Proving Boundedness of Entire Functions with Harmonic Components

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In summary, the conversation is about proving that uv=>0 is bounded in order to show that an entire function is constant. The person is asking for clarification on the conditions of u and v being harmonic and how they relate to the proof. They also mention using Louivilles theorem and realizing they were assuming that u and v were bounded.
  • #1
physicsjock
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Hey,

I'm trying to prove that uv=>0 is bounded so I can state that an entire function is constant when f = u + iv, when f is entire.

I have worked out the rest but I'm struggling to prove that its bounded,

Can you say u=>0, v=>0 then u + v => 0, and that bounded from below?
 
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  • #2
physicsjock said:
Hey,

I'm trying to prove that uv=>0 is bounded so I can state that an entire function is constant when f = u + iv, when f is entire.

I'm not sure what you are trying to prove - is it given that u,v are harmonic & uv>=0 ?
 
  • #3
They gave

f = u + iv is an entire function,

that means u and v are harmonic right?

and it asks to show f is constant if uv=>0 everywhere

I think i have done that part just using louivilles theorem, but i realized i was just assuming u and v were bounded
 

What does it mean to prove something is bounded?

Proving something is bounded means showing that it has a finite range or that it does not exceed a certain limit.

Why is it important to prove that something is bounded?

Proving that something is bounded is important because it allows us to make accurate predictions and draw conclusions about its behavior.

How do you prove that something is bounded?

To prove that something is bounded, you can use mathematical techniques such as limits, inequalities, or series convergence tests.

What happens if something is not bounded?

If something is not bounded, it means that it has an infinite range or that it exceeds any given limit. This can lead to unpredictable or unreliable results.

Can something be both bounded and unbounded?

No, something cannot be both bounded and unbounded. It is either one or the other. However, different aspects or properties of the same thing can be either bounded or unbounded.

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