# Laurent Series Expansion

Give the Laurent series development of the function $f(z)=\frac{1}{z(z-1)(z-2)}$ in the three rings $A_1=\{z:|z|<1\}$, $A_2=\{z:1<|z|<2\}$ and $A_3=\{z:2<|z|\}$.

I have gotten the partial fraction form $f(z)=\frac{1}{2z}-\frac{1}{z-1}+\frac{1}{z-2}$

For $A_1$, I think it's the standard Taylor expansion about 0 and I can do this for the second and third terms:

$-\frac{1}{z-1}=(1-z)^{-1}=\sum^\infty_{n=0}z^n$ and $\frac{1}{2(z-2)}=\frac{1}{4}\sum^\infty_{n=0}(\frac{z}{2})^n$. How do I get the Taylor expansion of the first term?

I foresee similar problems if I work out the expansion for $A_2$ and $A_3$ as well, so a general method is appreciated. Thanks!

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First of all, the partial fraction decomposition is $$f(z) = \frac{1}{2z} - \frac{1}{z - 1} + \frac{1}{2(z - 2)}.$$ The Laurent series converges on an annulus, so $A_1$ must not contain the origin: $$A_1 = \left\{z : 0 < |z| < 1 \right\}.$$ Then, you can just use the usual Taylor expansion around $z = 0$: \begin{align} f(z) &= \frac{1}{2z} + \frac{1}{1 - z} - \frac{1}{4(1 - \tfrac{z}{2})} \\ &= \tfrac{1}{2}z^{-1} + \sum_{n=0}^\infty z^n - \tfrac{1}{4} \sum_{n=0}^\infty \left(\frac{z}{2}\right)^n \\ &= \tfrac{1}{2}z^{-1} + \sum_{n=0}^\infty \left(1 - \frac{1}{2^{n+2}}\right)z^n \\ &= \sum_{n=-1}^\infty \left(1 - \frac{1}{2^{n+2}}\right)z^n \\ &= \tfrac{1}{2}z^{-1} + \tfrac{3}{4} + \tfrac{7}{8}z + \tfrac{15}{16}z^2 + \cdots \end{align}

There is actually an easier way to expand the series (around $z = 0$) which generalizes nicely for the other regions: $$f(z) = \frac{1}{z} \cdot \frac{1}{(z - 1)(z - 2)} = \frac{1}{z} \cdot \left( \frac{1}{1 - z} - \frac{1}{2 - z} \right).$$ Each of these is a geometric series that can be expanded inside or outside its radius of convergence. $$\frac{1}{1 - z} = \begin{cases} \phantom{-}\sum_{n = 0}^\infty z^n, &|z| < 1, \\ -\sum_{n = 1}^\infty \frac{1}{z^n}, &|z| > 1. \end{cases}$$ Similarly, $$\frac{1}{2 - z} = \frac{1}{2(1 - \tfrac{z}{2})} = \begin{cases} \phantom{-}\frac{1}{2}\sum_{n = 0}^\infty \left(\frac{z}{2}\right)^n = \frac{1}{2}\sum_{n = 0}^\infty \frac{z^n}{2^n} = \sum_{n = 0}^\infty \frac{z^n}{2^{n + 1}}, &|z| < 2, \\ -\frac{1}{2}\sum_{n = 1}^\infty \frac{1}{\left(\frac{z}{2}\right)^n} = -\frac{1}{2}\sum_{n = 1}^\infty \frac{2^n}{z^n} = -\sum_{n = 1}^\infty \frac{2^{n - 1}}{z^n}, &|z| > 2. \end{cases}$$ Now, for each of the three regions, choose the appropriate expansion, and combine them. For instance, on $A_2$, where $1 < |z| < 2$, \begin{align} f(z) &= \frac{1}{z} \left(-\sum_{n = 1}^\infty \frac{1}{z^n} - \sum_{n = 0}^\infty \frac{z^n}{2^{n + 1}} \right) \\ &= -\sum_{n = 1}^\infty \frac{1}{z^{n + 1}} - \sum_{n = 0}^\infty \frac{z^{n - 1}}{2^{n + 1}}. \end{align}

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For $\,A_1\,$:

$$|z|<1\implies\frac{|z|}2<1\implies$$

$$\frac1{2z}+\frac1{1-z}-\frac12\frac1{1-\frac z2}=\frac1{2z}+\left(1+z+z^2+\ldots\right)-\frac12\left(1+\frac z2+\frac{z^2}4+\ldots\right)=$$

$$=\frac1{2z}+\frac12+\frac{3z}4+\frac{7z^2}8+\ldots$$

Using that $\,\displaystyle{\frac1{1-w}=1+w+w^2+\ldots\;,\;\;\text{for}\;\;|w|<1}\,$ . I let you to write the above with a sum symbol, if you want.

For $\,A_2\,$ , again we get:

$$1<|z|<2\implies\frac1{|z|}<1\;,\;\;\frac{|z|}2<1 \;$$

so

$$\frac1{2z}-\frac1z\frac1{1-\frac1z}-\frac12\frac1{1-\frac z2}=\frac1{2z}-\frac1z\left(1+\frac1z+\frac1{z^2}+\ldots\right)+\frac12\left(1+\frac z2\frac{z^2}4+\ldots\right)=\ldots$$

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