# Inverse Laplace Transform and Branch Points

1. Oct 13, 2011

### smcro5

1. The problem statement, all variables and given/known data

Find the inverse Laplace Transform of $\frac{1}{s}\frac{\sqrt{s}-1}{\sqrt{s}+1}$

2. Relevant equations

The complex inversion formula (well known)

3. The attempt at a solution

The first thing is finding singularities and branch points and so on. From the $\frac{1}{s}$ part of the function, it seems as though s=0 is a simple pole (a pole of order one). However, it is known that each $\sqrt{s}$ has a branch point at s=0. Therefore the function has a branch point at s=0. Performing a substitution s=$\frac{1}{t}$ into $\sqrt{s}$ shows that the point at infinity is a branch point as well. I am about to start using the complex inversion formula, but am not sure about whether I have taken into account all the possible branch points/singularities.

Any ideas guys?

2. Oct 14, 2011

### jackmell

Isn't s=1 a singular point? That is, what is the multi-valued square root of 1?