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I am trying to find the image of $\{z \in \mathbb{C} : |z| < 1\}$ under $f(z) = \frac{1}{z}$

Let $$w = \frac{1}{z} \Rightarrow z = \frac{1}{w} = \frac{1}{u+iv} = \frac{u-iv}{u^2 + v^2}$$

if $w = u +iv$.

Then considering the boundary first, $$|z| =1 \Rightarrow \Big|\frac{u}{u^2+v^2} -i\frac{v}{u^2+v^2}\Big| =1 $$ Using Pythagoras gives $$\left(\frac{u}{u^2 +v^2}\right)^2 + \left(\frac{v}{u^2 +v^2}\right)^2 = 1$$ but this is not the equation of a unit circle because I have this factor of $(u^2 + v^2)^2$ lying about, which is what the answer should be...

What's happening?

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Of course, it makes no sense to talk about the image of $0$ under $\frac1z$, but the image of $\{z\in\mathbb{C}\,|\,\lvert z\rvert<1\}\setminus\{0\}$ is $\{z\in\mathbb{C}\,|\,\lvert z\rvert>1\}$.

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  • $\begingroup$ Why does my method not yield the unit circle as the boundary? $\endgroup$ – PhysicsMathsLove Jan 22 at 14:11
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    $\begingroup$ \begin{align}\left(\frac{u}{u^2 +v^2}\right)^2 + \left(\frac{v}{u^2 +v^2}\right)^2 = 1&\iff\frac{u^2+v^2}{(u^2+v^2)^2}=1\\&\iff\frac1{u^2+v^2}=1\\&\iff u^2+v^2=1.\end{align} $\endgroup$ – José Carlos Santos Jan 22 at 14:17

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