Is sudden appearance of poles possible?

Question

I was thinking about the following scenario:

For $|\epsilon|<1$ let $f_\epsilon(z)$ be a meromorphic function such that

• $f_0(z)$ has a pole at the origin

• for $\epsilon \neq 0$, $f_\epsilon(z)$ is entire.

Is this possible, if we assume the $f_\epsilon$ depends "nicely" on $\epsilon$ (e.g. continuously)? I want to say no, but I can't quite prove this. Thanks!

Answer 1

I'm going to go with no: if $f_{\epsilon}(z)$ is continuous in $\epsilon$, then the integrals $$ I_n(\epsilon) := \int_{\lvert z \rvert= r} z^n f_{\epsilon}(z) \, dz \quad n = 0,1,2,\dotsc $$ are

1. continuous functions of $\epsilon$,
2. zero for $\epsilon \neq 0$ by Cauchy's theorem.
3. One of them must be nonzero if $f_0(z)$ has a pole of order $m$ at zero, because it would pick out the coefficient of $z^{-m}$, which must be nonzero by definition.

Since these facts are unreconcilable, I don't think your scenario can occur.