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Math Help - complex inequalities

  1. #1
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    complex inequalities

    if |z|=3 is a circle, how do you know if the region outside the circle is
    |z|<3

    Many thanks
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  2. #2
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    Re: complex inequalities

    Quote Originally Posted by minicooper58 View Post
    if |z|=3 is a circle, how do you know if the region outside the circle is |z|<3 .
    |z-w| is the distance between z~\&~w.

    Thus \{z:|z|=|z-0|=3\} is a circle centered at the origin with radius 3.

    BUT the exterior of that circle is \{z:|z|=|z-0|>3\}, not it is not <.
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  3. #3
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    Re: complex inequalities

    Hi , the question and answer is in the attachments . sorry i didn't fully explain the transformation part . I don't understand how the region is outside the circle for (b) .

    Many thanks
    Attached Thumbnails Attached Thumbnails complex inequalities-question.png   complex inequalities-answer.png  
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  4. #4
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    Re: complex inequalities

    That's not at all the question you originally asked! The set |z|< 3 is the inside of the circle given by |z|= 3. But the function w= \frac{z}{z+ 1} maps the circle |z|= 3, with center at 0 and radius 3, into the circle with center at \frac{9}{8} and radius \frac{3}{8}. It also clearly maps 0 to 0. 0 is inside |z|< 3 but outside the circle with center \frac{9}{8} and radius \frac{3}{8}. Therefore, the interior of the first circle is mapped to the exterior of the second circle.
    Last edited by HallsofIvy; November 26th 2011 at 12:29 PM.
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