# If the negative and non-negative exponent terms of the Laurent series separately converge inside the annulus, why do we bother adding both together?

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?

• 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. – Lord Shark the Unknown Apr 24 '17 at 4:39
• 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? – user49404 Apr 24 '17 at 12:12
• Because $f_1+f_2$ equals $f$ on the annulus, but neither $f_1$ nor $f_2$ does in general. – Lord Shark the Unknown Apr 24 '17 at 15:07