TITLE: Conjugate of exponential imaginary number
QUESTION [15 upvotes]: The conjugate of $e^{-iwt}$ is $e^{iwt}$.
Then, what would be the conjugate of $e^{iwt}$? Would it be $e^{-iwt}$?
Also, for $|e^{iwt}|^2$, what would the value look like?

REPLY [19 votes]: Complex conjugation is an automorphism of order 2, meaning $\,\overline{\overline z}=z\,\,,\,\,\forall\,z\in\Bbb C\,$ , so if the conjugate of $\,e^{-iwt}\,$ is $\,e^{iwt}\,$ , then the conjugate of the latter is the former.
Also, writing the trigonometric version of $\,e^{ix}\,\,,\,x\in\Bbb R\,$ , you can check at once that $\,|e^{ix}|=1\,\,\,,\,\,\forall x\in\Bbb R\,$
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