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Thread: Complex function help

  1. #1
    Senior Member Danneedshelp's Avatar
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    Complex function help

    Q: Let $\displaystyle w=f(z)=i(\frac{1-z}{1+z})$. Show that $\displaystyle f$ maps the open unit disk $\displaystyle \{z\in{\mathbb{C}}:|z|<1\}$ into the upper half-plane $\displaystyle \{w\in{\mathbb{C}}:Im(w)>0\}$, and maps the unit circle $\displaystyle \{z\in{\mathbb{C}}:|z|=1\}$ to the real line.

    Im am not sure how to approach this. I found a problem similar to this one in my text book. Here is what I have worked out so far:

    $\displaystyle Im(f(z))=Im(i(\frac{1-z}{1+z}))=Im(i(\frac{|1-z|^{2}}{(1+z)(1-\bar{z})}))=Im(\frac{|1-z|^{2}i}{1-|z|^{2}})=\frac{|1-z|^{2}i}{1-|z|^{2}}$

    Since $\displaystyle |z|<1\Leftrightarrow\\|z|^{2}<1\Leftrightarrow\\0< 1-|z|^{2}$ and $\displaystyle |1-z|^{2}>0$ for all $\displaystyle z$, $\displaystyle f$ maps from the open disk of raduis 1 into the upper half plane.

    I am not sure if I interpreting what I read correctly, so any help would be appreciated. Thanks.
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    Q: Let $\displaystyle w=f(z)=i(\frac{1-z}{1+z})$. Show that $\displaystyle f$ maps the open unit disk $\displaystyle \{z\in{\mathbb{C}}:|z|<1\}$ into the upper half-plane $\displaystyle \{w\in{\mathbb{C}}:Im(w)>0\}$, and maps the unit circle $\displaystyle \{z\in{\mathbb{C}}:|z|=1\}$ to the real line.

    Im am not sure how to approach this. I found a problem similar to this one in my text book. Here is what I have worked out so far:

    $\displaystyle Im(f(z))=Im(i(\frac{1-z}{1+z}))=Im(i(\frac{|1-z|^{2}}{(1+z)(1-\bar{z})}))=Im(\frac{|1-z|^{2}i}{1-|z|^{2}})=\frac{|1-z|^{2}i}{1-|z|^{2}}$

    Since $\displaystyle |z|<1\Leftrightarrow\\|z|^{2}<1\Leftrightarrow\\0< 1-|z|^{2}$ and $\displaystyle |1-z|^{2}>0$ for all $\displaystyle z$, $\displaystyle f$ maps from the open disk of raduis 1 into the upper half plane.

    I am not sure if I interpreting what I read correctly, so any help would be appreciated. Thanks.


    Check carefully the denominator in the third equality of your "proof": do you think that $\displaystyle (1+z)(1+\overline{z})=1-|z|^2$?? I don't think so.
    A similar mistake happened with the numerator in the second expression...

    Tonio
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  3. #3
    Senior Member Danneedshelp's Avatar
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    what would be the correct approach? Should I write out z in the form x+iy and expand everyting out, distribute the i, and group?
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  4. #4
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    You have made a mistake in the fractions: $\displaystyle \dfrac{(1-z)}{(1+z)}\dfrac{(1+\overline{z})}{(1+\overline{z} )} $.
    Now $\displaystyle (1-z)(1+\overline{z})=1-2i\text{Im}(z)-|z|^2 $.
    And $\displaystyle (1+z)(1+\overline{z})=|1+z|^2 $


    So, $\displaystyle \displaystyle \text{Im}(i~f(z))=\dfrac{1-|z|^2}{|1+z|^2}$.
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  5. #5
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by Plato View Post
    You have made a mistake in the fractions: $\displaystyle \dfrac{(1-z)}{(1+z)}\dfrac{(1+\overline{z})}{(1+\overline{z} )} $.
    Now $\displaystyle (1-z)(1+\overline{z})=1-2i\text{Im}(z)-|z|^2 $.
    And $\displaystyle (1+z)(1+\overline{z})=|1+z|^2 $


    So, $\displaystyle \displaystyle \text{Im}(i~f(z))=\dfrac{1-|z|^2}{|1+z|^2}$.
    So, just to be clear, the conjugate of $\displaystyle (1+z)$ is $\displaystyle (1-\bar{z})$ and the conjugate of $\displaystyle (1-z)$ is $\displaystyle (1+\bar{z})$? If so, this means I alwars change the sign and take the conjugate of the complex number in the expression. Correct?

    Now, since $\displaystyle |z|<1\Leftrightarrow\\|z|^{2}<1\Leftrightarrow\\0< 1-|z|^{2}$ and $\displaystyle |1+z|^{2}>0$ for all $\displaystyle z$ in the domain, it follows that $\displaystyle f$ maps fromt he unit disk into the upper half-plane.

    Is this a sufficient answer to the question?

    Thanks for the help
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  6. #6
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    Quote Originally Posted by Danneedshelp View Post
    So, just to be clear, the conjugate of [tex](1+z)[\math] is $\displaystyle (1-\bar{z})$ and the conjugate of $\displaystyle (1-z)$ is $\displaystyle (1+\bar{z})$?
    That is completely wrong.
    First. To rationalize a complex fraction multiply by the conjugate of the denominator.

    So $\displaystyle \dfrac{z}{w}=\dfrac{z\overline{w}}{|w|^2} $

    Moreover, $\displaystyle \overline{1+z}=1+\bar{z} $ AND $\displaystyle \overline{1-z}=1-\bar{z} $
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