Q about the proof of periods of non-constant meromorphic functions

In summary, the theorem holds for a discrete subgroup of ##C##, and the set of periods of a non-constant meromorphic function is a discrete subset. The second case considers the set of periods ##\Omega = nw##, where ##n\neq 0##. In this case, it is shown that there exists ##w_1 \in \Omega /\{0\}## with the least value of ##|w_1|##. One step in the proof states that since ##\Omega## is discrete, there is an ##\epsilon > 0## such that any disc with radius ##\
  • #1
binbagsss
1,265
11

Homework Statement


[/B]
Theorem attached.

I know the theorem holds for a discrete subgroup of ##C## more generally, ##C## the complex plane, and that the set of periods of a non-constant meromorphic function are a discrete subset.

I have a question on part of the proof (showing the second type, integral muliples ##nw##)

On proving the second case ##\Omega##, the set of periods ##\Omega = nw ## , ##\Omega \neq \{0\}## is considered and we show that exists ##w_1 \in \Omega /\{0\} ## with the least value of ##|w_1|##. A step of doing this in my text is, and this is my problem:

Since ##\Omega ## is discrete, there is some ##\epsilon > 0 ## st for any disc ##|z| < \epsilon ## contains no elements of ##\Omega / \{0\}##, it follows that for any ##w \in \Omega ##, the disc ##|z-w|< \epsilon ## contains no elements of ##\Omega / \{w\} ##.


I don't understand why/ how this follows from the fact that is a group, why this 'translation' is possible with the same radius ##\epsilon## follows from the fact it is a group?

No idea where to start, any help much appreciated, but my guess would be that the proof will depend on what the operation of the group is, but this isn't specified in the theorem? or is just assumed to be addition?

Homework Equations


see above

The Attempt at a Solution


see above
[/B]
 

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  • #2
binbagsss said:

Homework Statement


[/B]
Theorem attached.

I know the theorem holds for a discrete subgroup of ##C## more generally, ##C## the complex plane, and that the set of periods of a non-constant meromorphic function are a discrete subset.

I have a question on part of the proof (showing the second type, integral muliples ##nw##)

On proving the second case ##\Omega##, the set of periods ##\Omega = nw ## , ##\Omega \neq \{0\}## is considered and we show that exists ##w_1 \in \Omega /\{0\} ## with the least value of ##|w_1|##. A step of doing this in my text is, and this is my problem:

Since ##\Omega ## is discrete, there is some ##\epsilon > 0 ## st for any disc ##|z| < \epsilon ## contains no elements of ##\Omega / \{0\}##, it follows that for any ##w \in \Omega ##, the disc ##|z-w|< \epsilon ## contains no elements of ##\Omega / \{w\} ##.


I don't understand why/ how this follows from the fact that is a group, why this 'translation' is possible with the same radius ##\epsilon## follows from the fact it is a group?

No idea where to start, any help much appreciated, but my guess would be that the proof will depend on what the operation of the group is, but this isn't specified in the theorem? or is just assumed to be addition?

Homework Equations


see above

The Attempt at a Solution


see above[/B]

bumpp, thank you
 

Related to Q about the proof of periods of non-constant meromorphic functions

1. What is a meromorphic function?

A meromorphic function is a complex-valued function that is holomorphic (analytic) everywhere except for isolated singularities, where it has poles. In other words, it is a function that is a ratio of two analytic functions.

2. How are periods related to meromorphic functions?

Periods are a measure of the "repetition" of a function. For meromorphic functions, periods are used to describe the behavior of the function over a particular interval, usually a complex plane or a torus.

3. Why is the proof of periods of non-constant meromorphic functions important?

The proof of periods of non-constant meromorphic functions is important because it helps us understand the behavior of these functions and their singularities. It also allows us to make connections between different branches of mathematics, such as complex analysis and algebraic geometry.

4. What are some applications of the proof of periods of non-constant meromorphic functions?

The proof of periods of non-constant meromorphic functions has applications in various fields of mathematics and physics, including algebraic geometry, differential geometry, number theory, and quantum mechanics. It also has practical applications in signal processing and image recognition.

5. What are some techniques used in the proof of periods of non-constant meromorphic functions?

Some techniques used in the proof of periods of non-constant meromorphic functions include the use of Riemann surfaces, complex analysis, and algebraic geometry. Other techniques may include the use of analytic continuation, the residue theorem, and the theory of elliptic functions.

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