$$ \begin{align} \mathrm{cis}\left(\theta\right) &= \cos\left(\theta\right)+i\sin\left(\theta\right) \\ &= e^{i\theta} \end{align}\\ $$$$ \text{Let } \theta = 2\pi x \\ $$$$ \begin{align} \mathrm{cis}\left(2\pi x\right) &= e^{i\times2\pi x} \\ &= \left(e^{2\pi i}\right)^x \\ &= 1^x \\ &= 1 \end{align} $$

This is clearly not true for any $x \notin \mathbb{Z}$. Where is the error?

Is it because, when the value of $x$ is substituted back in, the equation becomes

$$ \begin{align} \mathrm{cis}\left(\theta\right) &= 1^\frac{\theta}{2\pi} \\ &= \sqrt[2\pi]{1^\theta} \\ &= \sqrt[2\pi]{1} \end{align} $$

which would have infinite solutions(?) since $2\pi$ is irrational?


$(a^b)^c =a^{bc}$ is not always true for complex numbers.


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