Is the Singularity of F(z) at z=0 Removable or a Pole?

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Discussion Overview

The discussion revolves around the nature of the singularity of the function F(z) at z=0, specifically whether it is removable or a pole. Participants explore the implications of this singularity type on the normality of the family of functions defined by f_{n}=f(z/n) within the context of complex analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant notes that F(z) is analytic on the punctured disk and seeks to establish the conditions under which the sequence {f_n} is a normal family.
  • Another participant explains that if the singularity at z=0 is removable, then {f_n} converges uniformly to f(0), while if it is a pole, {f_n} converges uniformly to infinity.
  • The same participant references Picard's big theorem, suggesting that if z=0 is an essential singularity, then {f_n(z)} would assume almost all values for sufficiently large n, indicating non-uniform convergence.

Areas of Agreement / Disagreement

Participants have not reached a consensus on whether the singularity is removable or a pole, and the discussion includes multiple competing views regarding the implications of the singularity type on the normality of the function family.

Contextual Notes

The discussion does not clarify certain assumptions about the behavior of F(z) near the singularity, nor does it resolve the mathematical steps leading to the conclusions drawn about uniform convergence.

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

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