While I was working out a Laurent series of ${z^2 -2z+3 \over z-2}$ in the region $|z-1|>1$, I got stuck on a couple of things...
${z^2 - 2z +3 \over z-2} = {(z-1)^2 + 2 \over z-1}\cdot {z-1 \over z-2}\\ = {(z-1)^2 + 2 \over z-1} \cdot {1 \over 1 - {1 \over z-1}}\\ = \left[(z-1) + {2 \over z-1}\right] \cdot \left[1 + {1 \over z-1} + {1 \over (z-1)^2} + \dots \right]\\ = [(z-1) + 1 + 1/(z-1) + 1/(z-1)^2 + ...] + [ 2/(z-1) + 2/(z-1)^2 + ...] \qquad(a)\\ = (z-1) + 1 + [3/(z-1) + 3/(z-1)^2 + ...] \qquad(b)$
So what properties of complex series allows me to multiply each term of $1+1/(z-1) +\dots$ by $(z-1)$ and again each term of $1+1/(z-1)+\dots$ by $2/(z-1)$ to get $(a)$?
Also what properties of complex series allows me to add the terms together (like $1/(z-1)+2/(z-1)= 3/(z-1)$) to get $(b)$?
Is it absolute convergence? I have no idea! Unfortunately I just do these manipulations instinctively and no textbook seems to tell me!
Any help would be greatly appreciated!
Also if someone could point out to me where I can learn and format MathJax, I would be very grateful. It seems that every time I ask a question, it doesn't format!