(a) Find Mbius transforms that permute 0, 1, in all possible ways. That is, nd Mbius transforms f1 through f6 so that f1 (0) = 0 f1 (1) = 1 f1 () =...

(a) Find Mbius transforms that permute 0, 1, 1 in all possible ways. That

is, nd Mbius transforms f1 through f6 so that

f1(0) = 0 f1(1) = 1 f1(1) = 1

f2(0) = 0 f2(1) = 1 f2(1) = 1

f3(0) = 1 f3(1) = 0 f3(1) = 1

f4(0) = 1 f4(1) = 0 f4(1) = 1

f5(0) = 1 f5(1) = 1 f5(1) = 0

f6(0) = 1 f6(1) = 1 f6(1) = 0:

(You should be able to see patterns and use tricks; you don't need to

compute six independent functions.)

(b) Next we will try to understand as well as we can what one of these func-

tions does. Find and plot what f4 does to the 6 poles of the sphere

0; 1; i;????1;????i;1. Indicate where the natural circles through these 6

points go. That is, indicate what happens to the real axis, the imag-

inary axis, and the unit circle.

(c) There are two xed points of f4; nd and plot them.

(d) The real axis, the imaginary axis, and the unit circle divide the plane

into 8 regions. Indicate where each of these regions get mapped to.