From Liouville's theorem, we know that If a function $f$ is entire and bounded in the complex plane, then $f(z)$ is constant throughtout the plane

Is there a special meaning in the word constant? There are plenty of entire functions that have different values depending on different $z$. For example we know $sin(z)$ is entire and its range is not a single point. Is there something I am missing?

  • $\begingroup$ The sine function is not bounded. $\endgroup$ – user61527 Dec 15 '13 at 6:10

$\sin(z)$ is not bounded in the complex plane. Indeed, as we have $$\sin(z) = \frac{1}{2i}(e^{iz} - e^{-iz}),$$ it follows that $\sin(ix)$ (for $x \in \mathbb{R}$) grows exponentially as $x \to \infty$.

  • $\begingroup$ I read it wrong, I thought the condition is bounded by the complex plane, not bounded in complex plane, thanks for the answer $\endgroup$ – user2654176 Dec 15 '13 at 6:13
  • 2
    $\begingroup$ No thanks are necessary, just click the check mark and the upward arrow :) $\endgroup$ – Ryan Reich Dec 15 '13 at 6:14

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