Elementary Functions 257
Finally, suppose that IAI = IBI-::/:- 0 and AC= BC. Then Az+Bz+C =
0 defines a straight line that may either coincide, be parallel, or intersect
the line Lz + Mz + N = 0. The first case has been rule out since it requiresthat D1 = D2 = Da = 0. The second case occurs iff D 1 = 0, D 2 -::/:-0, and
D 3 -::/:-0. For z 1 on Az + Bz + C = 0 we have w{z 1 ) = 0, while for z 2 on
Lz + Mz + N = 0 we have w(z 2 ) = oo. At the point z = oo (which may
be regarded as common to the two parallel lines) w is undefined, and it is
easily seen that limz--+oo w( z) is undefined also, so that oo is not what is
called a removable singularity. As to the third case, suppose that the two
lines intersect at z 0 • This happens if D 1 =AM - BL-::/:-0, and from
we get
Azo + Bzo + C = 0
Lzo + Mzo + N = 0
and_ -Da
Zo= --
D1which implies that D1D2 + D1Da = 0. Hence w{zo) is undefined, and this
point is not a removable singularity, since in every neighborhood of z 0 there
are points where w = 0 and also points where w = oo.
From the preceding discussion it is clear that the mapping defined
by (5.13-1) is not in general one-to-one. For instance, we may have a
line all of whose points map into oo. Also, it follows that w( z) is defined at
every point of C iff Lz + Mz + N -::/:- 0 for every z, with the exception of the
case where Lz + Mz + N = 0 is satisfied at just one point z 0 and E 1 -::/:-O,
in which case we have w( z 0 ) = oo and, conversely, z( oo) = z 0 • Also, it
should be recalled that w( z) is defined at oo iff A = L = 0 or B = M = 0,
i.e., iff w(z) reduces either to the ordinary bilinear transformation or to
the conjugate bilinear transformation.
The conditions on the coefficients of the general bilinear transformation
that will ensure that the mapping is one-to-one are established with the
restriction set forth in the following theorem.
Theorem 5.10 Let G = {z: Lz + Mz + N-::/:-O}. Then the mapping
defined by (5.13-1) is one-to-one in Giff D1 = 0 and ID2I-::/:-IDal·
For the proof of this proposition, see [4].
Returning to (5.13-2), we note that the inverse of a general bilinear
transformation is not again a general bilinear transformation unless