4 0. Differentiability and the Cauchy-Riemann Equations
Suppose that / : y C, where V is an open subset of C. Suppose that
z G V. We say that / is differentiable at z if the limit
h-0
h
exists. This is what the word "differentiable" will mean throughout the
book; in the present chapter, since we are comparing this notion with another
sort of differentiability, we will refer to functions differentiable in this sense
as "complex-differentiable".
Since C =
R2,
there is something else that the word "differentiable"
could mean: Recall from advanced calculus that if V is an open subset of
R2
then a function / : V
R2
is said to be differentiable at the point z G V
if there exists an R-linear map T :
R2

R2
such that
f{z + h) = f(z)+Th + o(h)
as h 0; in this case the operator T is known as the Prechet derivative of
/ at z, or Df(z). Here we are using the Landau "little-oh"
notation1:
The
equation above means that
f(z + h) = f(z)+Th + E(h),
where the "error term" E{h) satisfies E(0) 0 and
v \\E(h)\\
h m
"
M
^ ; "
= o.
fc-o \\h\\
We will say that a function differentiable in this sense is real-differentiable.
Now if V is an open subset of C and / : V C then it is also true that
V is an open subset of
R2
and / : V
R2,
which raises the question of how
the two notions of differentiability are related. If we rearrange the definition
of
ff(z)
above we see that / is complex-differentiable at z, with derivative
f'{z) = a, if and only if
f(z + h) = f(z) + ah + o(h).
Since the mapping from C to C defined by h H- ah is certainly R-linear,
this shows that complex differentiability is a stronger condition than real
differentiability. Indeed, it is clear that the C-linear maps from C to C are
precisely the maps of the form h H-» ah for some complex number a.
1
Elsewhere we will use the similar "big-oh" notation: 0(F(x)) means "some function G(x)
such that \G(x)\ cF(x) for all x".
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