I'm confused by the textbook. They give a big formula for the Laurent series expansion, however, in the examples, they don't use the formula at all. Here is one of the examples:

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My questions: Why can we find the residue in this way? Which function, $f(z)$ or $g(z)$, should we pick to calculate the Maclaurin series representation? For example, I would like to find the residue of $\frac{1}{e^{2z}-1}$ and $\frac{z^2}{1-\cos(z)}$. Answer to any one of my questions is appreciated!

Definition of Laurent series representation (am I correct to call it the definition?):

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  • $\begingroup$ They chose $e^z$ because it is easy to remember and use, and then the just substituted $1/z^2$ for $z$, as is standard practice. But that's barely scratching the surface of your great question. $\endgroup$ – The Count Oct 31 '18 at 2:30
  • $\begingroup$ @TheCount Thanks for the comment. It's confusing because they give the big theorem and use several pages to prove it, but in the examples, I don't see how they use the theorem. And after reading those examples, I have no idea what to do when I am asked to find the residues of some "strange" functions. $\endgroup$ – user398843 Oct 31 '18 at 2:34
  • $\begingroup$ You are doing exactly the right thing, which is asking. I have not grokked this in general myself, or I would jump in. I am not an analyst like... at all. $\endgroup$ – The Count Oct 31 '18 at 2:39

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