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Thread: 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?

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

    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 $\displaystyle \displaystyle \textrm{Re}\,(z)$ and $\displaystyle \displaystyle \textrm{Im}\,(z)$.

    Now if $\displaystyle \displaystyle |z - i| < 3$

    $\displaystyle \displaystyle |x + iy - i| < 3$

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

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

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

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


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

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

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