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Math Help - prove f(z1)=f(z2) implies z1=z2 and maps to D(0;1)- Complex analysis

  1. #1
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    Question prove f(z1)=f(z2) implies z1=z2 and maps to D(0;1)- Complex analysis

    f(z)={z\over 1+|z|}

    (i) Prove that f(z_1)=f(z_2) implies z_1=z_2.

    (iv) Prove that f maps \mathbb{C} onto D(0;1).

    [The text suggests using polar coordinates for both of these problems.]
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  2. #2
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    Quote Originally Posted by JJMC89 View Post
    f(z)={z\over 1+|z|}

    (i) Prove that f(z_1)=f(z_2) implies z_1=z_2.


    Put z_1=re^{i\theta}\,,\,\,z_2=se^{i\phi}\,,\,\,r,\,s\  in\mathbb{R}^{+}\,,\,\,0\leq \theta,\,\phi<2\pi , so:

    \frac{z_1}{1+|z_1|}=\frac{z_2}{1+|z_2|}\Longrighta  rrow \left|\frac{z_1}{1+|z_1|}\right|=\left|\frac{z_2}{  1+|z_2|}\right|\Longrightarrow \frac{r}{1+r}=\frac{s}{1+s} .

    From here get that r=s , and then input in the original equality in polar form and get that also \theta=\phi and we're done.


    (iv) Prove that f maps \mathbb{C} onto D(0;1).


    This should be clear: what's the modulus of.. \frac{z}{1+|z|} ?

    Tonio


    [The text suggests using polar coordinates for both of these problems.]
    .
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