A <removable singularity> means that $f$ has a <holomorphic> extension to the full disk, assigning a finite value at zero. Equivalently its <Laurent series> has no negative-power terms. One direction follows by the <Taylor expansion> of the extension; the other by the convergent nonnegative-power series. Thus $\boxed{a_k=0\text{ for all }k<0}$ is the coefficient criterion.

Solution

Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2010 / Analysis / 5 / b / i / Solution

 Solution (source code)