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.
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