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</style><div role="note" class="hatnote navigation-not-searchable">"Fractional integration" redirects here; not to be confused with <a href="Autoregressive_fractionally_integrated_moving_average" title="Autoregressive fractionally integrated moving average">Autoregressive fractionally integrated moving average</a>.</div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation>

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</math></span><img src="./17d063dc86a53a2efb1fe86f4a5d47d498652766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}" loading="lazy"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;">

<ul><li><a href="Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul>

<ul><li><a href="Limit_of_a_function" title="Limit of a function">Limits</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuity</a></li></ul>

</div><div class="hlist">

<ul><li><a href="Rolle's_theorem" title="Rolle's theorem">Rolle's theorem</a></li>

<li><a href="Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li>

<li><a href="Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul>

</div></td></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Derivative" title="Derivative">Derivative</a> (<a href="Generalizations_of_the_derivative" title="Generalizations of the derivative">generalizations</a>)</li>

<li><a href="Differential_(mathematics)" title="Differential (mathematics)">Differential</a>

<ul><li><a href="Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li>

<li><a href="Differential_of_a_function" title="Differential of a function">of a function</a></li>

<li><a href="Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td>

</tr><tr><th class="sidebar-heading">

Concepts</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li>

<li><a href="Second_derivative" title="Second derivative">Second derivative</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit differentiation</a></li>

<li><a href="Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li>

<li><a href="Related_rates" title="Related rates">Related rates</a></li>

<li><a href="Taylor's_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist">

<li><a href="Product_rule" title="Product rule">Product</a></li>

<li><a href="Chain_rule" title="Chain rule">Chain</a></li>

<li><a href="Power_rule" title="Power rule">Power</a></li>

<li><a href="Quotient_rule" title="Quotient rule">Quotient</a></li>

<li><a href="L'H%C3%B4pital's_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li>

<li><a href="Inverse_function_rule" title="Inverse function rule">Inverse</a></li>

<li><a href="General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li>

<li><a href="Fa%C3%A0_di_Bruno's_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li>

<li><a href="Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Antiderivative" title="Antiderivative">Antiderivative</a></li>

<li><a href="Integral" title="Integral">Integral</a> (<a href="Improper_integral" title="Improper integral">improper</a>)</li>

<li><a href="Riemann_integral" title="Riemann integral">Riemann integral</a></li>

<li><a href="Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li>

<li><a href="Contour_integration" title="Contour integration">Contour integration</a></li>

<li><a href="Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Integration by</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Integration_by_parts" title="Integration by parts">Parts</a></li>

<li><a href="Disc_integration" title="Disc integration">Discs</a></li>

<li><a href="Shell_integration" title="Shell integration">Cylindrical shells</a></li>

<li><a href="Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="Euler_substitution" title="Euler substitution">Euler</a>)</li>

<li><a href="Integration_using_Euler's_formula" title="Integration using Euler's formula">Euler's formula</a></li>

<li><a href="Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li>

<li><a href="Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li>

<li><a href="Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li>

<li><a href="Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li>

<li><a href="Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Geometric_series" title="Geometric series">Geometric</a> (<a href="Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li>

<li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Power_series" title="Power series">Power</a></li>

<li><a href="Binomial_series" title="Binomial series">Binomial</a></li>

<li><a href="Taylor_series" title="Taylor series">Taylor</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

<a href="Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li>

<li><a href="Ratio_test" title="Ratio test">Ratio</a></li>

<li><a href="Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li>

<li><a href="Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li>

<li><br><a href="Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li>

<li><a href="Alternating_series_test" title="Alternating series test">Alternating series</a></li>

<li><a href="Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li>

<li><a href="Dirichlet's_test" title="Dirichlet's test">Dirichlet</a></li>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Gradient" title="Gradient">Gradient</a></li>

<li><a href="Divergence" title="Divergence">Divergence</a></li>

<li><a href="Laplace_operator" title="Laplace operator">Laplacian</a></li>

<li><a href="Directional_derivative" title="Directional derivative">Directional derivative</a></li>

<li><a href="Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Theorems</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Gradient_theorem" title="Gradient theorem">Gradient</a></li>

<li><a href="Green's_theorem" title="Green's theorem">Green's</a></li>

<li><a href="Stokes'_theorem" title="Stokes' theorem">Stokes'</a></li>

<li><a href="Divergence_theorem" title="Divergence theorem">Divergence</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading">

