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I want know if I will be able to say that follow expression is periodic. i.e $$e^{i x}=e^{ix+2i\pi n}?$$

where $n$ is a real number

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The answer is yes, so long as $n$ is an integer. By Euler's formula, $e^{i\theta}=\cos(\theta)+i\sin(\theta)$ when $\theta$ is a real number. In particular, $e^{2i\pi n}=\cos(2\pi n)+i\sin(2\pi n)=1$. From this it follows that $e^{ix}=e^{ix}e^{2i\pi n}=e^{i(x+2\pi n)}$. Note that it also follows that when $n$ is not an integer the equality will NOT hold.

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