Sinc function and Liouville's theorem

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?

• It is bounded on the real line, but not on the imaginary axis. – 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.
• 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. – Alex S Oct 30 '17 at 16:55