Complex analysis

[esisk]

Hello All,
Just when I thought I understood whatever there was to understand about Normal Families...

F(z) is analytic on the punctured disk and we define the sequence
f_{n}=f(z/n) for n \leq 1.

Trying (and failing) to show that {f_n} is a normal family on the punctured disk iff the singularity of f(z) at z=0 is removable or a pole

Any help is appreciated, thank you

 

[Eynstone]

Let D be a compact disk within the punctured disk. D doesn't contain the origin.
z/n -> 0 when n-> inf. So, the behaviour of {f_n} in D as in a neighbourhood of 0.
If the singularity is removable, {f_n} -> f(0) uniformly. If 0 is a pole,
{f_n} ->inf. uniformly ( because z^k .f_n(z) will be holomorphic for some k>=1).
Finally, Picard's big theorem guarantees that if 0 is an essential singularity,
f_n(z) assumes almost all values for sufficiently big n.Hence, the convergence can't be uniform.

 

[esisk]

I thank you very much Eynstone.
Need to study more...I am not prelim-ready yet
Regards

 

[esisk]

I thank you very much Eynstone.
Need to study more...I am not prelim-ready yet
Regards