Using Liouville's Theorem to Show f is Constant

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In summary, the conversation is discussing how to use Liouville's theorem to show that an entire function f, which satisfies the conditions f(z) = f(z+2\pi) and f(z) = f(z+2\pi i) for all z in the complex plane, is constant. The other person in the conversation suggests that since f is determined by its values in a square of side length 2\pi and maps that square onto its range of values, it can be shown that f is constant.
  • #1
smoothman
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Suppose f is an entire function such that [itex]f(z) = f(z+2\pi)[/itex]
and [itex]f(z)=f(z+2\pi i)[/itex] for all z [itex]\epsilon[/itex] C. How can you use Liouville's theorem to show f is constant..

any help on that please to get me started off.. thnx a lot :)
 
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  • #2
The two given relations tell you that [tex]f[/tex] is completely determined by its values in a square of side length [tex]2\pi[/tex]... what do you need to show about [tex]f[/tex] to use Liouville? Can you get it from this info now?
 
  • #3
not just completely determined, but actually that it maps that square onto its range of values.
 

FAQ: Using Liouville's Theorem to Show f is Constant

What is Liouville's Theorem?

Liouville's Theorem is a mathematical theorem that states that if a function is bounded and holomorphic (analytic and differentiable) in the complex plane, then it must be constant.

How does Liouville's Theorem relate to showing a function is constant?

Liouville's Theorem can be applied to show that a function is constant by proving that it is bounded and holomorphic in the complex plane.

What is the process for using Liouville's Theorem to show a function is constant?

The process involves first verifying that the function is bounded and holomorphic in the complex plane. Then, by applying Liouville's Theorem, it can be concluded that the function is constant.

Can Liouville's Theorem be used for functions in real variables?

No, Liouville's Theorem only applies to functions in complex variables. In real variables, a function can be bounded and differentiable without being constant.

Are there any limitations to using Liouville's Theorem to show a function is constant?

Yes, Liouville's Theorem only applies to entire functions, which means that they are defined and holomorphic in the entire complex plane. It cannot be used for functions that have singularities or are not defined in certain regions of the complex plane.

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