Page 70 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 7: Transformations and Conformal Mappings

a Notes
and thus, we get w = + z .
c 3
The above three auxiliary transformations are of the form
1
w = z + , w = , w = z (4)
z
Hence, every bilinear transformation is the resultant of the transformations in (4).
But we have already discussed these transformations and thus, we conclude that a bilinear
transformation always transforms circles and lines into circles and lines respectively, because
the transformations in (4) do so.

From (1), we observe that if c = 0, a, d  0, each point in the w plane is the image of one and only
d
one point in the z-plane. The same is true if c  0, except when z =  which makes the
c
d
denominator zero. Since we work in extended complex plane, so in case z =  , w =  and thus,
c
d
we may regard the point at infinity in the wplane as corresponding to the point z =  in the
c
zplane.

Thus, if we write
az b

T(z) = w = cz d , ad  bc  0 (5)

Then, T() = , if c = 0

a  d
and T() = , T    = , if c  0
c  c 
Thus, T is continuous on the extended z-plane. When the domain of definition is enlarged in this
way, the bilinear transformation (5) is oneone mapping of the extended z-plane onto the
extended w-plane.

Hence, associated with the transformation T, there is an inverse transformation T which is
1
defined on the extended w-plane as
T (w) = z if and only if T(z) = w.
1
Thus, when we solve equation (1) for z, then

 dw b

z = cw a , ad bc  0 (6)

and thus,
 dw b

T (w) = z = cw a , ad bc  0
1

Evidently, T is itself a bilinear transformation, where
1
T () =  if c = 0
1

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