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Thread: Two question from complex analysis

  1. #1
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    1. Find the general form of bilinear
    mapping that maps the upper halfplane $\displaystyle H^+=\{z : Imz >0\}$
    to yourself.

    2. Prove that for all $\displaystyle R: R>0$, $\displaystyle n \in N$ (large enough),
    all zeros of polynom
    $\displaystyle p_n(z)=\frac{1}{n!}z^n + ... + \frac{1}{2!}z^2 + 1$
    are in $\displaystyle \{z \in \mathbb{C}: |z| > R\}$

    Coment: For 2. $\displaystyle p_n -> e^z$, and... ?

    New question:

    $\displaystyle H^+ = \{z : Im z>0\}. $
    $\displaystyle f(z)=z^5. $

    Now, $\displaystyle f(H^+) = \{ z= r e^{i \theta} : r >0, 0 < \theta < 5\pi\} = \mathbb{C} \ \{x \leq 0\}$
    or $\displaystyle f(H^+) = \{z= r e^{i\theta} : r >0, 0 < \theta < \pi\}$ ?

    Is needed $\displaystyle mod 2\pi$ ?
    Last edited by mr fantastic; Apr 25th 2010 at 04:11 AM. Reason: Merged posts
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