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

Click For Summary
The discussion focuses on the nature of the singularity of the function F(z) at z=0, specifically whether it is removable or a pole. The sequence f_n = f(z/n) is analyzed in the context of normal families within a punctured disk that excludes the origin. It is established that if the singularity is removable, f_n converges uniformly to f(0), while if it is a pole, f_n diverges uniformly to infinity. Additionally, Picard's theorem indicates that if the singularity is essential, the convergence cannot be uniform as f_n will take on almost all values for large n. The inquiry highlights the complexities of understanding normal families in relation to singularities.
esisk
Messages
43
Reaction score
0
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
 
Physics news on Phys.org
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
 

Thread 'How does this derivative work?; and why the speed of light remains?'

Relativistic Momentum, Mass, and Energy Momentum and mass (...), the classic equations for conserving momentum and energy are not adequate for the analysis of high-speed collisions. (...) The momentum of a particle moving with velocity ##v## is given by $$p=\cfrac{mv}{\sqrt{1-(v^2/c^2)}}\qquad{R-10}$$ ENERGY In relativistic mechanics, as in classic mechanics, the net force on a particle is equal to the time rate of change of the momentum of the particle. Considering one-dimensional...
View full post »

Similar threads

Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Understanding the Order of Poles in Complex Functions
  • · Replies 4 ·
Replies
4
Views
2K
Classifying singularities of a function
  • · Replies 2 ·
Replies
2
Views
2K
Solving Removable Singularity in [z^(c-1)]/[exp(z)-1] at z=0
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Apparent counterexample to Cauchy-Goursat theorem (Complex Analysis)
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad What is the value of the integral for higher order poles in the real axis?
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
6K
Is f(z) =(1+z)/(1-z) a real function?
  • · Replies 17 ·
Replies
17
Views
3K