Show that entire function is constant.

Question

$f$ is an entire function and $f(z) = i$ when $z = \left(1+ \frac kn \right)+i$ for every positive integer $k$. $n$ is fixed.

How can we conclude that $f$ is constant?

Is there any result about entire function with infinite zeros.

(If it was true for all postive $n$ also then it was easy, I also doubt that there is typographical error in given question. But I want to confirm)

Answer 1

You can't conclude this. For instance we could have $f(z) = i e^{2\pi i n(z-i-1)}$ or similar.

Answer 2

No, we can’t: take $f(z)=\sin(2n\pi(z-1-i))+i$.

Answer 3

People have already given counterexamples. You might note that someone's well known theorem says that if $(z_n)$ and $(a_n)$ are any sequences with $z_n\ne z_m$ ($n\ne m$) and $z_n\to\infty$ then there exists an entire function $f$ with $f(z_n)=a_n$.

Maybe it's a typo; of course if you swap the roles of $n$ and $k$ it's true. It's also true if you add a suitable growth condition $|f(z)|\le ce^{\alpha|z|}$ for small enough $\alpha$ (I forget exactly how small, I think it's $\alpha<1/n$.)