Could someone take me through the process of plotting the locus of $\frac{1-iwt}{1+iwt}$ as $w$ varies on an Argand plane?

Question

Find the locus of $$\dfrac{1-iwt}{1+iwt}$$ where $w \in \mathbb{C}, t \in > \mathbb{R}$

My method so far was to split it into real and imaginary parts then set $x=\Re(z)$ and $y=\Im(z)$ then solve to get an equation in terms of $x$ and $y$ : $(x^2+y^2=1)$.

The problem is this doesn't tell me the boundaries. The plot should in fact only be below the Real axis - how do I get that boundary?

Thanks.

Answer 1

Let's check, assuming you meant $\;t\in\Bbb R\;$ and, of course, also $\;w\in\Bbb R^+\;$ :

$$\frac{1-wti}{1+wti}=\frac{(1-wti)(1-wti)}{|1+wti|^2}=\frac{1-w^2t^2-2wti}{|1+wti|^2}$$

$$=\frac{1-w^2t^2}{|1+wti|^2}-\frac{2wt}{|1+wti|^2}i$$

But then the above is in the lower complex semiplane iff $\;t>0\;$ ...