For \(0<\alpha \leq 2\), the \(\alpha\)-stable distribution on the real line is symmetric with characteristic function \(e^{-|\omega|^{\alpha}}\). For the sub-range \(0<\alpha<1\), Feller (1971, eqn. 6.5) gives the series expansion for the density

\[
p(t ; \alpha)=\Re \frac{i}{\pi t} \sum_{k=0}^{\infty}(-1)^{k+1} \frac{\Gamma(k \alpha+1)}{k !} t^{-k \alpha} e^{-\pi i k \alpha / 2}
\]

which is convergent for \(t>0\). The goal of this exercise is to simplify the density for \(\alpha=1 / 2\) by splitting the sum into four parts according to \(k(\bmod 4)\). Show that one of the four parts is zero, that the odd parts may be combined into a multiple of \(t^{-3 / 2} \sin (1 /(4 t)+\pi / 4)\), and that the remaining part is \(O\left(t^{-2}ight)\) as \(t ightarrow \infty\).