Analytic mapping from disk to disk must be rational

In summary, the conversation discusses a function f(x) that is analytic in the open unit disk and maps it onto itself k times. The task is to prove that f(x) must be a rational function and that the degree of its denominator cannot exceed k. It is suggested that for larger k, the Riemann mapping theorem can be used, but for smaller k, the focus is on automorphisms of the unit disk. The example of f(z)=z^2 is given, but it is noted that the preimages must be counted with multiplicity. The conversation ends with the acknowledgement that there may be a theorem needed to prove that any arbitrary analytic function with these properties must be rational.
  • #1
Grothard
29
0
Let f(x) be a function which is defined in the open unit disk (|z| < 1) and is analytic there. f(z) maps the unit disk onto itself k times, meaning |f(z)| < 1 for all |z| < 1 and every point in the unit disk has k preimages under f(z). Prove that f(z) must be a rational function. Furthermore, show that the degree of its denominator cannot exceed k.

If this was limited to k=1 I think we could use the Riemann mapping theorem, but for larger k I am quite lost. How does one go about proving that an arbitrary analytic function with these givens must be rational? I've seen a form of this question in several places, but I still can't grasp how one would tackle such a problem.
 
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  • #2
I think under these hypotheses, probably k must be 1. and then we are concerned with automorphisms of the unit disc, all fractional linear transformations.
 
  • #3
mathwonk said:
I think under these hypotheses, probably k must be 1. and then we are concerned with automorphisms of the unit disc, all fractional linear transformations.

What about [itex]f(z)=z^2[/itex]. This seems to satisfy the conditions? Except of course that 0 only has 1 preimage. It does have 2 preimage counting multiplicity, so I wonder whether to count the preimages with multiplicity or not.
 
  • #4
micromass said:
What about [itex]f(z)=z^2[/itex]. This seems to satisfy the conditions? Except of course that 0 only has 1 preimage. It does have 2 preimage counting multiplicity, so I wonder whether to count the preimages with multiplicity or not.

I do believe we should consider the multiplicity; my language was imprecise as I was paraphrasing the problem I've encountered in several sources before. I think [itex]f(z)=z^n[/itex] are precisely the types of functions we are looking for. But I have no clue how to show that any arbitrary analytic function that covers the unit disk in this way must be rational. I feel like there's a theorem I'm missing.
 
  • #5
i was using the fact that he did not use multiplicities for counting preimages.
 

1. What is analytic mapping from disk to disk?

Analytic mapping from disk to disk is a mathematical concept in complex analysis that refers to a function that maps points from one disk (in the complex plane) to another disk. This type of mapping is often used to study the behavior of complex functions and their properties.

2. Why must analytic mapping from disk to disk be rational?

Analytic mapping from disk to disk must be rational because it is a necessary condition for the function to be conformal, meaning that it preserves angles and shapes. This is a key property that allows for the use of analytic mapping in many applications, including in physics and engineering.

3. How is analytic mapping from disk to disk different from other types of mappings?

Analytic mapping from disk to disk is different from other types of mappings, such as polynomial mapping or rational mapping, because it is a type of holomorphic function. This means that it is infinitely differentiable and has a unique power series expansion, making it particularly useful in complex analysis.

4. What are some real-world applications of analytic mapping from disk to disk?

Analytic mapping from disk to disk has several real-world applications, including in fluid dynamics, where it is used to model the flow of fluids, and in computer graphics, where it is used to create smooth and realistic images. It is also used in the study of electrical circuits and in the design of electronic devices.

5. Is there a specific method for finding the analytic mapping from disk to disk?

There is no single method for finding the analytic mapping from disk to disk, as it depends on the specific problem at hand. However, there are several techniques that can be used, such as the Schwarz-Christoffel mapping, which uses conformal transformations to map one disk to another, or the Koebe approach, which uses power series expansions to find the mapping. Each method has its own advantages and limitations, and the choice of method depends on the complexity of the problem and the desired level of accuracy.

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