(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Singularities physics problem

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