Absolute value in double integral

Question

Would be appreciated if anyone could shed some lights on how to solve the double integral with absolute value in it.

\begin{align} \int_0^t\:du\int_0^\infty e^{-\mu\left|u-s\right|}Ae^{-\lambda s}\:ds,\:\:\left(\mu\neq \lambda\right).\tag{1} \end{align}

Thanks in advance!

Answer 1

I assume that $A$ is a constant. You divide the inner integral into the following:

$$ \int_0^{+\infty}e^{-\mu|u-s|}Ae^{-\lambda s}\,ds = \int_{0}^ue^{-\mu(u-s)}Ae^{-\lambda s}\,ds+\int_u^{+\infty}e^{-\mu(s-u)}Ae^{-\lambda u}\,ds $$ and just integrate. It seems that, in addition to your condition on $\mu$ and $\lambda$, (the real part of) $\mu+\lambda$ must be positive for the integral to converge.