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I'm trying to read a proof of Laurent expansions for holomorphic functions in a polyannuli in $\mathbb{C}^n$ and I'm having trouble justifying the passage outlined in green in the screenshot below. He does an induction step using Laurent expansions for functions obtained from $f : A \longrightarrow \mathbb{C}$ (where $A$ is the polyannulus where $f$ is holomorphic) by fixating one variable and varying the others ($f(z_1 , \bullet,...,\bullet)$), and by fixating all but one variable ($f(\bullet , z_2,...,z_n)$). Then he combines both expansions.

However, I'm not seeing how that is done, since by induction step we assume the functions have Laurent expansions, but they depend on the values of the variables we fix. And to be able to then just take out the infinite sum from the integrals, we need the uniform convergence of the Laurent series with respect to the domais of the variables we fixed, which is something I tried to prove but was unsuccessful.

He then proves that the final Laurent series converges uniformly (not present in the screenshot), which I understood. Could we use that to prove the first part? Or is it easier than I'm thinking and I'm just not seeing it?

$\mathbf{Alternatively, }$ $\mathbf{if}$ $\mathbf{someone}$ $\mathbf{could}$ $\mathbf{give}$ $\mathbf{me}$ $\mathbf{a}$ $\mathbf{reference}$ $\mathbf{for}$ $\mathbf{a}$ $\mathbf{cleaner}$ $\mathbf{proof}$ $\mathbf{of}$ $\mathbf{this}$ $\mathbf{theorem, }$ $\mathbf{I}$ $\mathbf{would}$ $\mathbf{appreciate}$ $\mathbf{it}$ $\mathbf{a}$ $\mathbf{lot!}$

Any help is appreciated. Thank you in advance :).

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  • $\begingroup$ Hi, Duarte Costa. I am currently studying this book of Volker Scheidemann. I am having trouble on some passages that he does, and I can't find people that has already studied it to discuss some questions. Could we talk about it? Perhaps you know how to answer my questions since you (apparently) studied this book. My email is [email protected] Thanks! $\endgroup$
    – lfsm
    Feb 4 at 2:08

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