1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Singularities physics problem Date
Convergence of a series with n-th term defined piecewise Feb 26, 2018
Singularity of e^(-1/z^2) Feb 1, 2018
Laurent series of z^2sin(1/(z-1)) Jan 11, 2018
Finding regular singular point Nov 13, 2017