Homework Help: Describing Singularities

1. Jan 21, 2012

Ted123

1. The problem statement, all variables and given/known data

3. The attempt at a solution

Both $\displaystyle \frac{\cos(z)-1}{z^2}$ and $\displaystyle \frac{\sinh(z)}{z^2}$ have 1 singular point at $z=0$.

For (a):

z=0 is a removable singularity since defining f(0)=1 makes it analytic at all $z\in\mathbb{C}$.

z=0 is isolated since f(z) is analytic for 0<|z|<1. But z=0 is not a pole since cos(0)-1 =0, and so z=0 is an essential singularity.

For (b):

z=0 is a removable singularity since defining f(0)=1 makes it analytic at all $z\in\mathbb{C}$.

z=0 is isolated since f(z) is analytic for 0<|z|<1. But z=0 is not a pole since sinh(0)=0, and so z=0 is an essential singularity.

Is this correct?

2. Jan 21, 2012

Dick

Parts of it might be true. You said basically the same thing about both functions and you didn't prove anything you said. Give some arguments. If f(z)=sinh(z)/z^2, why does defining f(0)=1 make it analytic on C?

3. Jan 21, 2012

Ted123

Probably because I'm not understanding the definitions correctly!

These are my set of definitions:

I think for both (a) and (b), z=0 is an isolated singularity but not a pole, so an essential singularity. But they probably aren't removable.

4. Jan 21, 2012

Dick

The definitions will be clearer to you if you look at a power series expansion of each function around z=0.

5. Jan 21, 2012

Ted123

I don't like how some of these definitions are given so if I use this definition of pole:

Clearly $z_0=0$ is an isolated singularity since it is the only singularity for both (a) and (b).

(a) $\displaystyle \lim_{z\to 0} \;(z-0)^N f(z) = \lim_{z\to 0} \; z^{N-2} (\cos(z)-1) = 0 \;\; \forall \;N>0$ so $z_0=0$ is not a pole. Hence it is an essential singularity.

(b) If N=1 then $\displaystyle \lim_{z\to 0} \;(z-0) f(z) = \lim_{z\to 0} \frac{\sinh(z)}{z} = 1 \neq 0$ so [itex]z_0=0[/tex] is a simple pole (of order 1). What would be the strength of the pole? It is not an essential singularity.

I'm not understanding how to see if 0 is a removable singularity in each case?

6. Jan 21, 2012

Dick

You know how to expand cos(z) and sinh(z) in a power series around z=0. Put those expansions into the two functions and simplify. See what you think. Then look back at the definitions.