Elementary Functions 219
- Circles through the origin are mapped into straight lines not containing
the origin.
By considering the straight lines as members of the family of circles (i.e.,
as circles passing through the point oo ), the four preceding statements can
be combined into the following theorem.
Theorem 5.3 The transformation w.= 1/z maps circles into circles.
This convention may be justified by considering the effect of the trans-
formation w = 1/z on the Riemann sphere. Under stereographic projection
lines and circles in the plane correspond to circles on the sphere, all lines
going into circles containing the point oo. Under w = 1/i circles on the
z-plane map onto circles on the w-sphere. The point z = 0 maps into
w = oo, and conversely, the point z = oo maps into w = 0.
The transformation w = 1/ z does not preserve the simple ratio of three
distinct points z1, z2, Z3 in C-{O}, since for the corresponding points w1,
w2, w 3 we have
W3 - W1 Z2 Z3 - Zl Z2
(w1,w2,w3) = = - --= -(z1,z2,z3) (5.4-1)
W3 - W2 Z1 Z3 - Z2 Z1However, the cross ratio (or anharmonic ratio) of four points, namely,
(z1,z2,z3,z4) = (z1,z2,z3): (z1,z2,z4)
is invariant under this transformation. In fact, (5.4-1) yields
(w1,w2,w3,W4) = (w1,w2,wa): (w1,w2,w4)
= (z1,z2,z3): (z1,z2,z4) = (z1,z2,z3,z4) (5.4-2)The reader will note that if the w-sphere is superimposed on the z-sphere,
the two points +1 and -1 are fixed or invariant under w = 1/z.
The reciprocal transformation is of great importance in complex analy-
sis. As will be seen later, this transformation is often used to investigate the
behavior of a function (or of some geometric figure) in the neighborhood
of infinity.
Because the transformation w = 1/ z is isogonal, the following definition
of the angle at oo between two arcs 11 and 12 extending to oo in I['.* will
be adopted.
Definition 5.1 Two arcs 11 and 12 in the extended complex plane axe
said to form an angle of 0 radians with vertex at oo if their images r 1
and r2 under the transformation w = 1/ z form an angle of 0 radians with
vertex at the origin.
Alternatively, the angle between "Yl and 12 at oo could be defined by con-
sidering the angle of their images "Yi and 1;, under stereographic projection,
at the north pole of the Riemann sphere.