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Math Help - Sketching Regions in the Complex Plane

  1. #1
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    Sketching Regions in the Complex Plane

    Hi, could someone please show me how to sketch this region?

    A = \left \{ Z\:\epsilon \: \mathbb{C}\, \left |  \ \left |Z - i  \right | < 3\:  and \:  Re\,  Z \geq 0 \}

    Also, when sketching something like this, do I label the axes x and y, or Re and Im?

    Any help would be very much appreciated.

    Thanks!
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  2. #2
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    I would label them \displaystyle \textrm{Re}\,(z) and \displaystyle \textrm{Im}\,(z).

    Now if \displaystyle |z - i| < 3

    \displaystyle |x + iy - i| < 3

    \displaystyle |x + (y - 1)i| < 3

    \displaystyle |x + (y - 1)i|^2 < 3^2

    \displaystyle [x + (y - 1)i][x - (y - 1)i] < 3^2

    \displaystyle x^2 + (y - 1)^2 < 3^2.


    So it's the region that's contained inside (but not including) the circle of radius \displaystyle 3 units, centred at \displaystyle (x, y) = (0, 1).
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  3. #3
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    Quote Originally Posted by rorosingsong View Post
    A = \left \{ Z\:\epsilon \: \mathbb{C}\, \left |  \ \left |Z - i  \right | < 3\:  and \:  Re\,  Z \geq 0 \}

    Also, when sketching something like this, do I label the axes x and y, or Re and Im?
    As far as labeling goes, that is strictly up to your instructor/textbook.

    Now if r\in\mathbb{R}^+ then |z-z_0|<r is the interior of a circle centered at z_0 with radius r.

    \{z:\Re(z)\ge 0\} is the closed right half-plane.
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  4. #4
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    Awesome, thanks so much!
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