$\mathcal{F}_t\left[E_{\alpha }\left(-\frac{\alpha t^{\alpha }}{1-\alpha
}\right)\right](w)=\\\mathcal{F}_t\left[\sum _{k=0}^{\infty } \frac{(-1)^k t^{k \alpha } (1-\alpha
)^{-k} \alpha ^k}{\Gamma (1+k \alpha )}\right](w)=\\\sum _{k=0}^{\infty }
\mathcal{F}_t\left[\frac{(-1)^k t^{k \alpha } (1-\alpha )^{-k} \alpha ^k}{\Gamma (1+k \alpha
)}\right](w)=\\\sum _{k=0}^{\infty } \frac{(-1)^k e^{\frac{1}{2} i k \pi \alpha } (1-\alpha )^{-k}
\alpha ^k \left| w\right| ^{-1-k \alpha } (-1+\text{sgn}(w)) \sin (k \pi \alpha )}{\sqrt{2 \pi
}}=\\\frac{i \left(-1+e^{2 i \pi \alpha }\right) (-1+\alpha ) \alpha \left| w\right| ^{-1+\alpha }
(-1+\text{sgn}(w))}{2 \sqrt{2 \pi } \left(e^{\frac{3 i \pi \alpha }{2}} \alpha -(-1+\alpha ) \left|
w\right| ^{\alpha }\right) \left(-\alpha +e^{\frac{i \pi \alpha }{2}} (-1+\alpha ) \left| w\right|
^{\alpha }\right)}$
HoldForm[
FourierTransform[
MittagLefflerE[\[Alpha], -\[Alpha]/(1 - \[Alpha])*t^\[Alpha]], t,
w] == FourierTransform[
Sum[((-1)^k t^(k \[Alpha]) (1 - \[Alpha])^-k \[Alpha]^k)/
Gamma[1 + k \[Alpha]], {k, 0, Infinity}], t, w] ==
Sum[FourierTransform[((-1)^k t^(
k \[Alpha]) (1 - \[Alpha])^-k \[Alpha]^k)/Gamma[1 + k \[Alpha]],
t, w], {k, 0, Infinity}] ==
Sum[((-1)^k E^(1/2 I k \[Pi] \[Alpha]) (1 - \[Alpha])^-k \[Alpha]^
k Abs[w]^(-1 - k \[Alpha]) (-1 + Sign[w]) Sin[k \[Pi] \[Alpha]])/
Sqrt[2 \[Pi]], {k, 0, Infinity}] == (
I (-1 + E^(2 I \[Pi] \[Alpha])) (-1 + \[Alpha]) \[Alpha] Abs[
w]^(-1 + \[Alpha]) (-1 + Sign[w]))/(
2 Sqrt[2 \[Pi]] (E^((3 I \[Pi] \[Alpha])/
2) \[Alpha] - (-1 + \[Alpha]) Abs[w]^\[Alpha]) (-\[Alpha] +
E^((I \[Pi] \[Alpha])/
2) (-1 + \[Alpha]) Abs[w]^\[Alpha]))] // TeXForm
Your Answer is correct:
Check:
Needs["FourierSeries`"];
f[t_] := MittagLefflerE[\[Alpha], -\[Alpha]/(1 - \[Alpha])*t^\[Alpha]];
NFourierTransform[f[t] /. \[Alpha] -> 1/3, t, -1] // Quiet
(* 0.20671 + 0.0489825 I *)
(I (-1 + E^(2 I \[Pi] \[Alpha])) (-1 + \[Alpha]) \[Alpha] Abs[
w]^(-1 + \[Alpha]) (-1 + Sign[w]))/(
2 Sqrt[2 \[Pi]] (E^((3 I \[Pi] \[Alpha])/
2) \[Alpha] - (-1 + \[Alpha]) Abs[w]^\[Alpha]) (-\[Alpha] +
E^((I \[Pi] \[Alpha])/
2) (-1 + \[Alpha]) Abs[w]^\[Alpha])) /. \[Alpha] -> 1/3 /.
w -> -1 // N
(* 0.20671 + 0.0489825 I *)
Outputs are the same.
Regards M.I.
