Find the image of the unit circle under the transformation $ f(z)=z^2 $
Answer 1
Emily Hall
The unit circle in the complex plane is given by $ |z| = 1 $, meaning any point $ z $ on the unit circle can be written as $ z = e^{i\theta} $ for some real number $ \theta $.
Under the transformation $ f(z) = z^2 $, the image of $ z $ is:
$ f(z) = (e^{i\theta})^2 = e^{i(2\theta)} $
Since $ e^{i(2\theta)} $ is still a point on the unit circle, the image of the unit circle under $ f(z) = z^2 $ is the unit circle itself.
Answer 2
Thomas Walker
The unit circle is represented by $ |z| = 1 $, so any point $ z $ can be written as $ z = e^{i heta} $.
Under $ f(z) = z^2 $, we have:
$ f(z) = (e^{i heta})^2 = e^{i(2 heta)} $
The image remains on the unit circle, meaning the unit circle maps to itself under the transformation.
Answer 3
Emma Johnson
The unit circle $ |z| = 1 $ transforms under $ f(z) = z^2 $ to:
$ f(z) = e^{i(2 heta)} $
Thus, the image is the unit circle.
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