image of $\dfrac{z-1}{z+1}$

Question

What is the image of the right plane $\lbrace z \in \mathbb{C}\mid \Re(z) > 0 \rbrace$ under the Möbius transformation $\dfrac{z-1}{z+1}$ and how can I see that?

Answer 1

Hint: The distance of points in the right half plane from the point -1 is always smaller than the distance to +1. The two distances are equal along the imaginary axis.

Answer 2

Möbius transformations take generalized circles to generalized circles.

We have $i\to i, 0\to-1$ and $\infty \to 1$. This shows that the imaginary axis goes to the unit circle.

Next use a test point, say $1$, to get that it's the interior of the circle.