How can I show that $$\overline{e^{i\theta}}=e^{-i\theta}$$
I know that if z is a complex number so
$$\overline{e^z}=e^\bar{z}$$
But i don't understand how to show this result.
Asked
Active
Viewed 180 times
3
-
Just by $\overline{i\theta}=-i\theta$. – Mar 12 '15 at 21:19
3 Answers
4
$$e^{i\theta}:=\cos\theta+i\sin\theta\implies\overline{e^{i\theta}}=\cos\theta-i\sin\theta=\cos(-\theta)+i\sin(-\theta)=:e^{-i\theta}$$
Timbuc
- 34,191
0
You could look at the power series expansion of $\exp$. It has only real coefficients, so conjugating only affects the variable $z$.
Once you have the second equation you get the first because $\overline i=-i$.
Rasmus
- 18,404
0
Well, if you already know that $\overline{e^z}=e^{\overline{z}}$, this is quite straightforward.
Simply take $z=i\theta$. Since $i$ is the imaginary number and $\theta$ has a real value, the complex conjugate of $z$ is simply: $\overline{z}=-i\theta$.
Therefore: $$\overline{e^{i\theta}}=e^{\overline{i\theta}}=e^{-i\theta}$$
Demosthene
- 5,420