1 sum. elementary transformation from z-plane to w-plane (complex number)

ssadi · #1 $Old$ December 3rd, 2008, 05:58 AM

For the transformation $w=z^2$ show that as z moves once round the circle centre O and radius 2, w moves twice round the circle center O and radius 4.

I have two problems for which I am stuck on the sum:

1. How do moving "once" and "twice" is represented on the equation? That is, if w lies on circle center O and radius 4, $|w|=4$ , but how do show that it moves twice?

2. If i take $|z|=2, w=z^2, z=\sqrt w, |z|=|\sqrt w|=2$ : how do I do this part: $2=|\sqrt w|$ , when w is a complex number?
And how do i carry out from there :help:

ThePerfectHacker · #2 $Old$ December 3rd, 2008, 07:42 AM

Quote:

Originally Posted by ssadi $View Post$

For the transformation $w=z^2$ show that as z moves once round the circle centre O and radius 2, w moves twice round the circle center O and radius 4.

I have two problems for which I am stuck on the sum:

1. How do moving "once" and "twice" is represented on the equation? That is, if w lies on circle center O and radius 4, $|w|=4$ , but how do show that it moves twice?

2. If i take $|z|=2, w=z^2, z=\sqrt w, |z|=|\sqrt w|=2$ : how do I do this part: $2=|\sqrt w|$ , when w is a complex number?
And how do i carry out from there :help:

The path that you take on the circle at origin of radius two can be expressed as $g(\theta) = 2e^{i\theta}$ for $0\leq \theta \leq 2\pi$ .
Under the transformation $z\mapsto z^2$ the path becomes mapped to $\left( 2e^{i\theta} \right)^2 = 4e^{2i\theta}$ for $0\leq \theta \leq 2\pi$ . Because of the presence of $2i\theta$ (rather than $i\theta$ ) it means the points moves twice around a circle of radius 4.

ssadi · #3 $Old$ December 3rd, 2008, 08:26 PM

Quote:

Originally Posted by ThePerfectHacker $View Post$

The path that you take on the circle at origin of radius two can be expressed as $g(\theta) = 2e^{i\theta}$ for $0\leq \theta \leq 2\pi$ .
Under the transformation $z\mapsto z^2$ the path becomes mapped to $\left( 2e^{i\theta} \right)^2 = 4e^{2i\theta}$ for $0\leq \theta \leq 2\pi$ . Because of the presence of $2i\theta$ (rather than $i\theta$ ) it means the points moves twice around a circle of radius 4.

Gotcha, thanks