I would like to find the maximal open set such that $(\frac{\sin(nz)}{n})_{n}$ converge uniformly on $\Bbb{C}.$
We have $$\sin(nz)=i\times\frac{e^{-inx}e^{ny}-e^{inx}e^{-ny}}{2}$$ so that $$\frac{\sin(nz)}{n}=i\times\frac{e^{-inx}e^{ny}-e^{inx}e^{-ny}}{2n}$$
Now I have $$\vert\frac{\sin(nz)}{n}\vert\le \frac{e^{ny}+e^{-ny}}{2n}$$
If $y\ge 0$ or $y\le 0$ then the RHS tend to $+\infty,$ so not sure what I have to do.