Consider $f(z)=\int_0^{\infty} \frac{e^{-t}}{t+z}dt.$
I'm trying to determine where this function has branch points, define suitable branch cuts, and determine the discontinuity across the cut. First of all, I believe it has an essential singularity along the entire negative real axis, correct?
I'm not sure how to handle the branch points. Intuitively it seems that 0 is a branch point, but I'm not sure what the cut or discontinuity would be--or how to justify it.