Let
$f(z)=\frac{1}{(z−1)(z−2)}$
and let
$R =\{z∣0<|z|<1\}$.
2026-03-31 01:46:07.1774921567
Finding the Laurent series of $f(z)=\frac{1}{(z−1)(z−2)}$ for $R =\{z∣0<|z|<1\}$
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Hint: $$f(z) \color{red}{=} -\frac{1}{z-1} + \frac{1}{z-2} \color{blue}{=} \frac{1}{1-z} - \frac{1}{2}\frac{1}{1 - \left(\frac{z}{2}\right)},$$ where $\color{red}{=}$ follows by partial fractions, $\color{blue}{=}$ is just a clever reorganization, notice that $|z|, |z/2| < 1$, and use the sum for the geometric series.