A Expansion with respect to ##z_1##

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The discussion centers on the manipulation of formal power series in the context of complex variables, specifically regarding the expansion of a function f in terms of z_1 and z'. It explains how f can be expressed as a sum over λ, where each term involves a function f_λ dependent on z'. The participants clarify that changing the order of factors in the series does not affect the overall expression due to the nature of formal power series, which do not require convergence. Additionally, the discussion raises questions about the ordering of terms in the expansion, emphasizing the need for a systematic approach to order elements in multi-index notation. Ultimately, the conversation highlights the flexibility and structure of formal power series in complex analysis.
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I'm reading "From Holomorphic Functions to Complex Manifolds" - Fritzsche & Grauert and I have something that I don't understand very well: If ##\nu \in \mathbb{N}_0^n, t \in \mathbb{R}^n_+## and ##z \in \mathbb{C}^n##, write ##\nu = (\nu_1, \nu'), t = (t_1, t')## and ##z = (z_1, z')##. \ An element ##f=\sum_{\nu \geq 0} a_\nu \mathbf{z}^\nu \in \mathbb{C} [\![ z ]\!]## (formal power series ring) can be written in the form ##f=\sum_{\lambda = 0}^\infty f_\lambda z_1^\lambda## where ##f_\lambda(z')=\sum_{\nu' \ge 0} a_{(\lambda, \nu')} (z')^{\nu'}##. Why can we write this? Changing the order of factors doesn't affect the formal power series? We have that ##f=\sum_{\nu \ge 0} a_\nu z^\nu = \sum_{(\nu_1, \nu')\ge 0} a_{(\nu_1, \nu')}z_1^{\nu_1} (z')^{\nu'} = \sum_{\nu_1 \ge 0} \sum_{\nu' \ge 0} a_{(\nu_1, \nu')}z_1^{\nu_1} (z')^{\nu'} = \sum_{\nu_1 \ge 0} \left(\sum_{\nu' \ge 0} a_{(\nu_1, \nu')} (z')^{\nu'}\right)z_1^{\nu_1} = \sum_{\lambda \ge 0} \left(\sum_{\nu' \ge 0} a_{(\lambda, \nu')} (z')^{\nu'}\right)z_1^{\lambda} = \sum_{\lambda \ge 0} f_\lambda z_1^{\lambda}##. But taking out that common factor ##z_1^{\nu_1}## doesn't change the formal power series? And another thing that I dont't understand: In what order the terms appear in the expansion ##f = \sum_{\nu \ge 0} a_\nu z^\nu##. For example, in ##\mathbb{C}[\![z_1,z_2,z_3]\!]## both terms ##a_{(1,2,3)} z^{(1,2,3)}## and ##a_{(1,3,2)} z^{(1,3,2)}## have "order" ##1+2+3=6## but who comes first in the expansion? I need to consider some order on ##\mathbb{N}_0^3## or something?
 
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The order of summation does not matter in formal power series (we aren't actually summing them, so issues of convergence do not arise). Hence<br /> \sum_{n=0}^\infty \sum_{m=0}^\infty a_{nm}z_1^nz_2^m = \sum_{n=0}^\infty\left(\sum_{m=0}^\infty a_{nm}z_2^m\right)z_1^n = \sum_{n=0}^\infty f_n(z_2) z_1^n.
 
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