Page 84 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 7: Transformations and Conformal Mappings

7.4 Keywords Notes

Transformation: The linear function.

w = Az
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( + )

Oneone transformation: 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

Bilinear transformation: The transformation
az b

w = , ad  bc  0
cz d

where a, b, c, d are complex constants, is called bilinear transformation or a linear fractional
transformation or Möbius transformation.
Conformal Mappings: Let S be a domain in a plane in which x and y are taken as rectangular
Cartesian co-ordinates. Let us suppose that the functions u(x, y) and v(x, y) are continuous and
possess continuous partial derivatives of the first order at each point of the domain S. The
equations
u = u(x, y), v = v(x, y)

7.5 Self Assessment

1. The general linear transformation ................. is evidently an expansion or contraction and
a rotation, followed by a translation.
2. Any point on the circle is mapped onto itself. The second of the transformation in (5) is
simply a ................ in the real axis.

3. It is natural to define a ................ w = T(z) from the extended z plane onto the extended w
plane by writing
T(0) = , T() = 0


4. The transformation w = az b , ad  bc  0 where a, b, c, d are complex constants, is called
cz d

................ or a linear fractional transformation or Möbius transformation.
5. Composition (or resultant or product) of two bilinear transformations is a ................
6. The points which coincide with their transforms under bilinear transformation are called
its fixed points. For the bilinear transformation ................, fixed points are given by w = z
az b

i.e. z = cz d

Since (1) is a quadratic in z and has in general two different roots, therefore, there are
generally two invariant points for a bilinear transformation.

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