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Sorry if this is dense, but I'm still trying to understand the motivation for Laurent series. I feel like I understand the theorems of how to find Laurent series etc., but I don't understand why they're more useful than series that only have positive or negative exponents.

We can split an analytic function $f(z)$ into $f(z) = f_1(z) + f_2(z)$, where $f_1$ is analytic for $|z-z_0|<R_1$, $f_2$ converges for $|z-z_0|>R_2$, and so the region of convergence is $R_2<|z-z_0|<R_1$, which is non-empty if $R_2\le R_1$.

But since $f_1$ and $f_2$ each are individually analytic on the annulus anyway, what's the purpose of adding them together to get a function $f$ that is only analytic on the annulus?

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  • $\begingroup$ In this situation we are starting with the function $f$, not finishing with it. We split $f$ as the sum of $f_1$ and $f_2$, one is represented by a power series in $z-a$ and the other as a power series in $(z-a)^{-1}$ where $a$ is the centre of the annulus. $\endgroup$ – Lord Shark the Unknown Apr 24 '17 at 4:39
  • $\begingroup$ Sure, but why can't we just define the Laurent series as the restriction of f1 xor f2 to the annulus, instead of using both? $\endgroup$ – user49404 Apr 24 '17 at 12:12
  • $\begingroup$ Because $f_1+f_2$ equals $f$ on the annulus, but neither $f_1$ nor $f_2$ does in general. $\endgroup$ – Lord Shark the Unknown Apr 24 '17 at 15:07

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