Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S1 / 2 / d / Solution

If $f(z)=\sum a_j{z}^j$, then $f(-z)=\sum (-1{)}^j{a}_j{z}^j$ and its residue at zero is $-a_{-1}$. The circle $|z-1|=2$ encloses zero and no other finite singularity. Thus the residue theorem givesfor positive orientation.