Singularities physics problem

1. Apr 9, 2007

moo5003

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.

2. Apr 9, 2007

Dick

For b) your idea to use series is fine. Just put in the expansion of e^z. What's the problem?

3. Apr 9, 2007

Dick

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?