Understanding Singularities in Complex Functions

In summary, the conversation discusses determining the type of singularities (removable, pole, or essential) for three different functions. Part a is easily simplified to show a removable singularity at z=0. In part b, the attempt to use power series to show a removable singularity at z=0 is unsuccessful. The conversation then suggests considering the limits as z approaches 0 from negative and positive real values for part c.
  • #1
moo5003
207
0

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. :bugeye:
 
Physics news on Phys.org
  • #2
For b) your idea to use series is fine. Just put in the expansion of e^z. What's the problem?
 
  • #3
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?
 

1. What is a singularity in physics?

A singularity in physics is a point in space where the laws of physics break down and our current mathematical models are unable to explain what is happening. It is often associated with extreme conditions such as infinite density, temperature, or curvature.

2. How are singularities related to black holes?

Black holes are one of the most well-known examples of singularities in physics. They occur when a massive star collapses under its own gravity, creating a singularity at the center. The singularity is surrounded by an event horizon, which is the point of no return for anything entering the black hole.

3. Are singularities real or just theoretical concepts?

Singularities are currently considered to be theoretical concepts, as they cannot be directly observed or measured. However, they are predicted by our current understanding of physics and play a crucial role in theories such as general relativity and quantum mechanics.

4. Can singularities be resolved or avoided?

There are some theories that suggest ways to resolve or avoid singularities, such as string theory and loop quantum gravity. However, these are still under development and require further research and experimentation to be confirmed.

5. What are the implications of singularities for our understanding of the universe?

Singularities challenge our current understanding of the laws of physics and the behavior of matter in extreme conditions. They also play a crucial role in theories of the origin of the universe, such as the Big Bang theory. Further research on singularities could potentially lead to a better understanding of the fundamental laws of the universe.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
934
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
355
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
812
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
Back
Top