Functions. Limits and Co11tinuity. Arcs and Curves 127
If we let z = rei^8 and w = f(z) = u +iv, the components u and v
become functions of r and 8. For example, we have
f(z) = z^3 = r^3 e^3 i^8 = r^3 ( cos 38 + i sin,38)
so that, in this case, u = r^3 cos 38 and v = r^3 sin 38.
Also, we may express w in the exponential form w = Reia. In such acase Rand a become functions (the latter not a single-valued function) of
r and 8. Thus in the example w = z^3 we have
and we obtain
R=r^3 and
Reio: = r3 e3i8a= 38 + 2k7r ( k any integer)
Although this is seldom done, we may as well. choose z = x + iy and
w =Reio:. Then Rand a become functions of x and.y [a= a(x, y) not a
single-valued function]. Thus in the example
w =Reio:=. (x + iy)^2 = (x^2 - y^2 ) + 2ixy
we find that
R = J(x2 _ y2)2 + 4x2y2 = xz + yz
and a is any solution of the system
xz -yz
cosa= 2 2 ,
x +y
whenever x^2 + y^2 f 0.
2xys1na = ---
x2 +yz
The discussion above shows that a complex function of a complex vari-
able is equivalent to two real-valued functions of two real variables, the
nature of these functions depending on the type of representation chosen
for z and w. There is a significant advantage in combining the two real
functions into a single complex function, as will become apparent later.
3.3 Geometric Representation of Complex Functions
A geometric representation of the function w = f ( z), or u +iv = f ( x + i y),
similar to that usually employed for real functions would require in general
a four-dimensional space since we have now four real variables involved.
Although it is possible to visualize to a certain extent four dimensional
figures by means of three-dimensional cuts (see [2],[3], and [4]); many of the
research and didactical advantages that the geometric intuition furnishes
are mostly lost.