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linear fractional transformation

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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?

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This solution illustrates a linear fractional transformation.

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Problem:

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 ...

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