\sin \frac{1}{z-1}= \displaystyle\sum_{n=0} ^\infty (-1)^{n}\frac{\frac{1}{z-1}^{2n+1}}{(2n+1)!}=\displaystyle\sum_{n=0} ^\infty\frac{(-1)^{n}}{(2n+1)!}\frac{1}{(z-1)^{2n+1}}