0. Differentiability and the Cauchy-Riemann Equations 5

Thus we have proved the following proposition:

Proposition 0.0. Suppose that V is an open subset of C, that f : V — C,

and z G V. Then f is complex-differentiate at z if and only if it is real-

differ entiable at z and the real derivative Df(z) is complex-linear.

Now suppose that / is real-differentiable at z. Let u and v denote the

real and imaginary parts of / , respectively. The entries in the 2 x 2 matrix

representing the real derivative Df(z) are given by the partial derivatives

of u and v:

D

\UX Uyl

[Vx Vy \

(Of course the existence of the partials uXl uyi vx and vy at a point does not

imply that / is real-differentiable at that point.)

On the other hand, it is easy to see that the real 2 x 2 matrix

la

b~\

\_c

d\

represents a complex-linear map from C to C if and only if a = d and c = —6,

and so we finally deduce

Proposition 0.1. Suppose that V is an open subset of C, that f : V — » C,

and z G V. Then f — u + iv is complex-differ entiable at z if and only

if it is real-differentiable at z and the real and imaginary parts satisfy the

u

Cauchy-Riemann equations"

UX{Z) = Vy(z), Uy(z) = -VX{Z).

This is the reason we introduced Frechet differentiability: Proposition

0.1 gives a characterization of complex differentiability at a single point in

terms of the Cauchy-Riemann equations, with no extra hypotheses. If we

try to characterize complex differentiability in terms of the Cauchy-Riemann

equations, but using only partial derivatives, the best we can do is to state

the following two facts; note that the converse of each of the following corol-

laries is false (see the exercises):

Corollary 0.2. Suppose that V is an open subset of C, that f : V — C,

and z G V; write f = u + iv. If f is complex-differ entiable at z G V then u

and v satisfy the Cauchy-Riemann equations at z.

Corollary 0.3. Suppose that V is an open subset of C, that f : V — C,

and z G V; write f = u + iv. If the first-order partial derivatives of u and

v are continuous at z G V and satisfy the Cauchy-Riemann equations there

then f is complex-differ entiable at z.