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    Liouville's theorem

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    Suppose that f(z) = u(x , y) + i*v(x, y) (where z = x + i*y) is ENTIRE and not constant. Use Liouville's theorem to prove that v(x, y) is unbounded. Under what conditions can a function that is harmonic everywhere be bounded?

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    https://brainmass.com/math/basic-calculus/liouvilles-theorem-281780

    Solution Preview

    Proof:
    Let . Since is entire, then is also entire.
    I claim that is unbounded.
    If is bounded, ...

    Solution Summary

    Liouville's theorem is applied.

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