Derivadas Generalizadas

ϕ[x_] := Exp[-x^2 / 2]

Plot[ ϕ[x], {x,-5,5}, PlotRange-> {0,1}]

dϕ[x_] :=D[ ϕ[x],x]

dϕ[x]

Plot[-(E^(-((x^2)/2))) x, {x,-5,5}, PlotRange->{-1,1}]

d2ϕ[x_] :=D[ dϕ[x],x]

d2ϕ[x]

Plot[-(E^(-((x^2)/2)))+(E^(-((x^2)/2))) (x^2), {x,-5,5}, PlotRange->{-1,1}]

Fϕ[k_]:= FourierTransform[ϕ[x], x, k]

Fϕ[k]

Plot[ Fϕ[k], {k,-5,5}]

InverseFourierTransform[I k (E^(-((k^2)/2))), k,x]

Plot[ (E^(-((x^2)/2))) x, {x,-5,5}]

df1x[x_] = InverseFourierTransform[I k (E^(-((k^2)/2))), k,x]

Plot[ df1x[x], {x,-5,5}]

df2x[x_] = InverseFourierTransform[(I k)^2 *  (E^(-((k^2)/2))), k,x]

Plot[ df2x[x], {x,-5,5}]

gdx[x_,s_] = InverseFourierTransform[(I k)^s *  (E^(-((k^2)/2))), k,x]

(1/(Sqrt[π]))(2^(-(1/2)+(s/2))) ((Sqrt[2]) Cos[((π s)/2)] Gamma[((1+s)/2)] Hypergeometric1F1[((1+s)/2),(1/2),-((x^2)/2)]+s x Gamma[(s/2)] Hypergeometric1F1[1+(s/2),(3/2),-((x^2)/2)] Sin[((π s)/2)])

analgdx[x_,s_]:= (1/(Sqrt[π]))(2^(-(1/2)+(s/2))) ((Sqrt[2]) Cos[((π s)/2)] Gamma[((1+s)/2)] Hypergeometric1F1[((1+s)/2),(1/2),-((x^2)/2)]+s x Gamma[(s/2)] Hypergeometric1F1[1+(s/2),(3/2),-((x^2)/2)] Sin[((π s)/2)])

Manipulate[Plot[ gdx[x,s], {x,-5,5}],{s,1,2}]

Manipulate[Plot[ analgdx[x,s], {x,-5,5}],{s,0.01,5}]