Essential singularity In shaum's outline complex analysis,definition of essential point is:
An isolated singularity that is not pole or removable singularity is called essential singularity
Now in the same book there is an excercise that;
Locate and name the singularity of sec(1/z)........it says that z=0 is essential singularity...but also it is non isolated...
I wonder that how can an essential singularity be a non isolated as according to definition essential singularity is isolated,I know that singularity z=0 is non isolated,but from my point of view it should not be essential as it not isolated
 A: There are two points of view here, which are frequently confused in textbooks.


*

*The name "essential singularity" is used only for analytic functions (whose image is in C), with isolated singularities. Then $\sec(1/z)$ has a non-isolated singularity at 0.

*In another context one considers meromorphic functions (as holomorphic maps to
the Riemann sphere). From this point of view, poles are not singularities at all,
and $\sec(1/z)$ has an essential singularity at $0$.
Text books usually gloss over this fine distinction. As well as over the question
whether "removable singularities" are singularities or not. For example,
what is the singularity of $$\frac{\sin(2/z)}{\sin(1/z)}=2\cos(1/z)$$ at zero?
Is it essential, or non-isolated (limit of removable singularities)?
Similar confusion applies to the expression "there exists a limit". The authors frequently forget to specify where this limit is supposed to be.
If the functions are supposed to map the domain into $C$ then he limit at a pole does not exist. If they are considered as maps to the Riemann sphere, it exists, and the difference between a pole and "removable singularity" disappears.
