Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 7 / d

Put $w=z-1$ and $a=1-i$. Partial fraction decomposition givesBoth terms admit a geometric series at $w=0$. The local Laurent series is actually a Taylor series, since the center is regular:The nearest pole is $z=i$, a distance $|1-i|=\sqrt{2}$ from the center; the other pole is at distance $4$. Thus the requested local radius of convergence is $\boxed{R=\sqrt{2}}$.

For completeness, specifying a Laurent series without a region can also mean either of the other maximal annuli. Expanding the first fraction in inverse powers and the second in positive powers givesExpanding both fractions in inverse powers instead givesThese are different Laurent series on different annuli, rather than extensions of the local Taylor series beyond its radius of convergence.