Complex Analysis: Theory and Applications · Problem 6.5.8

This problem tests whether you can recognize that $z^{-2}\sin z$ has a removable singularity at the origin—the key is expanding $\sin z$ as a power series and seeing what actually happens near $z=0$. Once you know the Laurent expansion, the residue theorem tells you immediately whether the integral is zero or nonzero, depending only on whether the pole at $z=0$ lies inside each contour. The radius of each circle determines everything.

This problem tests whether you can recognize that $z^{-2}\sin z$ has a removable singularity at the origin—the key is expanding $\sin z$ as a power series and seeing what actually happens near $z=0$. Once you know the Laurent expansion, the residue theorem tells you immediately whether the integral is zero or nonzero, depending only on whether the pole at $z=0$ lies inside each contour. The radius of each circle determines everything.