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

• Dec 3rd 2008, 04:58 AM
1 sum. elementary transformation from z-plane to w-plane (complex number)
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:
• Dec 3rd 2008, 06:42 AM
ThePerfectHacker
Quote:

Originally Posted by ssadi
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.
• Dec 3rd 2008, 07:26 PM
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.