Determine the largest real number $R>0$ such that the Laurent series of $$f(z)=\frac1{z-1} +\frac2{z-i}$$ about $z=1$ converges for $0<|z-1|<R$.
The singularities are $1$ and $i$. But in the region of the domain, $1$ is not inside the region so we should leave the first fraction how it is right?
Should find a taylor or Laurent expansion of the other fraction? It seems impossible to find a taylor of it but I think that is the one we need to find since the region is a circle excluding the point $1$ with radius $R$.
And then how would we find $R$?
The Laurent Series of this function will be analytic on an annulus with a hole at $1$ and which does not contain $i$. So, $R=|1-i|=\sqrt2$.