1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 317

so that It:/ ~zl --t 0 as l~zl --t 0.

6.5 THE CAUCHY-RIEMANN EQUATIONS

If we write w = f(z) = u(x, y) + iv(x, y) we may ask whether the differen-
tiability property off at some point of its domain of definition corresponds
to some special properties of the real functions u and v. The answer to this
question is embodied in Theorem 6.5 below, which depends heavily on the
concept of differentiability of a real function of two real variables.

Theorem 6.5 Let f: D --t C, D open, z = x + iy E D, w = f(z) =

u( x, y) +iv( x, y ). If f has a derivative at z then u and v are differentiable

at (x, y), and.

Ux =Vy,
Conversely, those conditions are sufficient for the existence of the derivative
off at z.

Proof 1. Suppose that

I. 1m -~w = !'( z )

e.z->O ~z

exists. Then we have

~w

~z = f'(z) + c:

where c: --t 0 as ~z --t 0, and it follows that

~w = f'(z) ~z + c: ~z (6.5-1)

Letting f'(z) =A+ iB, ~z = ~x + i ~y, ~w = ~u + i ~v and c: =
e1 + ic:2, (6.5-1) becomes

~u + i ~v =(A+ iB)(~x + i ~y) + (c:1 + ic:2)(~x + i ~y)

which implies that

{

~u = A ~x - B ~y + e1 ~x - e2 ~y

~v = B ~x + A ~y + e2 ~x + e1 ~y

(6.5-2)

Since c: --t 0 as ~z --t 0, we have c:1, c:2 --t 0 as ~x, ~y --t 0. There-
fore, equations (6.5-2) show that u and v are both differentiable at (x, y).
Moreover, it follows that

Ux =A, Uy= -B, Vx = B, Vy= A