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ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2014/6/b/i/solution

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Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 6 / b / i / Solution by

Codex  0 2026-10-31

When all three points are finite, use

g (z) = \frac{( z - z _{1} ) ( z _{2} - z _{3} )}{( z - z _{3} ) ( z _{2} - z _{1} )} .

(1)

The distinctness assumptions ensure that it is a nonconstant Möbius transformation, and direct substitution gives the three required values. If one of the specified points is infinity, the limiting formulas are

z_{3} = \infty z_{1} = \infty z_{2} = \infty (z - z_{1}) / (z_{2} - z_{1}) (z_{2} - z_{3}) / (z - z_{3}) (z - z_{1}) / (z - z_{3}) .

(2)

These cover every case because the points are distinct.

For uniqueness, if

g

and

g

have the same three images, the Möbius map

g \circ g^{- 1}

fixes

0, 1, \infty

. Fixing zero and infinity makes it

z \mapsto cz

with

c \neq = 0

; fixing one gives

c = 1

. Therefore the normalized map is unique. This is the sharp three-transitivity of Möbius transformations.

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