How would you prove $e^{-\pi ix}$ $\forall x \in \Re$ is not 1-periodic?
The Definition of 1-periodic function is: $f : \Re \to \Im$ be 1-periodic if $f(x+1) = f(x)$ $\forall x \in \Re$
What I have been able to do is:
$e^{-\pi i(x+1)} = e^{-\pi ix}$
$e^{-\pi ix}e^{-\pi 1i}$ = $e^{-\pi ix}$
$e^{-\pi 1i}$ = $e^{0}$
$e^{-\pi 1i}$ = $1$
I don't know how to proceed further or argue why this can't be 1-periodic. I have looked at Euler's formula but was not sure how to use it here.
Thanks