Formalisms</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Matrix_calculus" title="Matrix calculus">Matrix</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior</a></li>

<li><a href="Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td>

</tr><tr><th class="sidebar-heading">

Definitions</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Partial_derivative" title="Partial derivative">Partial derivative</a></li>

<li><a href="Multiple_integral" title="Multiple integral">Multiple integral</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li><a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li>

<li><a href="Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist">

<ul><li><a href="Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li>

<li><a href="Generalized_function" title="Generalized function">Generalized functions</a></li>

<li><a href="Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;">

<ul><li><a href="Fractional_calculus" title="Fractional calculus">Fractional</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin</a></li>

<li><a href="Stochastic_calculus" title="Stochastic calculus">Stochastic</a></li>

<li><a href="Calculus_of_variations" title="Calculus of variations">Variations</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Miscellanea</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;">

<ul><li><a href="Precalculus" title="Precalculus">Precalculus</a></li>

<li><a href="History_of_calculus" title="History of calculus">History</a></li>

<li><a href="Glossary_of_calculus" title="Glossary of calculus">Glossary</a></li>

<li><a href="List_of_calculus_topics" title="List of calculus topics">List of topics</a></li>

<li><a href="Integration_Bee" title="Integration Bee">Integration Bee</a></li>

<li><a href="Mathematical_analysis" title="Mathematical analysis">Mathematical analysis</a></li>

<li><a href="Nonstandard_analysis" title="Nonstandard analysis">Nonstandard analysis</a></li></ul></div></div></td>

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<p>In <a href="Fractional_calculus" title="Fractional calculus">fractional calculus</a>, an area of <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, the <b>differintegral</b> is a combined <a href="Differential_operator" title="Differential operator">differentiation</a>/<a href="Integral_operator" title="Integral operator">integration</a> operator. Applied to a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> ƒ, the <i>q</i>-differintegral of <i>f</i>, here denoted by

</p>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}f}</annotation>

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<p>is the <a href="Fractional_calculus#Historical_notes" title="Fractional calculus">fractional derivative</a> (if <i>q</i> > 0) or <a href="Fractional_calculus#Fractional_integrals" title="Fractional calculus">fractional integral</a> (if <i>q</i> < 0). If <i>q</i> = 0, then the <i>q</i>-th differintegral of a function is the function itself. In the context of fractional integration and differentiation, there are several definitions of the differintegral.

</p>

<div class="mw-heading mw-heading2"><h2 id="Standard_definitions">Standard definitions</h2></div>

<p>The four most common forms are:

</p>

<ul><li>The <a href="Riemann%E2%80%93Liouville_differintegral" class="mw-redirect" title="Riemann–Liouville differintegral">Riemann–Liouville differintegral</a><div class="paragraphbreak" style="margin-top:0.5em"></div>This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization of the <a href="Cauchy_formula_for_repeated_integration" title="Cauchy formula for repeated integration">Cauchy formula for repeated integration</a> to arbitrary order. Here, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=\lceil q\rceil }">

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<annotation encoding="application/x-tex">{\displaystyle n=\lceil q\rceil }</annotation>

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</math></span><img src="./ac19edc13c50d55ba22d2d607417b66793428427.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.628ex; height:2.843ex;" alt="{\displaystyle n=\lceil q\rceil }" loading="lazy"></span>. <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{}_{a}^{RL}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{a}^{t}(t-\tau )^{n-q-1}f(\tau )d\tau \end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{a}^{RL}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{a}^{t}(t-\tau )^{n-q-1}f(\tau )d\tau \end{aligned}}}</annotation>

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<li>The <a href="Grunwald%E2%80%93Letnikov_differintegral" class="mw-redirect" title="Grunwald–Letnikov differintegral">Grunwald–Letnikov differintegral</a><div class="paragraphbreak" style="margin-top:0.5em"></div>The Grunwald–Letnikov differintegral is a direct generalization of the definition of a <a href="Derivative" title="Derivative">derivative</a>. It is more difficult to use than the Riemann–Liouville differintegral, but can sometimes be used to solve problems that the Riemann–Liouville cannot. <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{}_{a}^{GL}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&=\lim _{N\to \infty }\left[{\frac {t-a}{N}}\right]^{-q}\sum _{j=0}^{N-1}(-1)^{j}{q \choose j}f\left(t-j\left[{\frac {t-a}{N}}\right]\right)\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{a}^{GL}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&=\lim _{N\to \infty }\left[{\frac {t-a}{N}}\right]^{-q}\sum _{j=0}^{N-1}(-1)^{j}{q \choose j}f\left(t-j\left[{\frac {t-a}{N}}\right]\right)\end{aligned}}}</annotation>

