I found the following contour integration done in an article, but I do not fully comprehend what has actually been calculated here? Contour integration
The argument of the function $s$ is supposed to be a positive number.
I included a small picture showing that they start with an integral for $c>0$ parallel to the imaginary axis and end up with a series (apparently from the residue theorem) and a contour integral encapsulating the negative real axis.
So the denominator of the initial integral has a pole at $t=0$. It appears as if they apply the residue theorem (to which integral?) and are left with some other integral around this negative real axis.
Is anyboy here able to provide a few details of what has been done?
