File:Fractional Derivative-Weyl.gif

f = E^-x^2;
coefficients = Table[Chop[{k, Evaluate[InverseFourierTransform[f, x, k]]}], {k, -5, 5, 0.1}];
frames1 = Table[Plot[{f, Evaluate[D[f, x]], Evaluate[D[f, {x, 2}]], 1/Sqrt[2 \[Pi] Dimensions[coefficients][[1]]] Sum[(I coefficients[[j, 1]])^\[Alpha] coefficients[[j, 2]] E^(I coefficients[[j, 1]] x), {j, 1,         Dimensions[coefficients][[1]]}]}, {x, -4, 4}, PlotStyle -> {Directive[Black, Dashed], Directive[Gray, Dashed], Directive[LightGray, Dashed], Purple}, PlotLegends -> LineLegend[{"f=\!\(\*SuperscriptBox[\(e\), \
\(-\*SuperscriptBox[\(x\), \(2\)]\)]\)", "\!\(\*SubscriptBox[\(\[PartialD]\), \(1\)]\) f", "\!\(\*SubscriptBox[\(\[PartialD]\), \(2\)]\) f", StringForm["\!\(\*SubscriptBox[\(\[PartialD]\), \(``\)]\) f",         NumberForm[\[Alpha], {2, 1}]]}], PlotRange -> {-2, 2}, PlotLabel -> "\!\(\*SubscriptBox[\(\[PartialD]\), \\(\[Alpha]\)]\)f(x)=\!\(\*SuperscriptBox[\(F\), \\(-1\)]\)[(ik\!\(\*SuperscriptBox[\()\), \(\[Alpha]\)]\)F[f]]", LabelStyle -> {Bold, Black}], {\[Alpha], 0.00001, 2, 0.02}];
ListAnimate[Join[Table[frames1[[1]], 4], frames1, Table[frames1[[-1]], 4], Reverse[frames1]]]