# Sketching Regions in the Complex Plane

• Mar 22nd 2011, 05:49 AM
rorosingsong
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! (Rofl)
• Mar 22nd 2011, 05:54 AM
Prove It
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 $\displaystyle \displaystyle (x, y) = (0, 1)$.
• Mar 22nd 2011, 05:58 AM
Plato
Quote:

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

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.
• Mar 23rd 2011, 03:22 AM
rorosingsong
Awesome, thanks so much!