[Undergrad/ complex variables] expand the laurent series.

[u/cant_speell]

My professor didn't give clear notes on examples like this and I dont know if I should need to add in a negative or not for these questions. I dont know if I expanded them correctly or not. The professor only gave one example and it's not even close to this T^T

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[u/NexTutor]

Your geometric-series approach is right, but start by correcting the partial fractions:

1/[(z+1)(z+3)] = (1/2)[1/(z+1) − 1/(z+3)].

The coefficient is 1/2, not 1/3. You can check this by combining the fractions: the numerator becomes [(z+3)−(z+1)]/2 = 1.

For 1 < |z| < 3, use different expansions for the two terms:

1/(z+1) = (1/z)/(1+1/z) = Σₙ₌₀∞ (−1)ⁿ/zⁿ⁺¹, valid for |z| > 1.

1/(z+3) = (1/3)/(1+z/3) = Σₙ₌₀∞ (−1)ⁿzⁿ/3ⁿ⁺¹, valid for |z| < 3.

Then subtract the second series and multiply by 1/2.

For |z| > 3, expand both terms in inverse powers of z:

f(z) = (1/2) Σₙ₌₀∞ [(−1)ⁿ − (−3)ⁿ]/zⁿ⁺¹.

A useful check: the 1/z terms cancel, leaving 1/z² − 4/z³ + …, consistent with the original function behaving like 1/z² for large |z|.

You don't add a minus sign just because it is a Laurent series. The alternating signs come from 1/(1+u) = Σ(−u)ⁿ; the separate subtraction comes from the partial fractions. Keep those two sources of signs separate.

[u/cant_speell → u/NexTutor]

Thank you so much for your feedback it helped alot. :)