hi Guys I am trying to determine a series for the following function:
$$f(x)=\frac{2z}{(z+i)(z-1)}$$ about $|z-1|<\frac{1}{2}$
I already developed partial fractions:
$$\frac{1 + i}{z + i} + \frac{1 - i}{z - 1}$$
but got a problem with: $$\frac{1 + i}{z + i}$$ since I cannot find any Laurent valid series there for the point $|z-1|<1/2$.
Assuming you actually mean an expansion centered at $z=1$, which will converge on $|z-1|<1/2$ (and a little more $|z-1|<\sqrt{2}$), expand $$\frac{1+i}{z+i}=\frac{1}{1+\frac{z-1}{i+1}}=\sum_{n\geq0}(-1)^n\frac{(z-1)^n}{(1+i)^n}$$
The point $z=1$ will have to be excluded due to the other term: $\frac{1-i}{z-1}$.