Page 68 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 7: Transformations and Conformal Mappings

Notes
z
y

Z

x
O
w
|z| = 1

1 1
Since lim =  and lim = 0,
z 0 z z z
it is natural to define a oneone transformation w = T(z) from the extended z plane onto the
extended w plane by writing
T(0) = , T() = 0

1
and T(z) =
z
for the remaining values of z. It is observed that T is continuous throughout the extended z
plane.
When a point w = u + iv is the image of a non-zero point z = x + iy under the transformation w
1 z
= , writing w = 2 results in
z |z|

x  y
u = x  y 2 , v = x  y 2 (6)
2
2
1 w

Also, since z = w |w| 2 , we get
u  v
x = 2 , y = (7)
2
2
u  v u  v 2
The following argument, based on these relations (6) and (7) between coordinates shows the
1
important result that the mapping w = transforms circles and lines into circles and lines.
z
When a, b, c, d are real numbers satisfying the condition b + c > 4ad, then the equation
2
2
a(x + y ) + bx + cy + d = 0 (8)
2
2
represents an arbitrary circle or line, where a  0 for a circle and a = 0 for a line.
If x and y satisfy equation (8), we can use relations (7) to substitute for these variables. Thus,
using (7) in (8), we obtain

d(u + v ) + bu  cv + a = 0 (9)
2
2

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