Classifying Singularities and the Laurent Series

[Physgeek64]

Homework Statement

Classify the singularities of ##\frac{1}{z^2sinh(z)}## and describe the behaviour as z goes to infinity

Find the Laurent series of the above and find the region of convergence

Homework Equations

N/A

The Attempt at a Solution

I thought these two were essentially the same thing, but I'm being given a fair few marks for each question, so I'm convinced I must be over-simplifying it

For the first part, I used to Laurent expansion to get:

##\frac{1}{z^2sinh(z)} = \frac{1}{z^2(z+\frac{z^3}{3!}+\frac{z^5}{5!}+...)}##
##\frac{1}{z^2(z+\frac{z^3}{3!}+\frac{z^5}{5!}+...)}= \frac{1}{z^3(1+\frac{z^2}{3!}+\frac{z^4}{5!}+...)}##
##\frac{1}{z^3(1+\frac{z^2}{3!}+\frac{z^4}{5!}+...)}=\frac{1}{z^3}(1+\frac{z^2}{3!}+\frac{z^4}{5!}+...)^{-1}##
##\frac{1}{z^3}(1+\frac{z^2}{3!}+\frac{z^4}{5!}+...)^{-1}=\frac{1}{z^3}(1-\frac{z^2}{3!}-\frac{z^4}{5!}+(\frac{z^2}{3!}+\frac{z^4}{5!})^2+...)##
##= \frac{1}{z^3} -\frac{1}{6z}+\frac{3z}{40}+...##

Hence, I got there to be an isolated singularity of order 3 at z=0, a simple pole of at z=0 and an essential singularity at ##z=\infinity##

So the behaviour as z goes to infinity is that the function is undefined? (I showed this by redefining a new variable ##\frac{1}{z}##

I just can't see what else to do, but for the number of marks it is worth, I don't feel I have done enough work

Also, I am not sure how to find the region of convergence

Many thanks

EDIT: Just had a thought- For the first part I could have written ##\frac{1}{z(i)(iz-\frac{1}{zi})}## ?

 

[stevendaryl]

Just a question for you: Is there only one value of [itex]z[/itex] where [itex]sinh(z) = 0[/itex]?

 

[Physgeek64]

Just a question for you: Is there only one value of [itex]z[/itex] where [itex]sinh(z) = 0[/itex]?

Oh no of course! There are also poles every ##n \pi i## With order ##\frac{1}{n^2 \pi^2}## ?

 

[stevendaryl]

Oh no of course! There are also poles every ##n \pi i## With order ##\frac{1}{n^2 \pi^2}## ?

Okay, that needs to be part of the answer to your homework problem. What is the expansion about [itex]z=n\pi\ i[/itex] for [itex]n \neq 0[/itex]?

 

[Physgeek64]

Okay, that needs to be part of the answer to your homework problem. What is the expansion about [itex]z=n\pi\ i[/itex] for [itex]n \neq 0[/itex]?

Can we not just evaluate it at these points? Since the rest of the function will be analytic?

 

[stevendaryl]

Can we not just evaluate it at these points? Since the rest of the function will be analytic?

You wrote down an expansion in powers of [itex]z[/itex]. You need an expansion in powers of [itex]z-n\pi i[/itex] for the other poles.

If you have a pole at [itex]z=p[/itex], then the Laurent series is in powers of [itex](z-p)[/itex], not powers of [itex]z[/itex]

 

[Physgeek64]

You wrote down an expansion in powers of [itex]z[/itex]. You need an expansion in powers of [itex]z-n\pi i[/itex] for the other poles.

If you have a pole at [itex]z=p[/itex], then the Laurent series is in powers of [itex](z-p)[/itex], not powers of [itex]z[/itex]

Okay, so we can expand ##\frac {1}{z}= \frac{1}{n \pi i +z- n \pi i} = \frac{1}{n \pi i} (1+\frac{z-n \pi i}{n\pi i})^{-1}##
## = \frac{1}{n \pi i}(1-\frac{z-n \pi i}{n\pi i}+\frac{(z-n \pi i)^2}{(n\pi i)^2}+...)##

How would we expand sinhx such that the brackets of ##z-n \pi i## remain in the denominator

Many thanks

 

[stevendaryl]

Okay, so we can expand ##\frac {1}{z}= \frac{1}{n \pi i +z- n \pi i} = \frac{1}{n \pi i} (1+\frac{z-n \pi i}{n\pi i})^{-1}##
## = \frac{1}{n \pi i}(1-\frac{z-n \pi i}{n\pi i}+\frac{(z-n \pi i)^2}{(n\pi i)^2}+...)##

How would we expand sinhx such that the brackets of ##z-n \pi i## remain in the denominator

Many thanks

Well, since [itex]sinh[/itex] is periodic, the expansion in powers of [itex]z[/itex] is the same as the expansion in terms of [itex]z - n \pi i[/itex].

 

[Physgeek64]

Well, since [itex]sinh[/itex] is periodic, the expansion in powers of [itex]z[/itex] is the same as the expansion in terms of [itex]z - n \pi i[/itex].

So ##sinhz= (z-n\pi i)+ \frac{(z-n \pi i)^3}{3!} +... ## ?

 

[stevendaryl]

So ##sinhz= (z-n\pi i)+ \frac{(z-n \pi i)^3}{3!} +... ## ?

Yes. You can see that by letting [itex]q = z - n \pi i[/itex]. Then [itex]sinh(z) = sinh(q + n \pi i) = sinh(q) = q + \frac{q^3}{3!} + ...[/itex]

 

[Physgeek64]

Yes. You can see that by letting [itex]q = z - n \pi i[/itex]. Then [itex]sinh(z) = sinh(q + n \pi i) = sinh(q) = q + \frac{q^3}{3!} + ...[/itex]

Ah okay. Does this mean that strictly speaking this only has a taylor series about ##n \pi i## ? Also when calculating the residue (slightly off topic I'm afraid) about ##n \pi i## do you need to expand ##sinhz## or can we simply 'read off' the residue?