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".