# conjugate of exponential imaginary number

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?

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 Your first formula is correct only if $w t$ is real... – Morgan Sherman Aug 9 '12 at 21:09 What are $\,w,t\,$? Real, complex...? – DonAntonio Aug 9 '12 at 21:10 Also, "linear-algebra" is not a good tag for this question (try instead something like complex-variables). – Morgan Sherman Aug 9 '12 at 21:12 Given the first, sure. – André Nicolas Aug 9 '12 at 21:13 In general $\overline{e^{z}} = e^{\overline{z}}$ for all $z \in \mathbb{C}$. – m. k. Aug 9 '12 at 21:28

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 $\,|e^{ix}|=1\,\,\,,\,\,\forall x\in\Bbb R\,$