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Bounded Harmonic Function Is Constant [Closed]

Chloe Bennett • • 1 min read
Bounded Harmonic Function Is Constant [Closed]
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Can you please help me to prove that bounded harmonic function is constant?

Thanks a lot!

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2 Answers

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E. Nelson, "A Proof of Liouville's Theorem", Proc. Amer. Math. Soc. 12 (1961) 995

9 lines long. Not the shortest paper ever, but maximizes importance/length

...

Edward Nelson's paper is freely and legally available here:

$\bullet\ $ pdf file,

$\bullet\ $ html page.

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If $u$ is a harmonic function then there exists a conjugate function $v$ and an analytic function $f=u+iv$. Thus $\exp(f)$ is bounded, applying the Liouville's Theorem shows that $\exp(f)$ is constant.It's easy to prove that $f$ is constant, as well as $u$.

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Chloe Bennett
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Chloe Bennett

Chloe Bennett explores the intersection of pop culture, streaming entertainment, digital trends, and contemporary lifestyle. Her weekly commentary reaches thousands of culture enthusiasts.