Thread: Sketching Regions in the Complex Plane

1. 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!

2. 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 $\displaystyle (x, y) = (0, 1)$.

3. Originally Posted by rorosingsong
$A = \left \{ Z\:\epsilon \: \mathbb{C}\, \left | \ \left |Z - i \right | < 3\: and \: Re\, Z \geq 0 \}$

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| is the interior of a circle centered at $z_0$ with radius $r$.

$\{z:\Re(z)\ge 0\}$ is the closed right half-plane.

4. Awesome, thanks so much!