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

In summary, the conversation discusses the concept of normal families within the context of analytic functions on a punctured disk. The sequence defined by f_n=f(z/n) is being examined, and it is being shown that the behavior of this sequence depends on the singularity of f(z) at z=0. If the singularity is removable, the sequence converges uniformly to f(0), but if it is a pole, the sequence diverges uniformly. In the case of an essential singularity, the sequence takes on almost all values for large n, making uniform convergence impossible. The conversation ends with the acknowledgement of the need for further study on the topic.
  • #1
esisk
44
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
  • #2
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.
 
  • #3
I thank you very much Eynstone.
Need to study more...I am not prelim-ready yet
Regards
 
  • #4
I thank you very much Eynstone.
Need to study more...I am not prelim-ready yet
Regards
 

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

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It involves the manipulation and analysis of functions that have complex-valued inputs and outputs.

2. Why is complex analysis important?

Complex analysis is important because it has many practical applications in various fields, such as physics, engineering, and economics. It also provides a powerful tool for solving problems in mathematics, including differential equations and harmonic analysis.

3. What are some key concepts in complex analysis?

Some key concepts in complex analysis include the Cauchy-Riemann equations, contour integration, analytic functions, and the Cauchy integral theorem. These concepts are used to study the behavior of complex functions and their properties.

4. What is the difference between real analysis and complex analysis?

The main difference between real analysis and complex analysis is the types of numbers that are being studied. Real analysis deals with functions of real numbers, while complex analysis deals with functions of complex numbers. Additionally, complex analysis involves the use of complex variables and techniques such as contour integration, which are not present in real analysis.

5. How is complex analysis used in other branches of science?

Complex analysis is used in a variety of scientific fields, such as physics, engineering, and economics. For example, in physics, complex analysis is used to study electromagnetism and fluid dynamics. In engineering, it is used in the design of electronic circuits and control systems. In economics, complex analysis is used to model and analyze economic systems and financial markets.

Similar threads

A Understanding the Order of Poles in Complex Functions
    • Calculus
Replies
4
Views
1K
I Question about uniform convergence in a proof
    • Calculus
Replies
1
Views
360
Classifying singularities of a function
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
I Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
    • Topology and Analysis
Replies
7
Views
1K
I What is the value of the integral for higher order poles in the real axis?
    • Calculus
Replies
9
Views
2K
B How can hyperreal numbers make infinitesimals logically sound in calculus?
    • Calculus
Replies
24
Views
3K
Is f(z) =(1+z)/(1-z) a real function?
    • Calculus and Beyond Homework Help
Replies
17
Views
1K
Solving Removable Singularity in [z^(c-1)]/[exp(z)-1] at z=0
    • Calculus
Replies
4
Views
2K
MHB Complex Analysis: Does $\int_{C(0,10)} f(z)$ Equal 0?
    • Topology and Analysis
Replies
2
Views
1K
MHB Is z_0 a Removable Singularity in Complex Analysis?
    • Topology and Analysis
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top