Elementary Functions 233
(c) M = Rei^6 (R-=/= 1,0-=/= 2k1r). In this subcase we have W = Rei^6 z,
and the transformation can be thought as the successive application of the
transformations
andthat is, as a homotecy followed by a rotation about the origin. Returning to
the original variables, this amounts to a hyperbolic transformation followed
by an elliptic one. Geometrically, this means a combination of the motions
indicated by arrows in Fig. 5.6b and 5.8, i.e., a translation along an arc
containing the fixed points followed by a rotation along a circle orthogonal
to that arc.
The bilinear transformations of this type are called loxodromic. In gen-
eral, a loxodromic transformation does not map a circle into itself, except
when 0 = ±7r. In this case an arc of a circle joining the fixed points z 1 and
z 2 is carried by the transformation into another arc, also joining the fixed
points, and making an angle ±7r with the first arc, i.e., into an arc of the
same circle. However, in this case the interior of a circle through the fixed
points is mapped onto its exterior.
To see what analytic condition is implied by a loxodromic transforma-
tion, let M = Rei^6 in:-i(5.7-5). We obtain
or
Rei6 + ..!_ e-i6 =(a+ d)2 - 2
R(5.8-5)The right-hand side of (5.8-5) is a nonreal complex number, except when
0 = (2k + l)?r. In this last case we have
Hence, in a loxodromic transformation a + d is always a nonreal com-
plex number. In the special case where the transformation carries circles
through the fixed points into themselves, a + d is pure imaginary.
(d) Next we consider the case where the transformation has a finite
fixed point z 1 = b/(d-a), while z 2 = oo. As seen in Section 5.7, this case
occurs when c = 0, H -=/= 0, which implies that ad= 1, a-=/= d. Then the
transformation can be written as
w - z1 = M(z - z1) (5.8-6)