How Can I Show That the Function is Bounded?

In summary, the conversation discusses how to use Liouville's theorem to show that an entire function is constant. The hint is to consider the restriction of the function to a specific square set, and it suffices to show that the function is bounded on this set. This is similar to checking the boundedness of sin(x) on [0,2pi].
  • #1
MidnightR
42
0
Hi,

Suppose f is an entire function such that f(z) = f(z+2pi) = f(z+2(pi)i) for all z E C.

Use Liouville's theorem to show that f is constant.

Obviously I need to show that the function is bounded but I'm unsure of how to approach it.

The hint is: Consider the restriction of f to the square S = {z = x + iy : 0 <= x <= 2Pi, 0<= y <= 2Pi}

Any help/hints appreciated to get me started, thanks
 
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  • #2
HINT: the set S is closed and bounded...
 
  • #3
Yea I realize that but f doesn't have to be restricted does it? Unless that's what they mean by the hint, in which case there's nothing to say?
 
  • #4
Well, it suffices to show that f is bounded on S. Since values not in S, can be brought back to a value in S.

It's like checking that sin(x) is bounded. It suffices to do that on [0,2pi], since any other value can be brought back to a value on [0,2pi].
 

FAQ: How Can I Show That the Function is Bounded?

What does it mean for a function to be bounded?

A bounded function is one that has a finite range, meaning that the output values of the function are limited to a specific range, rather than being able to increase or decrease without bound.

How is the boundedness of a function determined?

The boundedness of a function can be determined by analyzing the behavior of the function at its domain boundaries. If the function approaches a finite value at both the left and right boundaries, it is considered bounded.

What are the implications of a function being bounded?

A bounded function has a finite range, meaning that its output values are limited. This can have implications in various areas of mathematics and science, such as optimization problems or the convergence of series.

What is the difference between a bounded and an unbounded function?

An unbounded function has an infinite range, meaning that its output values can increase or decrease without bound. This is in contrast to a bounded function, where the output values are limited to a specific range.

How does the boundedness of a function relate to its domain and codomain?

The boundedness of a function is directly related to its domain and codomain. A function with a bounded domain and codomain will always be bounded, while a function with an unbounded domain or codomain may or may not be bounded.

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