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

where

.

Then from

we get

.

Thus,

.

When , let we see that is a removable singularity of since implies .

**Remark**. *The second derivative can be replaced by an and therefore should be . This is a question of UCLA QE in Winter 2007.*

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 .

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