I am considering the following problem:
$f_n$ is a sequence of entire functions and $f_n \rightarrow f$ uniformly on compact subsets of $\mathbb{C}$.Suppose each $f_n$ has only real zeros, then by Rouche's theorem, $f$ also has only real zeros.
Now my question is:Can we say $f$ must have real zeros? I have no idea about it, can anyone help me?
Best Answer
No, the limit function need not have zeros at all. A simple example is $$ f_n(z) = \left(1 - \frac zn\right) e^z $$ which converges locally uniformly to $f(z) = e^z$. Slightly more general, $$ f_n(z) = g\left(\frac zn\right) e^{h(z)} $$ where both $g$ and $h$ are entire functions, $g$ has only real zeros, and $g(0) \ne 0$.