1550251515-Classical_Complex_Analysis__Gonzalez_

90

or alternatively,

z = S(z) =

z1 - z2 z1z2 - z1z2

---z+-----

z1 - z2 z1 - z2

Chapter 2

The function S(z) on the right-hand side is called the Schwarz func-

tion for the straight line. Show that for any given point w the point

w* = S( w) is the symmetric of w with respect to that line (Davis [6]).

31. The Schwarz function for the circle of radius r and center at zo is given
    by
    rz
    z=S(z)=--+zo
    Z - Zo
    Show that for any point w, the point w* = S( w) is the inverse of w
    with respect to the circle.


32. Prove that the equation of the circle through three distinct points
    z1, z2, Z3 can be written in the form
    1 1 1 1
    z Z1 z2 Z3

=0

z 21 .Z2 za

zz Z1Z1 z2.Z2 Z3Z3

33. Let pi,pz,p 3 be nonnegative real numbers such that p~ + pz + p3 = 1,
    and suppose that the points z 1 , z 2 , and z 3 are not collinear. Prove that
    the point

zo = p1z1 + pzzz + p3Z3

lies in the triangle with vertices z 1 , Zz, z 3 •

34. Find the locus of the points z such that the three points z, iz, and i
    are always collinear.

35. If ]{ 1 and ]{ 2 are convex subsets of C, show that ]{ 1 nK 2 is also convex,

but that Ki U K2 need not be convex.

36. If A 1 and A 2 are starlike subsets of C with respect to the same center

zo, show that Ai U Az and Ai n Az are both starlike with respect to zo.

2.5 METRIC SPACES

We have already introduced in the complex number system several types of

"distances" between two points z and z^1 : the ordinary or Euclidean distance

lz -z'I, the chordal distance x(z, z'), and the spherical distance ds(zi, Zz).

In. general, a distance defined in a set S is a function d : S x S -+ JR+

subjected to certain conditions or axioms, and a metric space is a pair