Show Boundedness of Entire Function f: f(z) = f(z + 2π ) & f(z + 2π i)

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Discussion Overview

The discussion revolves around demonstrating the boundedness of an entire function \( f \) that satisfies the conditions \( f(z) = f(z + 2\pi) \) and \( f(z) = f(z + 2\pi i) \) for all \( z \in \mathbb{C} \). The focus is on applying Liouville's theorem to conclude that \( f \) is constant.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • Some participants propose that to show \( f(z) \) is constant, one must first demonstrate that \( f \) is bounded.
  • Others suggest using Liouville's theorem as a key step in the argument.
  • A participant emphasizes the importance of the set being closed and bounded, noting that this is crucial for applying the theorem to continuous functions.
  • There is a hint provided to consider the restriction of \( f \) to a specific square in the complex plane.
  • One participant expresses initial confusion about how to start the problem but later acknowledges understanding how to use the hint effectively.

Areas of Agreement / Disagreement

Participants generally agree on the approach of using Liouville's theorem and the importance of boundedness, but there are minor disagreements regarding the specifics of the set's boundaries.

Contextual Notes

Some assumptions about the periodicity and properties of the function are implied but not explicitly stated, and the discussion does not resolve all mathematical steps involved in the proof.

alvielwj
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How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C.
π is pi.
 
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alvielwj said:
How to show that if f is an entire function,such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C.
π is pi.
How to show "if ..."

but where is your conclusion? What do you want to prove?
 
need to prove f(z) is constant.
first show f is bounded,then by the Liouville's theorem, f is constant
 
let me post the whole question
Suppose that f is an entire function such that f(z) = f(z + 2π ) and f(z) = f(z + 2π i)
for all z belong to C. Use Liouville's theorem to show that f is constant.
Hint: Consider the restriction of f to the square {z = x + iy : 0 <x < 2π ; 0 < y <2π }
 
Looks like a good hint! Although wasn't it [itex]0\le x\le 2\pi[/itex], [itex]0\le y\le 2\pi[/itex]? The "=" part is important because that way the set is both closed and bounded and so any continuous function is bounded on it. Since f is "periodic" with periods [itex]2\pi[/itex] and [itex]2\pi i[/itex], the bounds on that square are the bounds for all z.
 
Thank you for your answer..
I finally know how to use the hint..
At the beginning i really don't know how to start..
 

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