Sketching Complex Sets Homework | Set Sketching Tips

[Soluz]

Homework Statement

I'm having some major trouble this these two questions.

Sketch the set, s, where s = {z| | z^2 - 1 | < 1 } ... z is a complex number
Sketch the set, s, where s = {Z| | Z | > 2 | Z - 1 | } ... Z is a complex number

2. The attempt at a solution

This is supposed to be done by hand apparently and I just cannot see the insight to what makes these simpler. I've tried substituting z = x + i y but then I just get bogged down by calculation. For the first one I get this: sqrt ( (x^2 - y^2 - 1)^2 + (2xy)^2 ) < 1. How am I supposed to graph that? I would like to take somehow take cases like if they were real numbers in relations but since these are complex numbers I'm sure I cannot do that.

3. Relevant equations

The only things he talked about were circles/discs. If it wasn't for that square in the first question or that inequality in the second one I would know what to do. Right now I'm very lost and I've looking at textbook examples that seem similar - I can't find any.

I know my prof likes all these intuitive geometric properties but I just cannot see it. If it was simpler things like s = {z| | z - 1 | < 1 } then I know what to do. But, with this square I'm sure things get a lot more different.

 

[dynamicsolo]

Homework Statement

I'm having some major trouble this these two questions.

Sketch the set, s, where s = {z| | z^2 - 1 | < 1 } ... z is a complex number
Sketch the set, s, where s = {Z| | Z | > 2 | Z - 1 | } ... Z is a complex number

2. The attempt at a solution

This is supposed to be done by hand apparently and I just cannot see the insight to what makes these simpler. I've tried substituting z = x + i y but then I just get bogged down by calculation. For the first one I get this: sqrt ( (x^2 - y^2 - 1)^2 + (2xy)^2 ) < 1.

If you multiply out the expression under the radical, you'll see that it can be simplified a bit. But I'll let someone else suggest how to proceed from there.

On the second one, keep in mind that | z | is the "length" of a vector from the origin in the Argand diagram to the point representing z . So | z - 1 | is the "length" of a vector from the point x = 1 (or ( 1, 0 ) ) to the same point for z . What sort of curve then satisfies | z | = 2 | z - 1 | ? That defines the boundary for the region that is described by the inequality. So where are the points for which | z | > 2 | z - 1 | ? (That is, more than twice as far from the origin than from ( 1, 0 ). )