For two independent L\'evy processes $\xi$ and $\eta$ an exponentially distributed random variable $\tau$ with parameter $q>0$, of $\eta$, the killed exponential functional is given by $V_{q,\xi,\eta} := \int_0^\tau \mathrm{e}^{-\xi_{s-}} \, \mathrm{d} \eta_s$. Interpreting case $q=0$ as $\tau=\infty$, $V_{q,\xi,\eta}$ a natural generalization $\int_0^\infty \eta_s$, law which well-studied in l...