Sinc function and Liouville's theorem

Question

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?

Answer 1

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.