d 9493:K 9489:+ 9486:) 9483:t 9480:( 9477:e 9464:t 9460:D 9453:i 9448:K 9444:+ 9441:) 9438:t 9435:( 9432:e 9426:p 9421:K 9417:= 9414:) 9411:t 9408:( 9405:u 9395:) 9393:t 9391:( 9389:e 9381:) 9379:t 9377:( 9375:u 9340:) 9338:t 9334:x 9332:( 9330:β 9325:) 9323:t 9319:x 9317:( 9315:α 9310:β 9306:α 9290:. 9287:u 9278:) 9268:( 9265:K 9259:= 9247:t 9238:u 9161:) 9158:t 9155:, 9152:x 9149:( 9146:C 9135:2 9132:1 9125:t 9121:d 9112:2 9109:1 9102:d 9096:= 9093:) 9090:t 9087:, 9084:x 9081:( 9078:C 9072:x 9069:d 9065:d 9053:) 9051:s 9049:, 9047:x 9045:( 9043:C 9038:s 9034:) 9032:s 9030:, 9028:x 9026:( 9024:C 9007:) 9004:s 9001:, 8998:x 8995:( 8992:C 8989:s 8986:= 8983:) 8980:s 8977:, 8974:x 8971:( 8968:C 8960:2 8956:x 8952:d 8946:2 8942:d 8898:t 8890:p 8880:) 8874:w 8863:+ 8858:s 8849:( 8834:1 8830:x 8823:) 8820:1 8817:+ 8811:( 8805:= 8801:) 8791:u 8771:( 8705:D 8689:) 8687:D 8685:( 8683:f 8650:, 8646:t 8643:d 8639:) 8636:t 8633:( 8630:f 8612:t 8606:1 8595:) 8591:x 8585:t 8581:( 8574:x 8569:0 8558:) 8552:( 8543:1 8540:+ 8524:x 8468:a 8451:. 8439:c 8428:, 8416:f 8395:c 8392:= 8389:x 8369:f 8349:a 8325:x 8305:f 8283:a 8216:F 8139:F 8115:, 8112:) 8109:k 8106:( 8103:} 8100:u 8097:{ 8092:F 8080:| 8077:k 8074:| 8066:= 8063:) 8060:k 8057:( 8053:} 8040:| 8037:x 8034:| 8024:u 8008:{ 8002:F 7978:, 7966:) 7963:1 7960:, 7957:0 7954:( 7925:, 7909:+ 7906:1 7902:1 7897:= 7894:) 7883:X 7872:e 7868:( 7864:E 7849:, 7837:) 7834:1 7831:, 7828:0 7825:( 7794:1 7784:, 7770:1 7766:E 7750:. 7713:) 7704:x 7697:( 7688:E 7681:1 7675:x 7665:. 7653:1 7650:= 7647:) 7644:0 7641:( 7632:E 7620:, 7584:0 7558:E 7547:, 7535:] 7532:1 7529:, 7526:0 7523:( 7494:, 7488:d 7484:) 7471:1 7460:) 7450:t 7447:( 7434:( 7424:E 7420:) 7414:( 7411:f 7406:t 7401:a 7390:t 7387:d 7383:d 7369:1 7364:) 7358:( 7346:= 7343:) 7340:t 7337:( 7334:f 7320:t 7315:D 7302:a 7295:B 7292:A 7279:D 7250:, 7244:d 7240:) 7227:1 7216:) 7206:t 7203:( 7190:( 7180:E 7176:) 7170:( 7163:f 7157:t 7152:a 7135:1 7130:) 7124:( 7112:= 7109:) 7106:t 7103:( 7100:f 7086:t 7081:D 7068:a 7061:B 7058:A 7045:D 7016:1 7012:C 6991:) 6988:t 6985:( 6982:f 6956:E 6935:. 6923:] 6920:1 6917:, 6914:0 6911:( 6902:, 6899:0 6893:a 6870:, 6864:d 6855:) 6842:1 6831:t 6818:( 6813:e 6806:) 6800:( 6793:f 6787:t 6782:a 6765:1 6761:1 6756:= 6753:) 6750:t 6747:( 6744:f 6730:t 6725:D 6712:a 6705:C 6692:D 6663:1 6659:C 6638:) 6635:t 6632:( 6629:f 6611:) 6609:ν 6607:( 6605:ϕ 6581:d 6576:] 6572:u 6569:d 6565:) 6562:u 6559:( 6552:f 6538:) 6534:u 6528:t 6524:( 6517:t 6512:0 6501:) 6492:1 6489:( 6481:) 6475:( 6465:[ 6459:b 6454:a 6446:= 6433:d 6428:] 6424:) 6421:t 6418:( 6415:f 6410:) 6404:( 6400:D 6395:[ 6391:) 6385:( 6377:b 6372:a 6364:= 6357:) 6354:t 6351:( 6348:f 6341:u 6333:n 6328:D 6320:b 6315:a 6306:D 6284:) 6282:t 6280:( 6278:f 6263:n 6251:) 6248:1 6242:n 6239:( 6235:u 6232:d 6228:) 6225:u 6222:( 6217:) 6214:n 6211:( 6207:f 6201:) 6198:1 6186:n 6183:( 6179:) 6175:u 6169:t 6166:( 6161:t 6156:0 6145:) 6136:n 6133:( 6126:1 6121:= 6118:) 6115:t 6112:( 6109:f 6100:D 6077:. 