Yes, z = 0 is a (simple) pole of . The principle part of the Laurent series terminates after 1 term: Wolfram|Alpha
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 . 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
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.
I'll add to this. RS probably means removable singularity.
To complete the taxonomy begun by HoI: An isolated singular point of is removable if the Laurent series expansion of about has no principle part (that is, there are no negative powers).
Alternatively, An isolated singular point of is removable if exists and is finite.
eg. has a removable singularity at .