# Finding singularities

1. Oct 17, 2009

### oddiseas

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

(1/s)Ln(s^2+1)

Find and classify all singularities in the "extended" complex plane. Draw but do not evaluate the contour you would use to find the inverse laplace transform.

2. Relevant equations

(1/s)Ln(s^2+1)

3. The attempt at a solution

Ln(s^2+1)=[Ln(s+i)+Ln(s-i)](1/s)

so the obvious singularities which are branch points are at s=0,i,-i. Now i am having trouble evaluating for singularities at infinity, and in addition drawing this contour.

Letting s=1/t:
t{Ln[(1/t)+i]+Ln[(1/t)-i]

=t{Ln[(i+ti)/ti]+Ln[(i-ti)/ti)]

=t[Ln(i+ti)-Ln(ti)+Ln(i-ti)-Ln(ti)]

=t[Ln(i+ti)+Ln(i-ti)-2Ln(ti)]

now at t=0 we get:

0[Ln(i)+Ln(i)-2Ln(o)]

Now when t=o we get the point i, which is a singularity and log(0) which is undefined, so i am not sure in this example if there is a branch point at positive or negative infinity>

Can anyone show me how to confirm if there is, and how i can draw the branch cuts. Since i already have one along the y axis from i to negative i.
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