Question

Could I please get an explicit example of a conformal transformation represented through only rotations on the riemann sphere to help show mobius transformations?

Answer 1

The Riemann sphere is an abstract idea implementing in a correct way the addition of a point ∞ to C. There are several models of this Riemann sphere, one just being yourC∞:=C∪{∞} (together with certain exception handling rules), and another one being S2 inheriting the conformal structure from its embedding in R3 and being related to C¯ via the stereographic projection

(resp. σ−1)

We take it for granted that rotations of S2, being conformal maps, are conjugated via σσ to certain special Moebius transformations

The question now is to characterize these transformations T.

A rotation of S2 which is different from the identity has two antipodal fixed points which are neither attracting nor repelling. We therefore have to find the Moebius transformations T with two fixed points that are carried by σ−1 to antipodal points on S2, and have a "scaling factor" λ of absolute value 1.Two points p, q∈C are carried to antipodal points by σ−1 if q=−1p¯. It follows that the Moebius transformations ′T:z↦z′ in question can be written as

Here eiϕ is the "scaling factor" mentioned above. Now solve (1) for z′, and you will obtain T in the form

with certain coefficients a, b, c, d.we will get after multiplying all coefficients with e−iϕ/21+|p|2.Conversely you should show that a T having coefficients as in your text has fixed points pp and −1/p¯and a "scaling factor" eiϕ