Page 83 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY
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Complex Analysis and Differential Geometry
Notes 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.
d(u + v ) + bu cv + a = 0
2
2
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.
The transformation
az b
w = , 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
In a bilinear transformation, a circle transforms into a circle and inverse points transform
into inverse points. In the particular case in which the circle becomes a straight line,
inverse points become points symmetric about the line.
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)
set up a correspondence between the points of S and the points of a set T in the (u, v) plane.
The set T is evidently a domain and is called a map of S. Moreover, since the first order
partial derivatives of u and v are continuous, a curve in S which has a continuously turning
tangent is mapped on a curve with the same property in T. The correspondence between
the two domains is not, however, necessarily a one-one correspondence.
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