I’m trying to calculate the integral $\int_{C(0,3/4)}\frac{\operatorname{Log}(1+z)}{(z-1/2)^3}dz$. Can I use the residues theorem here? I’m thinking the value of the integral would be given by $2\pi i {\rm Res}(f(z),1/2)$, where $1/2$ is a pole of order $3$ and $f(z)=\frac{\operatorname{Log}(1+z)}{(z-1/2)^3}$, is that correct?
P.S.: By $\operatorname{Log}(z)$ I mean the function $z \rightarrow \log|z|+i\operatorname{Arg}(z)$, where $\operatorname{Arg}(z)$ is the only argument of $z$ belonging to the interval $]-\pi,\pi[$.