Let us now define what complex differentiability is.
- Example 2.3.2
The function

is nowhere complex differentiable.
- Proof
Let
be arbitrary. Assume that
is complex differentiable at
, i.e. that

exists.
We choose

Due to lemma 2.2.3, which is applicable since of course
is open, we have:

But

a contradiction.
We can define a natural bijective function from
to
as follows:

In fact,
is a vector space isomorphism between
and
.
The inverse of
is given by

Theorem and definitions 2.3.3:
Let
be open, let
be a function and let
. If
is complex differentiable at
, then the functions

are well-defined, differentiable at
and satisfy the equations

These equations are called the Cauchy-Riemann equations.
- Proof
1. We prove well-definedness of
.
Let
. We apply the inverse function on both sides to obtain:

where the last equality holds since
is bijective (for any bijective
we have
if
; see exercise 1).
3. We prove differentiability of
and
and the Cauchy-Riemann equations.
We define

Then we have:

From these equations follows the existence of
, since for example

exists due to lemma 2.2.3.
The proof for

and the existence of
we leave for exercise 2.
- Let
be sets such that
, and let
be a bijective function. Prove that
.
- Let
be open, let
be a function and let
. Prove that if
is complex differentiable at
, then
and
exist and satisfy the equation
.
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