# Complex logarithm and $\mathbb{C}/2i\pi \mathbb{Z}$

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.

• 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 Nov 3, 2018 at 22:00
• I think in general, it would be a mistake to mention "additive group" or "metric space" in scientific courses. Nov 4, 2018 at 13:31
• 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 Nov 4, 2018 at 21:54
• 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? Nov 4, 2018 at 23:24
• 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. Nov 5, 2018 at 10:54