# Liouville's theorem

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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#### 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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