- #1
shebbbbo
- 17
- 0
Homework Statement
Find the Inverse Laplace Transform of
[itex]\frac{1}{s}[/itex]*[itex]\frac{\sqrt{s}-1}{\sqrt{s}+1}[/itex]
The Attempt at a Solution
for this question i found the singularities to be at 0 and when s = 1. (as the sqrt of 1 is ± 1) there is also a branch point that runs from 0→-∞. so if you take a contour that runs vertically to the right of all singularities. then arcs down towards the axis then along the branch point around the singularity at s=0, then back along the branch point and arc back towards the the vertical part, you should avoid crossing any branch points. can you take the contour integral over each part of the contour and then use the residue theorem to account for the singularity at s=1?
the other bit i am stuck on is the integral that runs either side of the branch point. i get:
- [itex]\int[/itex] e-xt* ((√x)i-1/ x(√x)+1) dx
and I am not sure if this is right or how to integrate from ∞ to 0
thanks for any help