- #1
moo5003
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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.