My attempt:
$$\frac{1}{z(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{A}{z}+\frac{B}{z-2}+\frac{C}{(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{(1/4)}{z}+\frac{(-1/4)}{z-2}+\frac{(1/2)}{(z-2)^2}$$
This is where I get stuck. The general idea I know is to get each of the 3 terms above into the form $D\cdot \frac{1}{1-z}$, where $D$ is a constant. But for the first term, it would be $\frac{1}{4}\cdot\frac{1}{(0-z)}$. I can't get rid of the zero.
On
Let $\zeta = z - 2$. We are trying to find the Laurent series for $1/(\zeta+2)\zeta^2$ centred at $\zeta = 0$, so $$ f = \frac{1}{(\zeta+2)\zeta^2} = \zeta^{-2} \frac{1}{\zeta + 2} $$ The latter is holomorphic in a neighbourhood of $0$, so let $(a_j)_{j=0}^\infty$ such that $$ \frac{1}{\zeta + 2} = \sum_{j=0}^\infty a_j \zeta^j, 1 = 2\sum_{j=0}^\infty a_j \zeta^j + \sum_{j=0}^\infty a_j \zeta^{j+1} $$ Therefore, $$ 1 = 2a_0 + \sum_{j=1}^\infty 2a_j \zeta^j + \sum_{j=1}^\infty a_{j-1} \zeta^j = 2a_0 + \sum_{j=1}^\infty (2a_j + a_{j-1}) \zeta^j $$ so $a_0 = \frac{1}{2}$, and we establish the recurrence relation $a_{j+1} = -\frac{1}{2}a_j$. Hence $a_j = (-1/2)^j$. The original series becomes $$ \zeta^{-2} \frac{1}{\zeta + 2} = \zeta^{-2} \sum_{j=0}^\infty (-\frac{1}{2})^j \zeta^j $$ Then substitute $\zeta = z-2$.
To find the region of convergence, notice that $f$ is holomorphic except at the poles $z = 0, z = 2$. The result follows easily from Laurent's theorem.
On
The function
\begin{align*} f(z)&=\frac{1}{z(z-2)^2} = \frac{1}{4z}-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\\ \end{align*} has a simple pole at $z=0$ and a pole of order two at $z=2$.
Since we want to find a Laurent expansion with center $2$, we look at the other pole $0$ and see we have to distinguish two regions.
\begin{align*} 0<|z-2|<2\qquad\quad\text{and}\qquad\qquad 2<|z-2| \end{align*}
The first region $ |z-2|<2$ is a disc with center $2$, radius $2$ and the pole $0$ at the boundary of the disc. In the interior of this disc the fraction with pole $0$ admits a representation as power series at $z=2$, while the fractions with pole $2$ admit a representation as principal part of a Laurent series at $z=2$.
The second region $|z-2|>2$ containing all points outside the disc with center $2$ and radius $2$ admits for all fractions a representation as principal part of a Laurent series at $z=2$.
A power series expansion of $\frac{1}{z+a}$ at $z=2$ is \begin{align*} \frac{1}{z+a}&=\frac{1}{(a+2)+(z-2)}=\frac{1}{a+2}\cdot\frac{1}{1+\frac{z-2}{a+2}}\\ &=\frac{1}{a+2}\sum_{n=0}^{\infty}(-1)^n\left(\frac{z-2}{a+2}\right)^n\tag{1}\\ &=\sum_{n=0}^{\infty}\frac{(-1)^n}{(a+2)^{n+1}}(z-2)^n\tag{2} \end{align*} The principal part of $\frac{1}{z+a}$ at $z=2$ is \begin{align*} \frac{1}{z+a}&=\frac{1}{(z-2)+(a+2)}=\frac{1}{z-2}\cdot\frac{1}{1+\frac{a+2}{z-2}}\\ &=\frac{1}{z-2}\sum_{n=0}^{\infty}(-1)^n\left(\frac{a+2}{z-2}\right)^n\tag{1}\\ &=\sum_{n=0}^{\infty}(-1)^n(a+2)^n\frac{1}{(z-2)^{n+1}}\\ &=\sum_{n=1}^{\infty}(-1)^{n-1}(a+2)^{n-1}\frac{1}{(z-2)^{n}}\tag{3} \end{align*}
We can now obtain the Laurent expansion of $f(z)$ at $z=2$ for both regions
- Region: $|z-2|<2$
\begin{align*} f(z)&= \frac{1}{4z}-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\\ &=\frac{1}{4}\cdot\frac{1}{2+(z-2)}-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\\ &=\frac{1}{4}\sum_{n=0}^{\infty}\frac{(-1)^n}{2^{n+1}}(z-2)^n-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\tag{4}\\ &=\frac{1}{8}\sum_{n=-2}^{\infty}\frac{(-1)^n}{2^n}(z-2)^n \end{align*}
- Region: $2<|z-2|$
\begin{align*} f(z)&=\frac{1}{4z}-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\\ &=\frac{1}{4}\frac{1}{(z-2)+2}-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\\ &=\frac{1}{4}\sum_{n=1}^{\infty}(-1)^{n-1}2^{n-1}\frac{1}{(z-2)^n}-\frac{1}{4(z-2)}+\frac{1}{2(z-2)^2}\tag{5}\\ &=\frac{1}{8}\sum_{n=3}^{\infty}(-1)^{n-1}2^{n}\frac{1}{(z-2)^n} \end{align*}
Comment:
In (1) we use the geometric series \begin{align*} \frac{1}{1+z}=\frac{1}{1-(-z)}=\sum_{n=0}^\infty (-z)^n=\sum_{n=0}^\infty (-1)^nz^n\qquad\qquad |z|<1 \end{align*}
In (4) we apply the series expansion of (2)
In (5) we apply the principal part representation of (3)
HINT:
Write $\frac1z$ as
$$\frac1z=\frac{1/2}{1+(z-2)/2}$$
Then, recall the sum of a geometric series.
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