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I learned the following $$ e^{ik2\pi}=1 $$ and I was wondering whether or not $k$ has to be an integer.


Thought 1: Since $e^{ik2\pi}=\cos(2k\pi)+i\sin(2k\pi)=1$, equating the real and imaginary parts we conclude that $k$ is an integer.

Thought 2: Since $e^{i2\pi}=1$, if we raise both sides to the power $k$ then we have $e^{ik2\pi}=1^k=1$ for any $k$. So $k$ does not have to be an integer.

I am more keen on Thought 1 but I would like to know what is wrong in Thought 2?

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marked as duplicate by Xander Henderson, TheSimpliFire, user21820, John Omielan, Jyrki Lahtonen Aug 30 at 16:43

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