Periodicity of the complex sine
= Periodicity of the complex sine
{title2=$\sin(z+2\pi)=\sin z$}
The kernel $2\pi i\mathbb Z$ of the <complex exponential function> gives
$$
\sin(z+2\pi)
=\frac{e^{iz}e^{2\pi i}-e^{-iz}e^{-2\pi i}}{2i}
=\sin z.
$$
The same period appears as the translation generated by the <monodromy of the complex inverse sine>.