To prove that a self-map of C is of the form f(z)=az+b, we need to show that the map preserves angles between curves and the orientation of the curves. This can be done by first showing that the map is a conformal map, meaning that it preserves angles locally, and then using the fact that any conformal map of C must be of the form f(z)=az+b.
A conformal self-map of C is a map that preserves angles between curves and the orientation of the curves. This means that if two curves intersect at a certain angle, their images under the map will also intersect at the same angle. It is a special type of self-map that is commonly used in complex analysis.
Proving that a self-map of C is of the form f(z)=az+b is important because it helps us understand the properties of the map. Knowing that a map is conformal allows us to use powerful techniques and theorems from complex analysis to analyze and solve problems involving the map.
The first step is to show that the map is conformal, which can be done by checking that it satisfies the Cauchy-Riemann equations. Then, we can use the fact that any conformal map of C must be of the form f(z)=az+b to conclude that the map is of this form. Finally, we may need to use some additional techniques, such as the Schwarz lemma, to further prove that the map is indeed of the given form.
Yes, a self-map of C can be of a different form than f(z)=az+b. For example, a self-map of C could be a rotation or a translation, which are not of the form f(z)=az+b. However, if a self-map of C is conformal, then it must be of the form f(z)=az+b, as stated by the Riemann mapping theorem.