Rotation of the Riemann sphere

Question

Show that the mapping $w = \frac{1 + z}{1 - z}$ corresponds to a $90$ degree counter-clockwise rotation of the Riemann sphere about the $y$-axis, $z$ is a complex number.

Answer 1

Hint : This is a Moebius transformation, hence determined uniquely by the image of 3 points. So you can simply compute the image of three different points and check it does agree with a rotation of $90°$ applied to these 3 points.

Answer 2

Hint: Write $w = f(z)$; calculate the fixed points of the Moebius transformation $f$ and the composition $f \circ f$.