Complex Analysis by Serge Lang

Author:Serge Lang
Language: eng
Format: epub
Publisher: Springer-Verlag Wien 2012
Published: 2014-06-21T00:00:00+00:00

Proof. Suppose c = 0. Then F(z) = (az + b)/d and F = Tβ ∘ Mα, with β = b/d, α = a/d. Suppose this is not the case, so c ≠ 0. We divide a, b, c, d by c and using these new numbers gives the same map F, so without loss of generality we may assume c = 1. We let β = d. We must solve

or in other words, az + b = α + γz + γd. We let γ = a, and then solve for α = b − ad ≠ 0 to conclude the proof.

The theorem shows that any fractional linear map is a composition of the simple maps listed above: translations, multiplication, or inversion.

Now let us define a straight line on the Riemann sphere S to consist of an ordinary line together with ∞. Theorem 5.2. A fractional linear transformation maps straight lines and circles onto straight lines and circles. (Of course, a circle may be mapped onto a line and vice versa.)

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.