find laurent series for :

given the constraints:

(a)

............... (b)

............... (c)

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**My Attempt: for part (a)**
first I break up the function using partial fractions:

...

**so for: 0 < |z-1| < 2:**

is this correct?

That looks completely correct to me.
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?

No, it doesn't need simplifying. It represents the one negative power of z-1 in that expansion.
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.

The reason that this is the correct expansion in this region is that the condition for the infinite series to converge is |(z-1)/2| < 1.
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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?