I'm reading a bit about the exponential function $e_k: \mathbb{R} \longrightarrow \mathbb{C}$ given by $$e_{k}: x \mapsto e^{i k x}=\cos k x+i \sin k x$$
The period is given by $2\pi/|k|$. I do not have an intuitive understanding of this. It's just a formular in my textbook. By plotting using WolframAlpha I got some understanding. But how does WolframAlpha plot for example $e_{2}: x \mapsto e^{i 2 x}=\cos 2 x+i \sin 2 x$ ? How am I suppose to do this by hand?
Kind regards
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