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In his book "Complex Analysis" (5.1.3), when talking about Laurent series, Ahlfors' shows that a complex function $f(z)$, which is analytic in an annulus $R_1 < |z-a| < R_2$, can be always written as a

[...] sum $f_1(z) + f_2(z)$ where $f_1$ is analytic for $|z-a|<R_2$ and $f_2$ is analytic for $|z-a|>R_1$ with a removable singularity at $\infty$.

where

$$f_1(z) = \frac{1}{2\pi i} \int_{|\zeta-a|=r} \frac{f(\zeta) d\zeta}{\zeta-z} \text{ for $|z-a| < r < R_2$ } $$

$$f_2(z) = - \frac{1}{2\pi i} \int_{|\zeta - a|=r} \frac{f(\zeta)d\zeta}{\zeta-z} \text{ for $R_1 < r <|z-a|$}$$

Then, as the value of $r$ is "irrelevant as long as the inequality is fulfilled", $f_1$ and $f_2$ are uniquely defined and represent analytic functions in $|z-a|<R_2$ and $|z-a|>R_1$ respectively (see also this question and answers).

I am trying to understand what does it mean for $f_1$ and $f_2$ to be uniquely defined. What if I take, for instance $f_3(z)=f_1(z)+z$ and $f_4(z)=f_2(z)-z$? It seems still true to me that $f_3(z) + f_4(z) = f(z)$ and $f_3(z)$ is analytic in $|z-a|<R_2$, while $f_4(z)$ seems analytic at $|z-a|>R_1$ (not sure what we can say at infinity, though; also, I don't see how I could write $-z$ as a sum of negative powers).

I am asking this also in light of what Penrose says in his book "The road to reality" (9.3), when (explaining frequency splitting on the Riemann sphere) he says:

We think of our splitting of $F(z)$ as expressing it as a sum of two parts, one of which extends holomorphically into the southern hemisphere—called the positive-frequency part of $F(z)$—as defined by $F^\mathbf{+}(z)$, together with whatever portion of the constant term we choose to include, and the other, extending holomorphically into the northern hemisphere—called the negative-frequency part of $F(z)$ as defined by $F^\mathbf{-}(z)$ and the remaining portion of the constant term. If we ignore the constant term, this splitting is uniquely determined by this holomorphicity requirement for the extension into one or other of the two hemispheres.

Here $F(z)$ is a function which is "holomorphic in some open region including the unit circle".

So, in this case, are $F^\mathbf{+}$ and $F^\mathbf{-}$ unique (apart from a constant term)? Is then $F^\mathbf{-}=f_1$ and $F^\mathbf{+}=f_2$? Maybe this also stems from the uniqueness of the Laurent development of $F$ (exercise from Ahlfors, same section), but I am not able to see how.

Thanks and sorry for the silly (maybe) question!

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The function $f_2$ has a removable singularity at $\infty$. This means that the limit $\lim_{z\to\infty}f_2(z)$ exists (in $\Bbb C$). If $f_4(z)=f_2(z)-z$, then it is not true that $\lim_{z\to\infty}f_4(z)$ exists (again, in $\Bbb C$).

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  • $\begingroup$ thanks! in light of this consideration, that should rule out my counter example, can we say that there are only two functions with the properties of $f_1$ and $f_2$ that are $f_1$ and $f_2$ themselves as defined by Ahlfors? $\endgroup$ Commented Dec 1, 2020 at 23:13
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    $\begingroup$ Yes, that is correct. $\endgroup$ Commented Dec 1, 2020 at 23:14

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