6071:d 6063:n 6057:1 6054:+ 6046:) 6036:t 6032:( 6026:) 6020:( 6015:) 6012:n 6009:( 6005:f 5996:t 5991:0 5980:) 5971:n 5968:( 5961:1 5956:= 5953:) 5950:t 5947:( 5944:f 5930:t 5925:D 5917:C 5908:D 5891:⌉ 5889:α 5885:n 5844:) 5841:t 5838:( 5835:f 5820:n 5815:b 5810:I 5802:t 5793:I 5778:n 5774:t 5770:d 5764:n 5760:d 5754:= 5744:) 5741:t 5738:( 5735:f 5726:) 5717:n 5714:( 5706:b 5701:D 5693:t 5684:D 5669:n 5665:t 5661:d 5655:n 5651:d 5645:= 5638:) 5635:t 5632:( 5629:f 5615:b 5610:D 5602:t 5593:D 5579:) 5576:t 5573:( 5570:f 5555:n 5550:t 5545:I 5537:a 5528:I 5513:n 5509:t 5505:d 5499:n 5495:d 5489:= 5479:) 5476:t 5473:( 5470:f 5461:) 5452:n 5449:( 5441:t 5436:D 5428:a 5419:D 5404:n 5400:t 5396:d 5390:n 5386:d 5380:= 5373:) 5370:t 5367:( 5364:f 5350:t 5345:D 5337:a 5328:D 5307:⌉ 5305:α 5301:n 5296:α 5292:n 5288:) 5286:α 5282:n 5280:( 5276:n 5272:α 5240:D 5236:D 5231:. 5205:d 5201:) 5195:( 5192:f 5187:1 5176:) 5166:t 5162:( 5155:t 5150:a 5139:) 5133:( 5127:) 5121:( 5106:+ 5103:) 5100:t 5097:( 5094:f 5088:) 5082:( 5065:1 5059:= 5056:) 5053:t 5050:( 5047:f 5033:t 5028:I 5015:a 5008:A 4995:I 4961:. 4957:a 4951:t 4947:, 4935:d 4929:) 4923:( 4920:f 4915:1 4904:) 4895:t 4883:( 4876:t 4871:a 4860:) 4854:( 4847:1 4842:= 4839:) 4836:t 4833:( 4830:f 4813:t 4807:D 4798:a 4788:D 4737:0 4734:= 4731:a 4717:b 4713:t 4707:a 4703:t 4679:d 4675:) 4669:( 4666:f 4661:1 4650:) 4646:t 4636:( 4629:b 4624:t 4613:) 4607:( 4600:1 4595:= 4585:) 4582:t 4579:( 4576:f 4562:b 4557:I 4549:t 4540:I 4530:= 4523:) 4520:t 4517:( 4514:f 4497:b 4492:D 4484:t 4475:D 4458:d 4454:) 4448:( 4445:f 4440:1 4429:) 4419:t 4415:( 4408:t 4403:a 4392:) 4386:( 4379:1 4374:= 4364:) 4361:t 4358:( 4355:f 4341:t 4336:I 4328:a 4319:I 4309:= 4302:) 4299:t 4296:( 4293:f 4276:t 4271:D 4263:a 4254:D 4231:b 4229:, 4227:a 4181:J 4177:β 4173:α 4153:) 4150:x 4147:( 4143:) 4139:f 4131:+ 4124:J 4119:( 4115:= 4105:s 4102:d 4098:) 4095:s 4092:( 4089:f 4084:1 4075:+ 4067:) 4063:s 4057:x 4053:( 4046:x 4041:0 4030:) 4024:+ 4018:( 4011:1 4006:= 3999:) 3996:x 3993:( 3989:) 3985:f 3976:J 3971:( 3966:) 3957:J 3953:( 3923:) 3917:+ 3911:( 3903:) 3897:( 3890:) 3884:( 3875:= 3872:) 3866:, 3860:( 3857:B 3854:= 3851:r 3848:d 3842:1 3832:r 3826:1 3815:) 3811:r 3805:1 3801:( 3794:1 3789:0 3758:s 3755:d 3750:) 3746:r 3743:d 3737:1 3727:r 3721:1 3710:) 3706:r 3700:1 3696:( 3689:1 3684:0 3675:( 3671:) 3668:s 3665:( 3662:f 3657:1 3648:+ 3640:) 3636:s 3630:x 3626:( 3619:x 3614:0 3603:) 3597:( 3591:) 3585:( 3578:1 3573:= 3570:) 3567:x 3564:( 3560:) 3556:f 3547:J 3542:( 3537:) 3528:J 3524:( 3512:r 3510:) 3508:s 3504:x 3500:s 3496:t 3491:r 3484:t 