[Math] the image of a unit circle under the map $f(z)=1+z^2$

complex-analysis

I am stuck on the following problem:

The image of a unit circle under the map $f(z)=1+z^2$ is :

  1. again the same unit circle

  2. another circle with different centre but same radius

  3. another circle with same centre but different radius

  4. not a circle

  5. None of the above

My Try: Consider the unit circle $|z|=1$. If we write this as $ \left\{ e^{i\theta} : 0 \leq \theta \leq 2 \pi \right\} $, then, its image under $z \mapsto {1+z^2} $ is readily seen to be
$$
\{ 1+e^{2i \theta}=2 (\cos \theta )e^{i \theta} : \theta \in [0,2\pi]\}
$$

(Using $e^{i\theta} + e^{-i\theta} = 2\cos(\theta)$)

Now, I am not sure how to progress further. Can anyone explain?

Best Answer

$z^2$ maps the unit circle onto itself. It actually maps the upper unit semicircle onto the whole unit circle and does the same with the lower unit semicircle. Then +1 shifts it to the right. So you get the circle $|z-1| = 1$.

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