1. ## pole or singularities

hi ,

is exp z / sin z a pole or singularities or neither ??

applying pole test,

lim z-->0

(z-0)( exp z / sin z)

= z*expz / sin z

sub 0 into equation, achieve 0 / 0,

using hopital rule, i get ( z*exp z + exp z ) / cos z

= 1 <--- can i conclued it's a pole ?

If i have to apply hopital rule and achieve a constant, does that mean that it's always a RS ??

many thanks ?

2. Originally Posted by Chris0724
hi ,

is exp z / sin z a pole or singularities or neither ??

applying pole test,

lim z-->0

(z-0)( exp z / sin z)

= z*expz / sin z

sub 0 into equation, achieve 0 / 0,

using hopital rule, i get ( z*exp z + exp z ) / cos z

= 1 <--- can i conclued it's a pole ?

If i have to apply hopital rule and achieve a constant, does that mean that it's always a RS ??

many thanks ?
Yes, z = 0 is a (simple) pole of $f(z) = \frac{e^z}{\sin z}$. The principle part of the Laurent series terminates after 1 term: Wolfram|Alpha

3. Originally Posted by mr fantastic
Yes, z = 0 is a (simple) pole of $f(z) = \frac{e^z}{\sin z}$. The principle part of the Laurent series terminates after 1 term: Wolfram|Alpha
hi mr fantastic,

thanks a lot. but another question,

if i were to apply L'hoptial rule, it doesn't necessary mean to be a RS , right ?

For question asking the nature of singularties

e.g

sin z / z

1st i try to use the pole test which gave me 0 which is not a pole but suspected to be a RS,

2nd , sub in 0 to get 0 / 0 then use hoptial rule to achieve a 1.

but for equation unable to get 0 / 0 is it conside as neither ?

many thanks

4. First, a pole is a "singularity". Did you mean an "essential singularity"? I don't know what you mean by "RS".

A function is "analytic" at a point if it can be written as a power series with only non-negative powers. It has a "pole" at at point if it can be written as a power series with a finite number of negative powers. It has an "essential singularity" at a point if it can be only be written as a power series with an infinite number of negative powers.

I have no idea what you mean by "applying L'Hopital's rule". Whether a point is a pole or an essential singularity will have a limit of infinity as you approach the point so "L'Hopital's rule" does not apply.

5. Originally Posted by HallsofIvy
First, a pole is a "singularity". Did you mean an "essential singularity"? I don't know what you mean by "RS".

A function is "analytic" at a point if it can be written as a power series with only non-negative powers. It has a "pole" at at point if it can be written as a power series with a finite number of negative powers. It has an "essential singularity" at a point if it can be only be written as a power series with an infinite number of negative powers.

[snip]
I'll add to this. RS probably means removable singularity.

To complete the taxonomy begun by HoI: An isolated singular point $z_0$ of $f(z)$ is removable if the Laurent series expansion of $f(z)$ about $z = z_0$ has no principle part (that is, there are no negative powers).

Alternatively, An isolated singular point $z_0$ of $f(z)$ is removable if $\lim_{z \rightarrow z_0} f(z)$ exists and is finite.

eg. $f(z) = \frac{\sin z}{z}$ has a removable singularity at $z = 0$.