Expand        in a Laurent series valid for

(i) and

(ii) .

To find the Laurent series expansion of f(z) = \frac{1}{(z-1)^{2}(z-3)} around different points, we can use partial fraction decomposition to rewrite the function as a sum of simpler terms. Then, we can expand each term as a power series.

First, perform partial fraction decomposition:

To find the constants A, B, and C, we multiply both sides of the equation by the denominator (z-1)^2(z-3) to clear the fractions:

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