Liouville's theorem says that every bounded and entire function should be a constant. $$ \operatorname{sinc}(x)=\frac{\sin(x)}{x}$$ is known as an entire function, and it seems to be bounded. Does this conflict with Liouville's theorem?

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    $\begingroup$ It is bounded on the real line, but not on the imaginary axis. $\endgroup$ – Jack D'Aurizio Oct 30 '17 at 16:06

This function is not bounded. If $x$ is a real number, then $$\text{sinc}(ix)=\frac{\sin(ix)}{ix}=\frac{1}{2i}\frac{e^{i\cdot ix}-e^{-i\cdot ix}}{ix}=\frac{\sinh(x)}{2x}.$$ As $x\to\infty$ you can show (using L'Hopital's rule, for instance) that this function tends to $\infty$, and so it is not bounded.

When using theorems such as Liouville's Theorem which are applicable to functions of complex variables, it is necessary to consider the behavior of the function over the whole complex plane, not only the real line.

  • $\begingroup$ Many thanks for the clarification. Just wondering, is there any other example that is bounded on the real axis, but is unbounded once is extended to the complex plane? For example, the density of standard normal distribution? $\endgroup$ – Yanhg Oct 30 '17 at 16:50
  • $\begingroup$ The density of the standard normal distribution is one example. $\sin$ and $\cos$ are two others. Any sum or product of two analytic functions bounded on the real line will also be analytic and bounded on the real line. $\endgroup$ – Alex S Oct 30 '17 at 16:55

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