conjugate of exponential imaginary number

Question

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?

Answer 1

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