1 Complex functions

Let the complex variable z be defined by z = x + i y where x and y are real variables and i is, as usual, given by i 2 = 1 . Now let a second complex variable w be defined by w = u + i v where u and v are real variables. If there is a relationship between w and z such that to each value of z in a given region of the z plane there is assigned one, and only one, value of w then w is said to be a function of z , defined on the given region. In this case we write

w = f ( z ) .

As a example consider w = z 2 z , which is defined for all values of z (that is, the right-hand side can be computed for every value of z ). Then, remembering that z = x + i y ,

w = u + i v = ( x + i y ) 2 ( x + i y ) = x 2 + 2 i x y y 2 x i y .

Hence, equating real and imaginary parts: u = x 2 x y 2 and v = 2 x y y .

If z = 2 + 3 i , for example, then x = 2 , y = 3 so that u = 4 2 9 = 7 and v = 12 3 = 9 , giving w = 7 + 9 i .

Example 1
  1. For which values of z is w = 1 z defined?
  2. For these values obtain u and v and evaluate w when z = 2 i .
Solution
  1. w is defined for all z 0 .
  2. u + i v = 1 x + i y = 1 x + i y x i y x i y = x i y x 2 + y 2 . Hence u = x x 2 + y 2 and v = y x 2 + y 2 .

    If z = 2 i then x = 2 , y = 1 so that x 2 + y 2 = 5 . Then u = 2 5 , v = 1 5 and w = 2 5 1 5 i .