<s>[INST] Laurent expansion Question is to write Laurent expansion of $f(z)=\dfrac{1}{(z-2)(z-1)}$ in the annulus $1<|z|<2$ based at $z=0$
I am aware of the method of partial fractions and writing expansions for both $\dfrac{1}{z-1}$ and $\dfrac{1}{z-2}$..
I am rying to do this using Laurent expansion formula..
We have $$f(z)=\sum_{n=-\infty}^{\infty}a_n z^n$$ where $$a_n=\frac{1}{2\pi i}\int_{|z|=r}\frac{f(z)}{z^{n+1}}dz$$
for any $1<r<2$. 
Easiest case is when $n=-1$ then it is just $$a_{-1}=\frac{1}{2\pi i}\int_{|z|=r}\frac{1}{(z-1)(z-2)}dz=\frac{1}{2\pi i}\left(\int_{|z|=r}\frac{1}{z-2}-\int_{|z|=r}\frac{1}{z-1}\right)$$
Now, as $r<2$ we see that $\frac{1}{z-2}$ is analytic inside the circle of radius $r$.. so first integral is zero and we are left with second integral..
$$a_{-1}=\frac{1}{2\pi i}\int_{|z|=r}\frac{1}{(z-1)(z-2)}dz=-\frac{1}{2\pi i}\int_{|z|=r}\frac{1}{z-1}$$
I got stuck after this...
 [/INST] I think i got it...
For $1<r<2$, we have to compute $$a_n=\frac{1}{2\pi i}\int_{|z|=r}\frac{1}{(z-2)(z-1)z^{n+1}}dz$$
For $n>0$ we have $$a_{-n}=\frac{1}{2\pi i}\int_{|z|=r}\frac{z^{n-1}}{(z-2)(z-1)}dz
=\frac{1}{2\pi i}\int_{|z|=r}\frac{z^{n-1}}{(z-2)}-\frac{z^{n-1}}{(z-1)}dz$$
As $r<2$, the function $\frac{z^{n-1}}{(z-2)}$ is analytic in the disk $\{|z|\leq r\}$ so first integral is zero. We then have  
$$a_{-n}=\frac{1}{2\pi i}\int_{|z|=r}\frac{z^{n-1}}{(z-2)(z-1)}dz
=-\frac{1}{2\pi i}\int_{|z|=r}\frac{z^{n-1}}{(z-1)}dz$$
The function $z^{n-1}$ is analytic so, by cauchy integral formula, we have
$$a_{-n}
=-\frac{1}{2\pi i}\int_{|z|=r}\frac{z^{n-1}}{(z-1)}dz=-z^{n-1}|_{z=1}=-1$$
So, for all $n>0$ we have $a_{-n}=-1$.
For $n>0$ we compute
 $$a_n=\frac{1}{2\pi i}\int_{|z|=r}\frac{1}{(z-2)(z-1)z^{n+1}}dz$$
By induction, i could see that 
$$\frac{1}{(z-2)(z-1)z^{n+1}}=\frac{1}{2^{n+1}}\frac{1}{z-2}-\frac{1}{2^{n+1}}\frac{1}{z}-\cdots-\frac{1}{2}\frac{1}{z^{n+1}}-\frac{1}{z-1}+\frac{1}{z}+\cdots+\frac{1}{z^{n+1}}$$
As $\frac{1}{z-2}$ is analytic in the disk $\{|z|\leq r\}$ the integral is zero. Then, by all integrals are zero except $\frac{1}{z}$ and $\frac{1}{z-1}$.. By cauchy integral formula, $\frac{1}{z-1}$ integral is $1$.. We then have 
$$a_n=
\frac{1}{2\pi i}\int_{|z|=r}\left[\frac{1}{2^{n+1}}\frac{-1}{z}-\frac{1}{z-1}+\frac{1}{z}\right]=\frac{-1}{2^{n+1}}-1+1=\frac{-1}{2^{n+1}}$$
So, we have $a_n=\frac{-1}{2^{n+1}}$ for all $n\geq 0$ and $a_{-n}=-1$ for all $n>0$.
We then have $$f(z)=\cdots-\frac{1}{z^2}-\frac{1}{z}-\frac{1}{2}\left(1+\frac{z}{2}+\frac{z^2}{2^2}+\cdots\right)$$
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