Problem:
Linear fractional transformation
Consider the transformation w = (i - z) / (i + z).
The upper half plane Im z > 0 maps to the disk |w| < 1 and the boundary of the half plane maps to the boundary of the circle |w| = 1.
1. Show that a point z = x is mapped to the point
w = [(1 - x^2) / (1 + x^2)] + i[(2x) / (1 + x^2)],
and use this to find the images of the points z = -1, 0, 1, infinity. Also, sketch a figure of the x-axis and the unit cirlce showing the images of the desired points and the orientations of the boundaries.
2. Is there an example of a linear fractional transformation that maps the unit disk to the upper half plane and takes the unit circle to the real axis?