Is it possible and is it a good idea to introduce the additive group and metric space $\mathbb{C}/2i\pi \mathbb{Z}$ very soon, at the same time as the complex logarithm $\log(r e^{i \theta}) = \ln(r)+i \theta, \theta \in (-\pi,\pi], r > 0$ ?

With $\log : \mathbb{C}^* \to \mathbb{C}/2i\pi \mathbb{Z}$ we can say $\log(ab) = \log(a)+\log(b), \log(a^n) = n \log(a)$, $\prod_n (1+a_n)$ converges iff $\sum_n \log(1+a_n)$ converges.

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    $\begingroup$ Programs, teaching, helping 18-21yo students in scientific cursus, the context where we use the complex logarithm without mentionning $\mathbb{C}/2i\pi \mathbb{Z}$ concretely @user683 $\endgroup$ – reuns Nov 3 '18 at 22:00
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    $\begingroup$ I think in general, it would be a mistake to mention "additive group" or "metric space" in scientific courses. $\endgroup$ – Gerald Edgar Nov 4 '18 at 13:31
  • $\begingroup$ Perhaps the main difficulty is the divison map $z \mapsto \frac{z}{m}, \mathbb{C}/2i\pi \mathbb{Z}\to \mathbb{C}/2i\pi m \mathbb{Z}$ that should be mentionned so that the students don't believe that $\log(a^{1/m}) = \frac{\log(a)}{m}$ @GeraldEdgar $\endgroup$ – reuns Nov 4 '18 at 21:54
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    $\begingroup$ I'm similarly feeling unsure about the context. You are introducing $\log(re^{i \theta}) = \ln(r) + i \theta$ in science classes for university students, and wondering whether to discuss additive groups, metric spaces, and convergence of infinite products via convergence of their associate log series? $\endgroup$ – Benjamin Dickman Nov 4 '18 at 23:24
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    $\begingroup$ After reading your question and then the comments, I agree with the restraint others have suggested. Indeed, my feeling is that, for your students, trying to understand what something like ${\mathbb C}/2i\pi{\mathbb Z}$ means (let alone what $z \mapsto \frac{z}{m}, \mathbb{C}/2i\pi \mathbb{Z}\to \mathbb{C}/2i\pi m \mathbb{Z}$ means) will be a MUCH HIGHER HURDLE than trying to understand (by examples and basic explanations) the various ideas that you actually want to get across. $\endgroup$ – Dave L Renfro Nov 5 '18 at 10:54

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