3480:) 3478:s 3476:( 3474:f 3453:s 3450:d 3445:) 3441:t 3438:d 3432:1 3421:) 3417:s 3411:t 3407:( 3400:1 3389:) 3385:t 3379:x 3375:( 3368:x 3363:s 3354:( 3350:) 3347:s 3344:( 3341:f 3336:x 3331:0 3320:) 3314:( 3308:) 3302:( 3295:1 3290:= 3280:t 3277:d 3273:s 3270:d 3266:) 3263:s 3260:( 3257:f 3252:1 3241:) 3237:s 3231:t 3227:( 3220:1 3209:) 3205:t 3199:x 3195:( 3188:t 3183:0 3173:x 3168:0 3157:) 3151:( 3145:) 3139:( 3132:1 3127:= 3117:t 3114:d 3110:) 3107:t 3104:( 3100:) 3096:f 3087:J 3082:( 3076:1 3066:) 3062:t 3056:x 3053:( 3048:x 3043:0 3032:) 3026:( 3019:1 3014:= 3007:) 3004:x 3001:( 2997:) 2993:f 2984:J 2979:( 2974:) 2965:J 2961:( 2919:. 2915:t 2912:d 2908:) 2905:t 2902:( 2899:f 2894:1 2885:+ 2877:) 2873:t 2867:x 2863:( 2856:x 2851:0 2840:) 2834:+ 2828:( 2821:1 2816:= 2806:) 2803:x 2800:( 2796:) 2792:f 2784:+ 2777:J 2772:( 2768:= 2758:) 2755:x 2752:( 2748:) 2744:f 2735:J 2730:( 2725:) 2716:J 2712:( 2708:= 2701:) 2698:x 2695:( 2691:) 2687:f 2678:J 2673:( 2668:) 2659:J 2655:( 2640:J 2621:. 2617:t 2614:d 2610:) 2607:t 2604:( 2601:f 2596:1 2585:) 2581:t 2575:x 2571:( 2564:x 2559:0 2548:) 2542:( 2535:1 2530:= 2527:) 2524:x 2521:( 2517:) 2513:f 2504:J 2499:( 2484:n 2470:, 2466:t 2463:d 2459:) 2456:t 2453:( 2450:f 2445:1 2439:n 2434:) 2430:t 2424:x 2420:( 2413:x 2408:0 2397:! 2394:) 2391:1 2385:n 2382:( 2378:1 2373:= 2370:) 2367:x 2364:( 2360:) 2356:f 2351:n 2347:J 2342:( 2308:, 2304:t 2301:d 2297:) 2293:s 2290:d 2286:) 2283:s 2280:( 2277:f 2272:t 2267:0 2258:( 2252:x 2247:0 2239:= 2229:t 2226:d 2222:) 2219:t 2216:( 2213:) 2210:f 2207:J 2204:( 2199:x 2194:0 2186:= 2179:) 2176:x 2173:( 2169:) 2165:f 2160:2 2156:J 2151:( 2124:. 2120:t 2117:d 2113:) 2110:t 2107:( 2104:f 2099:x 2094:0 2086:= 2083:) 2080:x 2077:( 2074:) 2071:f 2068:J 2065:( 2055:x 2049:x 2044:) 2042:x 2040:( 2038:f 1959:n 1939:} 1935:Z 1928:n 1920:n 1916:D 1912:{ 1898:, 1886:a 1858:} 1854:R 1847:a 1839:a 1835:D 1831:{ 1807:D 1795:. 1783:0 1777:n 1756:J 1734:n 1723:, 1711:0 1705:n 1684:D 1662:n 1651:, 1638:Z 1631:n 1606:a 1586:a 1562:a 1558:D 1518:2 1515:1 1508:D 1504:= 1499:D 1471:. 1468:) 1465:) 1462:) 1456:) 1453:f 1450:( 1445:n 1435:D 1429:( 1426:D 1423:( 1420:D 1417:( 1414:D 1407:= 1397:) 1394:f 1391:( 1388:) 1383:n 1373:D 1361:D 1355:D 1349:D 1342:( 1339:= 1332:) 1329:f 1326:( 1321:n 1317:D 1292:D 1279:, 1267:f 1243:D 1207:, 1203:s 1200:d 1196:) 1193:s 1190:( 1187:f 1182:x 1177:0 1169:= 1166:) 1163:x 1160:( 1157:f 1154:J 1135:J 1109:, 1105:) 1102:x 1099:( 1096:f 1090:x 1087:d 1083:d 1078:= 1075:) 1072:x 1069:( 1066:f 1063:D 1044:D 1004:e 997:t 990:v 568:) 513:) 509:( 498:) 434:) 242:) 154:) 151:a 148:( 145:f 139:) 136:b 133:( 130:f 127:= 124:t 121:d 117:) 114:t 111:( 104:f 98:b 93:a 62:. 46:)

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Knowledge

Fractional calculus

Index