What is the Nature of Singularity in the Function f(x)=exp(-1/z)?

[ion santra]

what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number?
now i arrive at two different results by progressing in two different ways.
1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity.
2) now again if i take the limit of the function from either sides of zero, i see it exists finitely, therefore not a singular point at all.

 

[micromass]

It is an essential singularity.

2) now again if i take the limit of the function from either sides of zero, i see it exists finitely, therefore not a singular point at all.

Can you expand on this? Don't forget that $z$ is a complex number, so just taking the two sided limits in $\mathbb{R}$ is not good enough.

 

[mathman]

what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number?
now i arrive at two different results by progressing in two different ways.
1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity.
2) now again if i take the limit of the function from either sides of zero, i see it exists finitely, therefore not a singular point at all.

For Real z < 0, the limit as z ->0 is infinite.