On the first passage time of a given level for processes with independent increments

The distribution of the first passage time of a non-negative level for a homogeneous process with independent increments $\xi(t)$ is studied, $\xi(t)$ having a bounded variation, and its characteristic function being of the form $\mathbf Me^{i\alpha\xi(t)}=e^{i\psi(\alpha)}$, where
$$ \psi(\alpha)=i\alpha a+\int_{-\infty}^0(e^{i\alpha x}-1)\,dM(x)+\int_0^\infty(e^{i\alpha x}-1)\,dN(x). $$

The double transformation of the distribution considered is shown to be
$$ \theta(s,\alpha)= \begin{cases} -\frac{\varkappa^+(s,0)}{\pi^+(s,\alpha)}&(a\le0), \\ -\frac1{1-i\alpha a}\cdot\frac{\varkappa^+(s,0)}{\varkappa^+(s,\alpha)}&(a>0), \end{cases} $$
where $\varkappa^+(s,\alpha)$ is determined by the factorization identity
$$ \frac{s-\psi(\alpha)}{1-i\alpha a}=\varkappa^+(s,\alpha)\varkappa^-(s,\alpha)\quad(s>0,\ -\infty<\alpha<\infty). $$