The function of a real variable defined by $f(x)=\sin(x)$ is analytic everywhere and bounded because $|\sin(x)|\leqslant 1$ for all $x$ but it is certainly not a constant. Should this contradict Liouville's theorem? Why is it not a contradiction?
$\endgroup$ 22 Answers
$\begingroup$The inequality $|\sin(x)|\leq 1$ is only true for real values of $x$. In fact, $\sin$ is unbounded when considered as a complex function. Note that we have $$ \sin(z)=\frac{e^{iz}-e^{-iz}}{2i} $$ so $\sin$ grows unbounded along the imaginary axis.
$\endgroup$ $\begingroup$The function $\sin$ isn't bounded in $\Bbb C$.
$\endgroup$
