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Math Help - Mapping question

  1. #1
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    Mapping question

    The question

    For the mapping f(z) = \frac{z - i}{z + i}, find the image of \{z \eo C : im(z) > 0\}

    My tutor did this a tricky way, by noticing that:

    |f(z)| = |\frac{z - i}{z + i}| < 1

    Therefore, |f(z)| < 1 (which is just the open unit circle).

    But how do I attempt this if I don't notice such nifty tricks? I keep making a mess of it.

    Thanks.
    Last edited by Glitch; June 13th 2011 at 06:04 AM. Reason: Changed less than equal to less than
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  2. #2
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    Quote Originally Posted by Glitch View Post
    The question

    For the mapping f(z) = \frac{z - i}{z + i}, find the image of \{z \eo C : im(z) > 0\}

    My tutor did this a tricky way, by noticing that:

    |f(z)| = |\frac{z - i}{z + i}| < 1

    Therefore, |f(z)| < 1 (which is just the open unit circle).

    But how do I attempt this if I don't notice such nifty tricks? I keep making a mess of it.
    This mapping f(z) is an example of a fractional linear transformation (sometimes called a Möbius transformation). These have the property that they take (lines and circles) to (lines or circles).

    The boundary of the domain D = \{z \in \mathbb{C} : \text{Im}(z) > 0\} is the real axis. This is a line, so the mapping f will take it to a line or a circle. A circle is determined by three points on it, so take three points on the real line, say 0, 1 and –1, and see what f does to them. You find that f(0) = –1, f(1) = –i and f(–1) = i. The circle through the three points –1, i and –i is the unit circle.

    Therefore f takes the boundary of D to the unit circle. The circle splits the complex plane into two regions, namely the inside and the outside of the circle. So f must take D to one or other of those regions. To find out which, choose a point in D and see where f takes it. If you take the point i, you see that f(i) = 0. From that, you conclude that f takes D to the inside of the unit circle, namely the open unit disc.

    That method is not nearly as slick as the trick used by the tutor, but it is a standard method for keeping track of what a fractional linear linear transformation does to a half-plane or a disc.
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  3. #3
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    Re: Mapping question

    Thank you.
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