220 Chapter 55.5 The Bilinear Function w = (az + b)/(cz + d)
This function is also known as the bilinear transformation, the linear
fractional transformation, the M iibius transformation (after the German
mathematician A. F. Mobius, 1790-1868), and the homographic transfor-
mation. It is assumed that a, b, c, d are given complex constants such
that the determinant
1: ~I= ad-be (5.5-1)
is not zero. Of course, this condition implies that not both elements in
any one row or column of the matrix of the determinant are zero, and also
that the elements on any given line of the matrix are not in proportion to
those on a parallel line.
To see what the condition implies for the given function, suppose first
that c = 0. Then we must have a =f 0 and d =f 0, so the function reduces in
this case tow= (a/d)z + (b/d), which is the nonconstant linear function
discussed in (5.3).
If c =f 0, we may always write
az+b a
w=--=-+
CZ+ d Cbe-ad 1
c2 z+d/c
(5.5-2)which shows that if ad - be =f 0 the function is not a constant either.
Letting k = (be - ad)/c^2 , equation (5.5-2) also shows that the bilinear
function may be thought as a composition of all or some of the following
elementary transformations:
- The translation z 1 = z + d/ e
2. The inversion and symmetry z 2 = 1/ z 1
- The similitude z 3 = kz 2 (k =f 0)
- The translation w = (a/e) + za
Clearly, ad - be = 0 implies that w = a/e (a constant, possibly the
zero constant). We may also note that if e =f 0, a =f 0 and ad - be = O,
the fraction
az+ b
cz+d
a(z + b/a)
e(z + d/e)is not in simplest terms since then bf a = d/c and it reduces to a/e in
C-{-b/a}. Completing the definition of the function at -bf a by continuity
we have w = a/e in C. In what follows it is assumed throughout that the
condition ad - be =f 0 is satisfied by the bilinear transformation under
consideration.