1550251515-Classical_Complex_Analysis__Gonzalez_

250 Chapter 5

In general, only six of the twenty-four ratios are different. In fact, if the

value of one of them is r, the values of the six different cross ratios are:

r, r--i, 1-r, (1-r )-1, r(r -1)-^1 , and r-^1 (r -1). However, if r = -1,

those ratios are equal in pairs, and the cross ratio with value -1 is said

to be harmonic. Another special case arises when three of the ratios are

equal in value (as well as the other three). In this case the cross ratio

r is said to be equi-anharmonic. Find r for the equi-anharmonic case.

13. Suppose that T represents either a hyperbolic or a loxodromic trans-
    formation. Show that Tnz (n a positive integer) converges to one of

the fixed points as n---+ oo, while T-nz converges to the other, provide

that z 'f z1, z2; If T is parabolic, and z1 its fixed point, then Tn z ---+ Z1

as n ---+ oo.

14. Find the bilinear transformations that represent rotations of the
    Riemann sphere.


15. Show that the conjugation J z = z is not a bilinear transformation.


16. Show that if z1, z 2 , z 3 , z 4 are consecutive vertices of a quadrilateral
    inscribed in a circle, then

lz1 - z3llz2 - z4I = lz1 - z2llz3 - z4I + lz2 - z3llz1 -z4I

and conversely (Ptolemy's theorem).

17. Prove that any four distinct points z1, z2, Z3, Z4 can be mapped by

a bilinear transformation into the points 1, -1, u, -u, the value of u

depending on the given points. Show that there are two values for u

and that they are inverses of each other with respect to the unit circle.

18. Find the bilinear transformation that maps the circle lzl = 3 into the
    circle lw - II = 1, carrying the point 3 into O, and 0 into +l.


19. Find a bilinear transformation that maps the circles lz I = 2 and lz+ I I =
    VlO into two concentric circles.
    *20. Prove that if a bilinear transformation carries a ring G = { z : r 1 :=;;

lzl ~ r2} into another ring I<= {w: Ri:::;; lwl:::;; R2}, the ratio of the

radii remains constant.

21. Let z1, z2, z 3 , Z4 be distinct points on a circle. Prove that the ordered
    triplets [z1, z2, Z3] and [z1, z2, z4] determine the same orientation on

the circle iff (z1, z2, z3, z4) > 0.

22. Find a bilinear transformation that maps the point z = 9 into the origin
    w = O, the circle lzl = 1 into a line parallel to the imaginary axis, and
    leaves the circle lzl = 3 invariant.

23. If T and S are two different bilinear transformations, prove that T and

ST s-^1 are of the same type.

24. If a and bare points symmetric with respect to the circle C, show that

the ratio (z -a)/(z -b) has a constant absolute value when the point

z describes C.