Finding Laurent series of this function

Question

I need to find the Laurent series and find the region of convergence for: \begin{equation} \frac{2z+i}{z(z+i)} \end{equation} About point $z=i$, which I've split into partial fractions to get: \begin{equation} \frac{1}{z}+\frac{1}{z+i} \end{equation} But I'm unsure on how to apply the definition of a Laurent series to obtain it for this equation at $z=i$.

Answer 1

You should rewrite both of the terms such that they can be expressed as a geometric series. For $\frac{1}{z+\mathrm i}$ this would be

$$\frac{1}{z+\mathrm i}=\frac{1}{2\mathrm i +(z-\mathrm i)}=\frac{1}{2\mathrm i}\frac{1}{1-\frac{\mathrm i}{2}(z-\mathrm i)}.$$

Then use the known formula

$$\sum_{k=0}^\infty q^k=\frac{1}{1-q}$$

for $\vert q\vert<1$. The other term works similarly.