Why are there infinitely many values if a complex number is raised to an irrational power? I have to prove this, and don't know how. I am thinking Cauchy sequences, but I don't know how to start. Could you you please help? Thanks!
2026-03-27 22:52:37.1774651957
Complex number raised to irrational power
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It's a matter of definition. If $a,b\in\mathbb C$ and $a\neq0$, then $a^b$ is any number of the form $\exp(b\log a)$, where $\log a$ is any logarithm of $a$. But if $l_a$ is one logarithm of $a$, then$$\{\text{logarithms of }a\}=\{l_a+2\pi im\,|\,m\in\mathbb{Z}\}.$$So, if $b$ is irrational and if $m\neq n$, then$$\exp\bigl(b(l_a+2\pi im))\neq\exp\bigl(b(l_a+2\pi in)).$$