or

If (x + iy) = u+ iv, then show that u/x + v/y = 4(x2 - y2)

Solution:

             Given,

                    (x+iy)3= u + iv

                  or,  x3 + 3x2yi + 3xi2y2 + i3y3= u + iv

                   or,  x3 – 3xy2 + 3x2iy – iy3= u + iv

                     or, x(x2 – 3y2) + y(3x2 – y2)i= u+iv

Comparing Corresponding Terms,

        u = x(x2 – 3y2)                   and              v = y(3x2 – y2)

    Now,

       LHS = u/x + v/y

                 = (x(x2 – 3y2))/x + (y(3x2 – y2))/y

                   = (x2 – 3y2) + (3x2 – y2)

                     = 4x2 – 4y2

                                  = 4(x2 – y2)

                        = RHS

                                                                      Proved.......