Elementary Functions 229
and Hf:. 0, and the transformation can be written as in (5.7-2). Letting
W-Z1
W=S(w)= --,
w-z2
Z=S(z)= (5.8-1)
the transformation reduces to the simple form
W = MZ (5.8-2)
If we write M = Rei^9 , the following subcases occur: (a) M = R (0 =
2k7r). In this subcase the multiplier is a positive real number other than 1,
since M = 1 would imply H = 0 by (5.7-5). The transformation W = RZ
defines a homotecy, as seen in Section 5.2, and has the following properties:
- The points 0 and oo are fixed points of the transformation. A straight
line through the origin (a circle through 0 and oo) is mapped into
itself. The half-planes determined by such a line are transformed into
themselves. Also, a half-ray from the origin is mapped into itself. - Every circle with center at the origin is mapped into another circle with
the same center. The family of such circles is orthogonal to the family
of straight lines through the origin. The points 0 and oo are inverses
of each other with respect to any one of those circles (Fig. 5.6a).
Letting w = L(z), W = T(Z), in view of (5.8-1) we have
w = s-^1 (W) = s-^1 T(Z) = s-^1 TS(z)
so that
Now suppose that a figure A (e.g., line, circle, region) is carried into
B by T, i.e.,
B=T(A)
Then it follows that the figure s-^1 (A) is carried into s-^1 (B) by L. In fact,
LS-^1 (A) = (s-^1 TS)S-^1 (A) = s-^1 T(A) = s-^1 (B)
Hence passing (5.8-2) back to the original variables, and considering that
s, as well as s-1, are bilinear transformations, we have the following:
- A circle C 1 passing through the fixed points z1 and z2 is mapped into
itself. The regions determined by such a circle are transformed into
themselves. Each arc ilZ2 is mapped into itself. - The circles C 2 that are orthogonal to those passing through the fixed
points are mapped into orthogonal circles of the same family. The fixed
points z 1 and z 2 are inverses of each other with respect to any one of
the circles C2 (Fig. 5.6b ).