this is a post connected to my original question; see here for full explaination: Solution to diffusion/Smoluchowski equation using an inverse Laplace transform
Basically, I am working on a new way to solve the diffusion-Smoluchowski equation using complex analysis. My issue lies in obtaining the Ornstein-Uhlenbeck solution. In this particular instance, one has an exponent with a factor of \begin{equation} \operatorname{Arsinh}(\tfrac{x_1}{\sqrt{H}}) = \log \left(\cfrac{x_1 + \sqrt{H+x_1^2}}{\sqrt{H}} \right) \end{equation} to deal with, where $H$ is our Laplace parameter and a Bromwich integral is needed. I am confident with handling the other terms in the integral, but I think this one requires more delicate care. I think we have 3 branch points, $0, -x_1^2$ and an $\infty$. So far, I have been plugging in the values of the square roots when approaching above and below cuts, which I have chosen to start at 0 and span to negative real infinity, giving $\pm \exp(i \pi/2)$ factors attached to the square roots. So my questions are: Can I chose my branch to span to negative infinity in this case? Also, even how I have written this $\log$ term leaves me slightly uneasy. I think this logarithm factor may have something more to say about the phase on either side of the branch cut.
