# Homework Help: Laurent Series

1. Jul 9, 2008

### mathfied

i get the idea of the laurent expansion but i get confused with the constraints and how they change the way you work with the expansion.
by now you can prob. tell im trying to get to grasp with complex analysis as a whole.

for example i have this :

find laurent series for :$$f(z) = \frac{z} {{z^2 - 1}}$$

given the constraints:
(a) $$0 < \left| {z - 1} \right| < 2$$ ............... (b) $$\left| {z + 1} \right| > 2$$ ............... (c)$$\left| z \right| > 1$$

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My Attempt: for part (a)

first I break up the function using partial fractions:
$$\frac{z} {{z^2 - 1}} = \frac{z} {{(z - 1)(z + 1)}} = \frac{A} {{(z - 1)}} + \frac{B} {{(z + 1)}}$$

$$z = A(z + 1) + B(z - 1)$$

setting: z=1:
$$1 = 2A,A = \frac{1} {2}$$

setting: z=-1:
$$- 1 = B( - 2),B = \frac{1} {2}$$

$$so:\frac{z} {{z^2 - 1}} = \frac{1} {{2(z - 1)}} + \frac{1} {{2(z + 1)}}$$

so for: 0 < |z-1| < 2:
$$\begin{gathered} \frac{1} {{2(z - 1)}} + \frac{1} {{2(z + 1)}} \hfill \\ = \frac{1} {{2(z - 1)}} + \frac{1} {2}\left[ {\frac{1} {{2 - ( - (z - 1)}}} \right] \hfill \\ = \frac{1} {{2(z - 1)}} + \frac{1} {2}\frac{1} {2}\left[ {\frac{1} {{1 - \left[ { - (\frac{{z - 1}} {2})} \right]}}} \right] \hfill \\ = \frac{1} {{2(z - 1)}} + \frac{1} {4}\left[ {\frac{1} {{1 - \left[ { - (\frac{{z - 1}} {2})} \right]}}} \right] \hfill \\ = \frac{1} {{2(z - 1)}} + \frac{1} {4}\sum\limits_{n = 0}^\infty {\left[ {( - 1)^n \frac{{(z - 1)^n }} {{2^n }}} \right]} \hfill \\ \frac{1} {{2(z - 1)}} + \sum\limits_{n = 0}^\infty {\left[ {( - 1)^n \frac{{(z - 1)^n }} {{2^{n + 2} }}} \right]} \hfill \\ \end{gathered}$$

is this correct? i'm predicting that i may have missed out two parts:
1) the first part of the final equation : 1/2(z-1) : can this be simplified and integrated into the sum formula?

2) the constraint for part (a) was 0 < |z-1| < 2. I didn't know how to interpret |z-1| being between 0 and 2.

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for the other two parts - the constraints are (part (b) |z+1|>2 , part (c) |z|>1) - please could you advise me on how the constraints are meant to be used and how the final answer changes? is there a trick to this?