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Thread: Complex analysis

  1. #1
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    Complex analysis

    Hey guys.

    I have this problem, I need to show that it's true and I don't have a clue.
    I tried to do like alpha = x+yi but it got me nowhere, any ideas?

    Thanks.
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  2. #2
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    Quote Originally Posted by asi123 View Post
    Hey guys.

    I have this problem, I need to show that it's true and I don't have a clue.
    I tried to do like alpha = x+yi but it got me nowhere, any ideas?

    Thanks.
    For $\displaystyle |\alpha| < 1, \alpha \in \mathbb{C}$ define $\displaystyle f(z) = \frac{z-\alpha}{1 - \overline{\alpha} z}$.

    It is defined for $\displaystyle |z|\leq 1$, this is because $\displaystyle |1 - \overline{\alpha}z| \geq \bigg||1| - |\overline{\alpha}||z|\bigg| \geq 1 - |\alpha| > 0$.

    Therefore, on the closed disk $\displaystyle |z|\leq 1$ the function $\displaystyle f$ is a continous function that is analytic on the open disk $\displaystyle |z|<1$. By the maximum modulos theorem theorem the largest value that $\displaystyle |f|$ attains happens on the boundary of the disk i.e. on $\displaystyle |z|=1$.

    We know that $\displaystyle |f|^2 = f ~ \overline{f}$, therefore, if $\displaystyle |z|= 1$ we have:

    $\displaystyle |f(z)|^2 = \frac{z-\alpha}{1-\overline{\alpha}z} \cdot \overline{ \frac{z-\alpha}{1-\overline{\alpha} z} } = \frac{z-\alpha}{1-\overline{\alpha} z} \cdot \frac{\overline{z}-\overline{\alpha}}{1-\alpha \overline{z}} = 1$.

    Therefore, $\displaystyle |f| \leq 1$ for all $\displaystyle |z|\leq 1$ and furthermore $\displaystyle |f| = 1$ if and only if $\displaystyle |z|=1$.
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