Question

*10. Möbius transformations. Let a,b,c,d EC with ad-bc70. The function T(2) = 2 a2 + b cz + d à (2 +-d/c) is called a Möbius transformation (or linear fractional transformation). Show that (a) lim T(z) = 20 if c=0; (b) lim T(z) =-. and lim T(z) = oo if c# 0. 200 200 2+-d/c
10. Mobius transformations. Let a, b, c, d $\in$ ad-bc $\neq$ 0 . The function is called a Mobius transformation (or linear fractional transformation). Show that

a) lim z->inf T(z) = inf if c=0;

b)kim z-> inf T(z) = a/c and lim z-> d/c T(z) = inf if c$\neq$0

Answer 1

Mobius transformation

T(z) = 9z+b z A - Fud/c. Cz+ 17 If C=0. T(z) = az +b = 22 +b lim T(z) = lim 9z+b - OD Z- 2 -) . "] lim 270 (z) = lim ztb 27 cztd - lim z (a+b/z) 2- co zrcadlz) - lim a+b/2 Ztes et d/z - atto pim T(z) - lim 027b za-dle cz rd. 2-)-dlc 91-4%) + b ptp e (-9%) + di