1. ## Complex Numbers III

One final problem I have to solve is:

If w=(z-i)/(z+i) and z lies below the real axis, show that w lies outside the unit circle |w|=1.
How will w move as z travels along the real axis from -infinity to +infinity.

Many thanks once more to those who are able to provide help or advice.

2. Let z = x+yi, where x and y are real and y < 0. Substitute this into w, simplify, and show that |w| > 1.

EDIT: I’ve just tried the problem, and I think a simpler method is to express z in terms of w and let w = u+vi instead. After substituting and simplifying, you should get

$\mathrm{Im}(z)\ =\ \frac{1-u^2-v^2}{(1-u)^2+v^2}$

Since Im(z) < 0, the rest is straightforward.

EDIT again: For the second part of the question
How will w move as z travels along the real axis from -infinity to +infinity.
set Im(z) = 0.

3. Thanks to Jane Bennet for your help, however, I attempted the question and could not work out how the equation simplifies to your expression.

Smiler

4. Originally Posted by smiler
could not work out how the equation simplifies to your expression
This may not help but:
$w = \frac{{z - i}}{{z + i}}\left( {\frac{{\overline z - i}}{{\overline {z + i} }}} \right) = \frac{{z\overline z - zi - \overline z i - 1}}{{\left| {z + i} \right|^2 }} = \frac{{x^2 + y^2 - 2xi - 1}}{{x^2 + \left( {y + 1} \right)^2 }}$ where $z = x + yi$.