Sketching Regions in the Complex Plane

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March 22nd 2011, 05:49 AM
rorosingsong

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! (Rofl)
March 22nd 2011, 05:54 AM
Prove It

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 $\displaystyle (x, y) = (0, 1)$ .
March 22nd 2011, 05:58 AM
Plato

Quote:

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

$\{z:\Re(z)\ge 0\}$ is the closed right half-plane.
March 23rd 2011, 03:22 AM
rorosingsong

Awesome, thanks so much!

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