The following question was proposed in NUS under the QE in AY 2007-2008:
Consider the punctured disk . Suppose is an analytic function such that
for all . Is it true that f has a removable singular point at ?
Proof. Denote the Laurent expansion of by
When , let we see that is a removable singularity of since implies .
We also have a similar question proposed in a QE of Indiana University. It says that if is an analytic function such that
for all . Then .
Proof. As above, one has
When , letting we have for all which implies is a removable singularity of . In other words, can be extended to an analytic function of the unit disk. Since when , by the Maximum Modules Principle we obtain .