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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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