Liouville's Theorem

smoothman

Suppose f is an entire function such that [latex]f(z) = f(z+2\pi)[/latex]
and [latex]f(z)=f(z+2\pi i)[/latex] for all z [latex]\epsilon[/latex] 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 :)

Xevarion

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?

mathwonk

not just completely determined, but actually that it maps that square onto its range of values.

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smoothman	Liouville's Theorem Tweet Suppose f is an entire function such that [latex]f(z) = f(z+2\pi)[/latex] and [latex]f(z)=f(z+2\pi i)[/latex] for all z [latex]\epsilon[/latex] 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 :)

Xevarion	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?

mathwonk Recognitions: Homework Help Science Advisor	not just completely determined, but actually that it maps that square onto its range of values.