1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 65

A pseudochordal distance defined for pair of points on the unit open disk

with center at the origin is defined by

lz1 -z2I

v(z1,z2) = ll -Z1Z2 _ I'. (1.17-17)

This represents a distance invariant in a certain model of non-Euclidean

geometry. For further details the reader is referred to Caratheodory [3],

Vol. 1, Chap. 3. ·

Remark The stereographic projection of the complex plane on the sphere

can also be accomplished by letting the complex plane be the plane of the

equator, and then using as a center of projection either the north or the

south pole. Any other surface topologically equivalent to the sphere could

also be used to represent the extended complex number system.

EXERCISES 1.7

1. Find the coordinates of the points on the Riemann sphere correspond-
   ing to the following complex numbers.
   (a) -i (b) -l + i (c) 1 + Vsi
   ../2


2. Find the complex numbers corresponding to the vertices of a cube
   inscribed on the Riemann sphere with edges parallel· to the coordinate
   axes.


3. Show that x(z1,z2) = x(z1,z2) = x(l/zi, l/z2).


4. Show that z 1 and z 2 map on diametrically opposite points of the sphere

iff 1 + Z1Z2 = 0.

5. Let B be a bounded set in the complex plane, i.e., such that lzl < M

for all z E B. If z1, Z2 E B, show that for z1 '/= z2,

x(zi, z2) < lz1 -z2 I < (1 + M^2 )x(z1, z2)

6. The sector 0 ::=; lzl ::=; r, a ::=; Arg z ::=; (3, is projected stereographically
   on the sphere. Find the area of the projection.


7. Prove that under stereographic projection the circles and straight lines
   of the plane map into circles of the sphere, and conversely. Find the

condition on a straight line ax + by + c = 0 to map into a great circle

of the sphere.

8. Find the radius of the spherical image of a circle in the plane with
   center at z 0 and radius R.


9. Prove that the stereographic projection is an isogonal transformation
   (i.e., it preserves angles).