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<li>The <a href="Weyl_differintegral" class="mw-redirect" title="Weyl differintegral">Weyl differintegral</a><div class="paragraphbreak" style="margin-top:0.5em"></div> This is formally similar to the Riemann–Liouville differintegral, but applies to <a href="Periodic_function" title="Periodic function">periodic functions</a>, with integral zero over a period.</li>

<li>The Caputo differintegral<div class="paragraphbreak" style="margin-top:0.5em"></div>In opposite to the Riemann-Liouville differintegral, Caputo derivative of a constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

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</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>. <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{}_{a}^{C}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&={\frac {1}{\Gamma (n-q)}}\int _{a}^{t}{\frac {f^{(n)}(\tau )}{(t-\tau )^{q-n+1}}}d\tau \end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{a}^{C}\mathbb {D} _{t}^{q}f(t)&={\frac {d^{q}f(t)}{d(t-a)^{q}}}\\&={\frac {1}{\Gamma (n-q)}}\int _{a}^{t}{\frac {f^{(n)}(\tau )}{(t-\tau )^{q-n+1}}}d\tau \end{aligned}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Definitions_via_transforms">Definitions via transforms</h2></div>

<p>The definitions of fractional derivatives given by Liouville, Fourier, and Grunwald and Letnikov coincide.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> They can be represented via Laplace, Fourier transforms or via Newton series expansion.

</p><p>Recall the <a href="Continuous_Fourier_transform" class="mw-redirect" title="Continuous Fourier transform">continuous Fourier transform</a>, here denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

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</math></span><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle F(\omega )={\mathcal {F}}\{f(t)\}={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}\,dt}</annotation>

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</p><p>Using the continuous Fourier transform, in Fourier space, differentiation transforms into a multiplication:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\left[{\frac {df(t)}{dt}}\right]=i\omega {\mathcal {F}}[f(t)]}</annotation>

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</p><p>So,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{n}f(t)}{dt^{n}}}={\mathcal {F}}^{-1}\left\{(i\omega )^{n}{\mathcal {F}}[f(t)]\right\}}</annotation>

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which generalizes to

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}f(t)={\mathcal {F}}^{-1}\left\{(i\omega )^{q}{\mathcal {F}}[f(t)]\right\}.}</annotation>

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</p><p>Under the <a href="Bilateral_Laplace_transform" class="mw-redirect" title="Bilateral Laplace transform">bilateral Laplace transform</a>, here denoted by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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</math></span><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"></span> and defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}[f(t)]=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt}">

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}[f(t)]=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}\left[{\frac {df(t)}{dt}}\right]=s{\mathcal {L}}[f(t)].}</annotation>

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</p><p>Generalizing to arbitrary order and solving for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{q}f(t)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}f(t)={\mathcal {L}}^{-1}\left\{s^{q}{\mathcal {L}}[f(t)]\right\}.}</annotation>

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</p><p>Representation via Newton series is the Newton interpolation over consecutive integer orders:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{q}f(t)=\sum _{m=0}^{\infty }{\binom {q}{m}}\sum _{k=0}^{m}{\binom {m}{k}}(-1)^{m-k}f^{(k)}(x).}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}f(t)=\sum _{m=0}^{\infty }{\binom {q}{m}}\sum _{k=0}^{m}{\binom {m}{k}}(-1)^{m-k}f^{(k)}(x).}</annotation>

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</p><p>For fractional derivative definitions described in this section, the following identities hold:

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}(t^{n})={\frac {\Gamma (n+1)}{\Gamma (n+1-q)}}t^{n-q}}</annotation>

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</math></span><img src="./12bf525246668f90b28594c7775d8794ff418058.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.131ex; height:6.509ex;" alt="{\displaystyle \mathbb {D} ^{q}(t^{n})={\frac {\Gamma (n+1)}{\Gamma (n+1-q)}}t^{n-q}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}(\sin(t))=\sin \left(t+{\frac {q\pi }{2}}\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} ^{q}(e^{at})=a^{q}e^{at}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Basic_formal_properties">Basic formal properties</h2></div>

