\qquad q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Psi}_{c}^{+}(i,j;q_1,\ldots,q_n)\end{aligned}$$ are well-defined for $q_{k}\in\bC^+:= {\{z\in\bC {\;|\;}\Re(z)>0\}},k=1,\ldots,n$, as being the Laplace transforms of $\Pi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})$ and $\Psi_{c}^{+}(i,j;t_{1},\ldots,t_{1}+\ldots+t_{n})$, respectively.

All the above leads to the following result, which is our main theorem, and where we make use of the inverse multivariate Laplace transform. We refer to the Appendix for the definition and the properties of the inverse multivariate Laplace transform relevant to our set-up.

\[th:main\] We have that

$$\begin{aligned}
\label{eq:invLaplacePi+}
\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})=\cL^{-1}\left(q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)\right)(s_{1},s_{2}-s_{1},\ldots,s_{n}-s_{n-1}),\end{aligned}$$

for any $i\in\mathbf{E}^{-}$, $j\in\mathbf{E}^{+}$, and

$$\begin{aligned}
\label{eq:invLaplacePsi+}
\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})=\cL^{-1}\left(q_{1}^{-1}\cdots q_{n}^{-1}\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)\right)(s_{1},s_{2}-s_{1},\ldots,s_{n}-s_{n-1}),\end{aligned}$$

for any $t>0$, $i,j\in\mathbf{E}^{+}$, where $\cL^{-1}$ is the inverse multivariate Laplace transform.

It needs to be stressed that we can compute the values of $\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$ and $\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)$ only for positive values of $q_{i}$’s. Thus, Theorem \[th:main\] may not be directly applied to compute $\Pi_{c}^{+}(i,j;s_{1},\ldots,s_{n})$ and $\Psi_{c}^{+}(t,i,j;s_{1},\ldots,s_{n})$. However, we can approximate these functions, as explained in Section \[subsubsec:specialcase\] by using only the values of $\widehat{\Pi}_{c}^{+}(i,j;q_1,\ldots,q_n)$ and $\widehat{\Psi}_{c}^{+}(t,i,j;q_1,\ldots,q_n)$ for positive values of $q_{i}$’s.

Numerical Example {#sec:Numerical}
=================

In this section we will illustrate our theoretical results with a simple, but telling example. We first describe a numerical method to approximate $\Pi_{c}^{+}$ and $\Psi_{c}^{+}$, and then we proceed with its application to a concrete example.

Numerical Procedure to approximate $\Pi_{c}^{+}$ and $\Psi_{c}^{+}$
