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I need to find the principal part of the Laurent series for $f(z) = \frac{e^{2z}}{1-\cos(z)}$, around $z = 0$. Also, I have to use the undetermined coefficient method.

I don't know how to proceed. Should I write down the series for each part and then divide it, this is probably the long way? What is this method in the instruction?

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The "undetermined coefficient method" is probably the method where you make an ansatz

$$\frac{f(z)}{g(z)} = \sum_{n=-k}^\infty a_n z^n$$

and determine the coefficients $a_n$ by computing the Cauchy product

$$\sum_{r=0}^\infty \varphi_r z^r = \left(\sum_{n=-k}^\infty a_n z^n\right)\cdot \left(\sum_{m=0}^\infty \gamma_m z^m\right),$$

where the $\varphi_r$ and $\gamma_m$ are the coefficients of the Taylor series of $f$ resp. $g$,

$$f(z) = \sum_{r=0}^\infty \varphi_r z^r;\qquad g(z) = \sum_{m=0}^\infty \gamma_m z^m.$$

In this ansatz, $k$ is the order of the pole, it does not work for essential singularities.

Since $k$ is small here, and you are only interested in the principal part, it is not much work.

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