# 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 '18 at 22:00
• I think in general, it would be a mistake to mention "additive group" or "metric space" in scientific courses. Nov 4 '18 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 '18 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 '18 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 '18 at 10:54