<ul><li><i><a href="Linear_operator" class="mw-redirect" title="Linear operator">Linearity</a> rules</i> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} ^{q}(f+g)=\mathbb {D} ^{q}(f)+\mathbb {D} ^{q}(g)}">

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<li><i>Product rule</i> <span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {D} _{t}^{q}(fg)=\sum _{j=0}^{\infty }{q \choose j}\mathbb {D} _{t}^{j}(f)\mathbb {D} _{t}^{q-j}(g)}">

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</mrow>

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</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {D} _{t}^{q}(fg)=\sum _{j=0}^{\infty }{q \choose j}\mathbb {D} _{t}^{j}(f)\mathbb {D} _{t}^{q-j}(g)}</annotation>

</semantics>

</math></span></span></li></ul>

<p>In general, <i>composition (or <a href="Semigroup" title="Semigroup">semigroup</a>) rule</i> is a desirable property, but is hard to achieve mathematically and hence is <b>not always completely satisfied</b> by each proposed operator;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> this forms part of the decision making process on which one to choose:

</p>

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</mrow>

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</msup>

<msup>

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</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbb {D} ^{a}\mathbb {D} ^{b}f=\mathbb {D} ^{a+b}f}</annotation>

</semantics>

</math></span><img src="./e1e6ac9346a88a02cceb2cb00cc03bf70b230ad7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.815ex; height:3.009ex;" alt="{\textstyle \mathbb {D} ^{a}\mathbb {D} ^{b}f=\mathbb {D} ^{a+b}f}" loading="lazy"></span> (ideally)</li>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

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<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbb {D} ^{a}\mathbb {D} ^{b}f\neq \mathbb {D} ^{a+b}f}</annotation>

</semantics>

</math></span><img src="./80c4bd7169c2bdbe88e181c6679b459b3fa93e7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.815ex; height:3.176ex;" alt="{\textstyle \mathbb {D} ^{a}\mathbb {D} ^{b}f\neq \mathbb {D} ^{a+b}f}" loading="lazy"></span> (in practice)</li></ul>

<ul><li><a href="Fractional-order_integrator" title="Fractional-order integrator">Fractional-order integrator</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFHerrmann2011" class="citation book cs1">Herrmann, Richard (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA11"><i>Fractional Calculus: An Introduction for Physicists</i></a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789814551076</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">See <cite id="CITEREFHerrmann2011" class="citation book cs1">Herrmann, Richard (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mPXzp1f7ycMC&pg=PA11"><i>Fractional Calculus: An Introduction for Physicists</i></a>. p. 16. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789814551076</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">See <cite id="CITEREFKilbasSrivastavaTrujillo2006" class="citation book cs1">Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uxANOU0H8IUC&pg=PA75">"2. Fractional Integrals and Fractional Derivatives §2.1 Property 2.4"</a>. <i>Theory and Applications of Fractional Differential Equations</i>. Elsevier. p. 75. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780444518323</bdi>.</cite></span>

</li>

</ol></div></div>

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</style><div class="refbegin" style="">

<ul><li><cite id="CITEREFMiller1993" class="citation book cs1">Miller, Kenneth S. (1993). Ross, Bertram (ed.). <i>An Introduction to the Fractional Calculus and Fractional Differential Equations</i>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-58884-9</bdi>.</cite></li>

<li><cite id="CITEREFOldhamSpanier1974" class="citation book cs1">Oldham, Keith B.; Spanier, Jerome (1974). <i>The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order</i>. Mathematics in Science and Engineering. Vol. V. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-525550-0</bdi>.</cite></li>

<li><cite id="CITEREFPodlubny1998" class="citation book cs1">Podlubny, Igor (1998). <i>Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications</i>. Mathematics in Science and Engineering. Vol. 198. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-558840-2</bdi>.</cite></li>

<li><cite id="CITEREFCarpinteriMainardi1998" class="citation book cs1">Carpinteri, A.; Mainardi, F., eds. (1998). <i>Fractals and Fractional Calculus in Continuum Mechanics</i>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-211-82913-X</bdi>.</cite></li>

