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Thread: Complex Mapping

  1. #1
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    Complex Mapping

    Find the image of the following region:




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


    I am lost on this one. Thanks for any advice with this one.
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  2. #2
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    Quote Originally Posted by hilbertcube182 View Post
    Find the image of the following region:




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


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

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

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

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