Page 67 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes where A is non zero complex constant and z  0. We write A and z in exponential form as
A = ae , z = re i
i
Then w = (ar) e i( + ) (2)
Thus we observe from (2) that transformation (1) expands (or contracts) the radius vector
representing z by the factor a = | A | and rotates it through an angle  = arg A about the origin.
The image of a given region is, therefore, geometrically similar to that region. The general
linear transformation,

w = Az + B (3)
is evidently an expansion or contraction and a rotation, followed by a translation. The image
region mapped by (3) is geometrically congruent to the original one.

Now we consider the function,
1
w  (4)
z
which establishes a one to one correspondence between the non zero points of the zplane and
the wplane. Since z z = | z | , the mapping can be described by means of the successive
2
transformations
1
Z = |z| 2 z, w = Z (5)

Geometrically, we know that if P and Q are inverse points w.r.t. a circle of radius r with centre
A, then
(AP) (AQ) = r 2
Thus a and b are inverse points w.r.t. the circle | z  a | = r if

(  a) (  a) = r 2



Q

a P
r A

where the pair  = a,  =  is also included. We note that , , a are collinear. Also points  and
 are inverse w.r.t a straight line l if  is the reflection of a in l and conversely. Thus, the first of
the transformation in (5) is an inversion w.r.t the unit circle | z | = 1 i.e. the image of a non-zero
point z is the point Z with the properties
1
z Z = 1, | Z | = |z| and arg Z = arg z

Thus, the point exterior to the circle | z | = 1 are mapped onto the non-zero points interior to it
and conversely. Any point on the circle is mapped onto itself. The second of the transformation
in (5) is simply a reflection in the real axis.

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