Consider the expression appearing in the Levy-Kchinchine formula:
$$\varphi(t) = e^{ita - \sigma^2/(2t)+\int_{-\infty}^\infty [e^{itx} - 1 -itx \mathbf{1}_{|x|<1}]\,{\rm d}\nu(x)},$$ but with $a =0, \sigma = 0$ and $d\nu(x) = \frac{c}{|x|^{1+\alpha}}$, i.e., $$\varphi(t) = e^{\int_{-\infty}^\infty [e^{itx} - 1 -itx \mathbf{1}_{|x|<1}]\frac{c}{|x|^{1+\alpha}}\,{\rm d}x},$$
Is there a way to confirm, without going through tedious integral computations that it corresponds to the characteristic function of a symmetric $\alpha$-stable distribution?
I got lost in computations when arrived at $\Gamma(-\alpha)$ floating around.