Timeline for Complex logarithm base 1

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6 hours ago	comment	added	M.G.		This answer is a bit of a mess :-) First of all, your usage $(\alpha \beta)^y = \alpha^y \beta^y$ in the first equation implies that you've made a choice to use the principal branch of $\log$, I guess. Since $y$ is complex, multiplicative distributivity does not hold in general in the complex numbers (branches and all). Secondly, again since $y$ is complex, the expression $\cos(2 \pi k y) + i \sin(2 \pi k y)$, which itself is simply $e^{2 \pi k y}$, certainly need not be of absolute value $1$. In other words, you've implicitly assumed that $y$ is real.
6 hours ago	history	answered	Dan	CC BY-SA 4.0