I wanted to do an exercise and it turns that I didn't understand something about branch of logarithm. First, and I think for this I now get it, we know that if it exists $\log$ a determination of logarithm on a open set $D$, we have : $\log(z) = \ln(|z|) + i\arg(z)$, $\arg$ a function to determine of course. But I also thought we should have $\arg(D) \subset ]a;b[$ with $b-a = 2\pi$. But it turns that it not necessarily the case (as the example below, with $I$). But sometimes it solves the problem, for example when $D \cap R_{-} = \emptyset$ (then we can define arg on $]-\pi;\pi[$), actually it's when we can remove a ray on a complex plane. Until here, I'm right ?

But then, when we have a simply connected set $D$ which verify $0 \notin D$, we can also define a branch of logarithm. So, for example, here, I consider $I = \{ e^{\theta + i\theta} \; | \; \theta \in \mathbb{R} \}$, and $D = \mathbb{C}\backslash I $, $D$ is simply connected and we define a branch of logarithm such as $\log(e) = 1$. We want to determine $\log(e^{15})$. Actually, we should have $\log(15) = 15 + i\arg(15)$, all the difficulties are to determine $\arg(15)$. One suggest to write $\log(z) = \log(e) + \delta(z)$, where $\delta$ is the change of $\log(z)$ over a path which connect $z=e$ and $z=e^{15}$, and $\delta$ is a continuous function which verify $\delta(z) = \delta_{ln}(z) + i\delta_{arg}(z)$ (where $\delta_{ln}$ is a change for the $ln(|z|)$ part and $\delta_{arg}$ for the $\arg(z)$ part) . But then, from $e \in [e^0;e^{2\pi}]$ and $e^{15} \in [e^{4\pi};e^{6\pi}]$, it's written that we can deduce : $\delta_{arg}(z) = 4\pi$. And I just don't understand this part.

If someone can help me, thank you very much !

Edit : Oh maybe, for the last question, as we can see that the spiral intersect the positive real axis at $\{e^{2n\pi} \; | \; n \in \mathbb{Z}\}$, we have to count how many times the spiral intersect the real axis between $e$ and $e^{15}$, as we add $2\pi$ at each passage (by continuity of the argument). But then, what I don't understand, is why we must have $n\pi$, $n \in \mathbb{N}$ for $\delta_{arg}$, I mean, even if we add $4\pi$, then in order to reach $e^{15}$ we should still add something for the argument, why we whould have $arg(e^{15}) = arg(e) + n\pi$ ?

Edit 2 : Wow, sorry, actually $e$ and $e^{15}$ are real so are on real axis, so we just have to count how many times the spiral intersect the real axis between $e$ and $e^{15}$ to find $\delta_{arg}$, right ?


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