While solving a partial differential equation, I came across the following four functions:
$$ F_1(s) = e^{ k \sqrt{ q^2 + cs^2 } } \tag 1$$ $$ F_2(s) = e^{ -k \sqrt{ q^2 + cs^2 } } \tag 2$$ $$ F_3(s) = e^{ k \sqrt{ q^2 - cs^2 } } \tag 3$$ $$ F_4(s) = e^{ -k \sqrt{ q^2 - cs^2 } } \tag 4$$
where $k$, $q$ and $c$ are constants, and $s$ is the Laplace transformation variable. Because these are not tabular expressions for the inverse Laplace transformation, I started researching ways to evaluate the inverse Laplace transformations in such cases.
Currently, I am studying the residue theorem (along with other parts of complex analysis). According to that theorem, I need to find the poles of my function so I can use it. As I understand it, poles are points at which functions are not defined. This is a problem for my case because all four functions I wrote down are defined for every value of $s$. Does this mean I can not find the inverse Laplace transform for any one of them by using the residue theorem?
To summarize, I am trying to find the inverse Laplace transformation of the functions $(1)$, $(2)$, $(3)$, and $(4)$. My questions are:
$1.$ Can I use the residue theorem to do that (at least for one of them)?
$2.$ If not, is there some other way I can find the inverse Laplace transformation of these functions?