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Finding singularities

  1. Oct 17, 2009 #1
    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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  2. jcsd
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