1550251515-Classical_Complex_Analysis__Gonzalez_

126 Chapter^3

At times we shall also consider functions from en (n ~ 2) into C. These

are called complex functions of several complex variables. Unless otherwise

stated, we shall agree in what follows that the term function means a

single-valued function of a complex variable.

3.2 The Components of a Complex Function

If w = u + iv is the value of the function f at the point ~ = x + iy on its

domain of definition, we write

w = f(z) or u +iv= J(x + iy)

That given z the function f determines w is equivalent to saying that

the pair (x, y) determines w. But once w is determined, the real numbers

u = Rew and v = Im w are also determined. Hence each of the variables

u and v is a function, in fact, a composite function of the real variables x

and y. That is, we have u = Ref((x,y)) = u(x,y) and v = lmf((x,y)) =

v(x, y), and we write

f(z) = u(x, y) + iv(x, y) (3.2-1)

The functions u(x, y) and v(x, y) are called the real components off.

For instance, if f(z) = z^3 , we have

f(z) = (x + iy)^3 = (x^3 - 3xy^2 ) + i(3x^2 y - y^3 )

so, in this case,:

u(x,y) = x^3 - 3xy^2 and v(x,y) = 3x^2 y-y^3

Conversely, given two real functions u( x, y) and v( x, y) of the real vari-

ables x and y, equation (3.2-1) may be used to define a function of the

complex variable z. In general, the result appears as a function of z and

z obtained by substituting x =^1 /2(z + z), y = (z - z)/2i, so it becomes a

composite function of z. However, for the class of analytic functions, to be
introduced later, the result does not depend on z.

Examples

1. If u = x^2 + y^2 , v = xy, we have

f(z) = (x2 + y2) + ixy = zz + i;4z2 - i;4z2

2. If u = x^2 - y^2 , v = 2xy, we have

f(z) = (x^2 - y^2 ) + 2ixy = z^2