Solution
= Solution
Here the zeros accumulate only at the boundary point zero. An explicit <holomorphic>, nonzero example on the <upper half-plane> is
$$
\boxed{f(z)=\sin(\pi i/z).}
$$
At $z_n=i/(n+1)$ its argument is $\pi(n+1)$, so it vanishes. It is <holomorphic> because the only excluded point of its formula is zero; it is not identically zero, for instance at $z=2i$. Thus the answer is \b[yes].