if |z|=3 is a circle, how do you know if the region outside the circle is

|z|<3

Many thanks

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- Nov 26th 2011, 08:03 AMminicooper58complex inequalities
if |z|=3 is a circle, how do you know if the region outside the circle is

|z|<3

Many thanks - Nov 26th 2011, 08:12 AMPlatoRe: complex inequalities
$\displaystyle |z-w|$ is the distance between $\displaystyle z~\&~w$.

Thus $\displaystyle \{z:|z|=|z-0|=3\}$ is a circle centered at the origin with radius 3.

BUT theof that circle is $\displaystyle \{z:|z|=|z-0|>3\}$, not it is not $\displaystyle <$.**exterior** - Nov 26th 2011, 08:27 AMminicooper58Re: complex inequalities
Hi , the question and answer is in the attachments . sorry i didn't fully explain the transformation part . I don't understand how the region is outside the circle for (b) .

Many thanks - Nov 26th 2011, 12:16 PMHallsofIvyRe: complex inequalities
That's not at all the question you originally asked! The set |z|< 3 is the

**inside**of the circle given by |z|= 3. But the function $\displaystyle w= \frac{z}{z+ 1}$ maps the circle |z|= 3, with center at 0 and radius 3, into the circle with center at $\displaystyle \frac{9}{8}$ and radius $\displaystyle \frac{3}{8}$. It also clearly maps 0 to 0. 0 is inside |z|< 3 but**outside**the circle with center $\displaystyle \frac{9}{8}$ and radius $\displaystyle \frac{3}{8}$. Therefore, the interior of the first circle is mapped to the exterior of the second circle.