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Singularities physics problem

  1. Apr 9, 2007 #1
    1. The problem statement, all variables and given/known data
    Determine if the following are removable, pole (with order), or essential singularities.

    a) f(z) = (z^3+3z-2i)/(z^2+1) a=i

    b) f(z) = z/(e^z - 1) a=0

    c) e^e^(-1/z) a=0

    2. The attempt at a solution

    Part a is pretty straightforward, just simplify it down to (z-i)(z+2i)/(z+i) and the sing is removable with value 0.

    Part b is where I'm having some trouble. I'm pretty sure its also removable since when I graphed it the limit looks like it converges to 1. Though when I expand it out into a power series I cant seem to get it to work.

    z = Sigma (0 to inf over n) delta(n-1)z^n
    delta = Kroniker delta function, 1 at delta(0) and 0 everywhere else.

    e^z = Sigma (z^n/n!)
    -1 = -Sigma (d(n)z^n)

    After failing to come up with anything usefull with that method I decided to show that the actual limit was one. I couldnt seem to come up with a delta such that given an epsilon |z|<d => |f(z) - 1|<epsilon.

    Overall, I was wondering if you guys could give me some hints on how to tackle the problem. :bugeye:
  2. jcsd
  3. Apr 9, 2007 #2


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    For b) your idea to use series is fine. Just put in the expansion of e^z. What's the problem?
  4. Apr 9, 2007 #3


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    BTW for c), you might want to consider the limits as z->0 for z negative real and z positive real. What do you learn from considering these two limits?
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