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Complex Analysis and Differential Geometry
Notes which also represents a circle or a line. Conversely, if u and v satisfy (9), it follows from (6) that
x and y satisfy (8). From (8) and (9), it is clear that
(i) a circle (a 0) not passing through the origin (d 0) in the z plane is transformed into a
circle not passing through the origin in the w plane.
(ii) a circle (a 0) through the origin (d = 0) in the z plane is transformed into a line which does
not pass through the origin in the w plane.
(iii) a line (a = 0) not passing through the origin (d 0) in the z plane is transformed into a circle
through the origin in the w plane.
(iv) a line (a = 0) through the origin (d = 0) in the z plane is transformed into a line through the
origin in the w plane.
1
Hence, we conclude that w = transforms circles and lines into circles and lines respectively.
z
Remark : In the extended complex plane, a line may be treated as a circle with infinite radius.
7.1.1 Bilinear Transformation
The transformation
w = az b , ad bc 0
(1)
cz d
where a, b, c, d are complex constants, is called bilinear transformation or a linear fractional
transformation or Möbius transformation. We observe that the condition ad bc 0 is necessary
for (1) to be a bilinear transformation, since if
b d
ad bc = 0, then and we get
a c
a(z b/a) a
w = i.e. we get a constant function which is not linear.
c(z d/c) c
Equation (1) can be written in the form
cwz + dw az b = 0
(2)
Since (2) is linear in z and linear in w or bilinear in z and w, therefore, (1) is termed as bilinear
transformation.
When c = 0, the condition ad bc 0 becomes ad 0 and we see that the transformation reduces
to general linear transformation. When c 0, equation (1) can be written as
a (z b/a) a b/a d/c
w = c (z d/c) c 1 z d/c
a bc ad 1
=
(3)
c c 2 z d/c
We note that (3) is a composition of the mappings
d 1 bc ad
z = z + c , z = z 1 , z = c 2 z 2
1
2
3
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