Complex Functions: A complex valued function 
of a complex variable is a rule that assigns to each complex number 
in a set 
one and only complex value 
. We write 
and call 
the image of 
under 
. The set 
is called the domain of the definition of 
and the set of all images 
is called the range of 
.
Usually, the real and imaginary parts of 
are denoted by 
and 
, and those of the image point 
are denoted by 
and 
respectively, so that 
, where 
and 
are real valued functions of 
.
Example: Consider the function 
for 
. This function assigns to each complex number 
in 
one and only complex value 
. The real and imaginary parts of 
are given by
