1. ## complex inequalities

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

Many thanks

2. ## Re: complex inequalities

Originally Posted by minicooper58
if |z|=3 is a circle, how do you know if the region outside the circle is |z|<3 .
$\displaystyle |z-w|$ 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 the exterior of that circle is $\displaystyle \{z:|z|=|z-0|>3\}$, not it is not $\displaystyle <$.

3. ## Re: 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

4. ## Re: 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.