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Suppose that $f(x+iy) = u(x,y) +iv(x,y)$ is differentiable on $\mathbb{C}$ and $u$ is bounded on $\mathbb{C}$. Use Liouville's theorem to show that $f$ is constant on $\mathbb{C}$.

Hint: what can you say about $e^f$?

I understand the idea of the liouville's theorem but don't know how to apply it on this

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Use the hint: $$ e^{u+iv} = e^u(\cos v+ i\sin v) $$ and since $v$ is real, that trigonometric factor is bounded. So if $u$ is bounded, then the whole thing is bounded.

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