Understanding Singularities in Complex Functions

[moo5003]

Homework Statement

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 can't 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 couldn't 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.

 

[Dick]

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

 

[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?