From the cosine integral (int cos (omega t) e^{-|omega|^{alpha}} d omega), deduce that the (alpha)-stable density has

Question:

From the cosine integral $\int \cos (\omega t) e^{-|\omega|^{\alpha}} d \omega$, deduce that the $\alpha$-stable density has a Taylor series at the origin which begins

\[
\log p(t ; 1 / 2)=\text { const }-60 t^{2}+O\left(t^{4}ight) \text {. }
\]

Find the general term in this expansion and deduce the radius of convergence.

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