I'm scheduled to host a prep course for the complex analysis qualifying exam at my university this summer, so I'm going through old qualifying exams to prepare. I'm stuck on the following:
Show that there exists a sequence $\{f_n\}$ of entire functions with the following property: For each rational number $q\geq 0$, the sequence converges to $\sqrt{q}$ uniformly on compact subsets of the line $\{z\in\mathbb{C}\,:\,\text{Re }(z)=q\}$.
I wish I could share my thoughts on this problem, but I honestly don't even know where to start. I imagine it uses the Mittag-Leffler or Weierstrass factorization theorem in some way but I can't put together how. I also assume that the countability of $\mathbb{Q}$ is exploited in some way, however the square root seems a bit mysterious.
Would anyone mind nudging me in the right direction? Thanks!
Best Answer
Enumerate the nonnegative rationals as $q_n$. Use Runge's theorem to find $f_n$ so $|f_n(q_k+iy) - \sqrt{q_k}| \le 1/n$ for $k=1\ldots n$ when $|y| \le n$.
The square root is a red herring. Any function on the nonnegative rationals would do.