Linear fractional transformations Cornell （线性分式变换康奈尔）.pdfVIP

Math 418 Spring 2006 Linear fractional transformations 0.1. Definitions. We study a special class of maps T : C ∪ {∞} −→ C ∪ {∞}. A linear fractional transformation of C ∪ {∞} is a map of the form az + b (0.1.1) w = T (z) = , a, b, c, d ∈ C. cz + d We think of the transformation as depending on the 2 × 2 matrix a b (0.1.2) g := c d , and write Tg for the transformation. We assume that g = 0, or else we would be dividing 0 for all z. In fact, if c = d = 0, T would be undefined 0 for all z. Note that if we multiply the coefficients by the same λ = 0, we get the same transformation. In particular, if ad − bc = 0, then there is cancellation, and the transformation becomes T (z) = constant except for maybe one value of z where the transformation is undefined. We assume that ad − bc = 0. The following properties are verified by direct calculation. 1: Tg = Tg if and only if there is λ = 0 such that a = λa, b = λb, c = λc, d = λd. This can be written as a b λ 0 a b = · c d 0 λ c d 2: Tg1 ·g2 = Tg1 ◦ Tg2 , where · is usual matrix multiplication, and ◦ is composition of transformations/functions. 3: If g = 1 0 , then T (z) = z. 0 1 g In view of this, we define dividing any nonzero number by 0 to be ∞. With this convention, we define a