<li><cite id="CITEREFMainardi2010" class="citation book cs1">Mainardi, F. (2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120519174508/http://www.worldscibooks.com/mathematics/p614.html"><i>Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models</i></a>. Imperial College Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84816-329-4</bdi>. Archived from <a rel="nofollow" class="external text" href="http://www.worldscibooks.com/mathematics/p614.html">the original</a> on 2012-05-19.</cite></li>

<li><cite id="CITEREFTarasov2010" class="citation book cs1">Tarasov, V.E. (2010). <a rel="nofollow" class="external text" href="https://www.springer.com/physics/complexity/book/978-3-642-14003-7"><i>Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media</i></a>. Nonlinear Physical Science. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-14003-7</bdi>.</cite></li>

<li><cite id="CITEREFUchaikin2012" class="citation book cs1">Uchaikin, V.V. (2012). <a rel="nofollow" class="external text" href="https://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-642-33910-3"><i>Fractional Derivatives for Physicists and Engineers</i></a>. Nonlinear Physical Science. Springer. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013fdpe.book.....U">2013fdpe.book.....U</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-33910-3</bdi>.</cite></li>

<li><cite id="CITEREFWestBolognaGrigolini2003" class="citation book cs1">West, Bruce J.; Bologna, Mauro; Grigolini, Paolo (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EgyTpQZOga0C&pg=PR7"><i>Physics of Fractal Operators</i></a>. Springer Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95554-2</bdi>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/FractionalCalculus.html">MathWorld – Fractional calculus</a></li>

<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/FractionalDerivative.html">MathWorld – Fractional derivative</a></li>

<li>Specialized journal: <a rel="nofollow" class="external text" href="http://www.diogenes.bg/fcaa/">Fractional Calculus and Applied Analysis (1998-2014)</a> and <a rel="nofollow" class="external text" href="http://www.degruyter.com/view/j/fca">Fractional Calculus and Applied Analysis (from 2015)</a></li>

<li>Specialized journal: <a rel="nofollow" class="external text" href="https://archive.today/20120712033445/http://fde.ele-math.com/">Fractional Differential Equations (FDE)</a></li>

<li>Specialized journal: <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180421124535/http://www.nonlinearscience.com/journal_2218-3892.php">Communications in Fractional Calculus</a> (<a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2218-3892">2218-3892</a>)</li>

<li>Specialized journal: <a rel="nofollow" class="external text" href="http://fcag-egypt.com/Journals/JFCA/">Journal of Fractional Calculus and Applications (JFCA)</a></li>

<li><cite id="CITEREFLorenzoHartley2002" class="citation web cs1">Lorenzo, Carl F.; Hartley, Tom T. (2002). <a rel="nofollow" class="external text" href="https://www.techbriefs.com/component/content/article/tb/techbriefs/information-sciences/2264">"Initialized Fractional Calculus"</a>. <i>Information Technology</i>. Tech Briefs Media Group.</cite></li>

<li><a rel="nofollow" class="external free" href="https://web.archive.org/web/20040502170831/http://unr.edu/homepage/mcubed/FRG.html">https://web.archive.org/web/20040502170831/http://unr.edu/homepage/mcubed/FRG.html</a></li>

<li><a rel="nofollow" class="external text" href="http://www.tuke.sk/podlubny/fc_resources.html">Igor Podlubny's collection of related books, articles, links, software, etc. </a></li>

<li><cite id="CITEREFPodlubny2002" class="citation journal cs1">Podlubny, I. (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060407100616/http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf">"Geometric and physical interpretation of fractional integration and fractional differentiation"</a> <span class="cs1-format">(PDF)</span>. <i>Fractional Calculus and Applied Analysis</i>. <b>5</b> (4): <span class="nowrap">367–</span>386. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.CA/0110241">math.CA/0110241</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001math.....10241P">2001math.....10241P</a>. Archived from <a rel="nofollow" class="external text" href="http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2006-04-07<span class="reference-accessdate">. Retrieved <span class="nowrap">2004-05-18</span></span>.</cite></li>

<li><cite id="CITEREFZavada1998" class="citation journal cs1">Zavada, P. (1998). "Operator of fractional derivative in the complex plane". <i>Communications in Mathematical Physics</i>. <b>192</b> (2): <span class="nowrap">261–</span>285. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/funct-an/9608002">funct-an/9608002</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998CMaPh.192..261Z">1998CMaPh.192..261Z</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs002200050299">10.1007/s002200050299</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1201395">1201395</a>.</cite></li></ul></div><div><div class="zim-footer">

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