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I'm looking for an intuitive motivation for Liouville's theorem from complex analysis. If somebody could illustrate this with a simple example, that would be great. Thank you so much.

T A O
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    See also https://mathoverflow.net/questions/116896/liouvilles-theorem-with-your-bare-hands and https://www.jstor.org/stable/2323342 – lhf Apr 11 '20 at 21:01
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    this is a subtle property as there are entire non-constant functions that are bounded on any ray through the origin, so the materials linke din the mathoverflow answer are usueful – Conrad Apr 11 '20 at 21:09
  • For more detailed answers, see this post – David Raveh Jul 14 '23 at 23:20

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My point of view may be wrong. An extremely important fact about analytics functions is the mean formula : the value of f at $z$ is its mean on every circle centered around $z$. So, for an entire function f, the radii of theses circles can be chosen arbitrary large. Thus, if f is bounded, the value of f at every point $z$, will be its means "at the infinity". Consequently f is constant.

jvc
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Much like polynomials on the real line, (non-constant) complex analytic functions blow up. After all, an analytic function is sort of an infinite version of a polynomial.

  • what about $\sin x$ on the real line –  Apr 11 '20 at 21:01
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    This is flawed. – Mark Viola Apr 11 '20 at 21:02
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    @piombino $\sin x$ isn't a polynomial. –  Apr 11 '20 at 21:11
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    this is much subtler than that as I noted above there are entire nonconstant functions that are bounded on every line through the origin – Conrad Apr 11 '20 at 21:30
  • @Conrad that the analogy is flawed is duly noted. Thanks. –  Apr 11 '20 at 21:48
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    I wouldn't say that the analogy is flawed, only incomplete as for functions of finite order (so the Taylor coefficients $|a_n|$ decrease fast enough) it has some value, since then the function must be unbounded on many lines and then rotating it, one can assume it is unbounded on the reals (eg $\cos$ becomes $\cosh$) – Conrad Apr 11 '20 at 22:02
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Liouville's theorem and few other related theorems depict the rigid behaviour of an entire function. In contrast to the real-analytic functions where boundedness can be imposed solely, the complex analytic functions are sensitive to boundedness and adapt constant behaviour. This rigidity can also be seen in Identity theorem and Open mapping theorem.

Nitin Uniyal
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