Complex Question (1 Viewer)

[_ShiFTy_]

If w = (z + 2i)/(2iz - 1),show that when z describes the circle | z | = 1, completely in one direction, then w describes | w | = 1 in another direction

 

[_ShiFTy_]

Noone?

I can find that if |z| = 1, then |w| = 1, but i dno how to prove it goes in the other direction...

 

[Riviet]

Just a thought, but wouldn't the direction not matter if the locus is a circle?

 

[_ShiFTy_]

Just a thought, but wouldn't the direction not matter if the locus is a circle?

That was what i was thinking, but the question was exactly that

 

[Roobs]

What is the question asking, by sayiong :in one direction, or "another direction" thats the bit i dont understand?

 

[_ShiFTy_]

Hmmm, i think i figured it out. If you plot points, you can tell its going in the other direction.

 

[icycloud]

Well basically, you have w = (z+2i)/(2iz-1)
And so arg(w) = arg(z+2i) - arg(2iz-1)

Then you plot the vectors z+2i and 2iz-1. Notice that it is implied that w must be parallel to z+2i rotated clockwise by arg(2iz-1).

Now, as z rotates anticlockwise from -i to i, we have arg(z+2i) moving from pi/2, increasing as z approaches 1 then decreasing back to pi/2 as z reaches i. At the same time, arg(2iz-1) increases from 0 to pi. Therefore, we have arg(w)=arg(z+2i)-arg(2iz-1) decreasing from pi/2 to -pi/2, clockwise. Thus as z moves anticlockwise from -i to i, w moves from i to -i -- the opposite direction. Using the same method, it is shown that when z moves from i to -i, w moves from -i to i, i.e. the opposite direction.

Thus, we have shown that as z traces the unit circle in one direction, w traces the unit circle in the other.

(Note: The unit circle part you have already proven. Just express z in terms of w, then take the modulus of both sides, equating it to 1.)