# Math Help - Complex Mapping

1. ## Complex Mapping

Find the image of the following region:

under the action of
$f(z)=e^{\bar{z}}$
(e raised to z conjugate)

I am lost on this one. Thanks for any advice with this one.

2. Originally Posted by hilbertcube182
Find the image of the following region:

under the action of
$f(z)=e^{\bar{z}}$
(e raised to z conjugate)

I am lost on this one. Thanks for any advice with this one.
Let $\psi$ be the angle $y=\tfrac{x}{2}$ makes and $\phi$ be the angle $y=3x$ makes.

Your region can be described as, $\{ re^{i\theta} | r \in \mathbb{R}^{\times}, \psi <\theta < \phi \}$.

This means, $f(re^{i\theta}) = \exp ( \overline{re^{i\theta}} ) = \exp ( r e^{-i\theta}) = e^{r\cos\theta}e^{-ir\sin \theta}$.

Now $r\cos \theta,r\sin\theta$ ranges through all values of $\mathbb{R}^{\times}$. Therefore, the range is $r'e^{i\theta'}$ where $\theta'\in \mathbb{R}^{\times}$ and $r'>0,r'\not = 1$.