2.2 Proof of the Cauchy–Riemann Converse Theorem
The proof of the Cauchy–Riemann Converse Theorem is rather involved and may require more than one reading.
We will need two results from real analysis. The first result is known as the Mean Value Theorem.
Theorem 6 Mean Value Theorem
Let be a real function that is continuous on the closed interval
and differentiable on the open interval
. Then there is a number
such that

To appreciate why this theorem is true, imagine pushing the chord between and
in Figure 13 parallel to itself until it becomes a tangent to the graph of
at a point
, where
lies somewhere between
and
. Clearly, the gradient of the original chord must be equal to the gradient of the tangent, so

Multiplication by gives
. Notice that this equation is also true if
.


The second result that we will need is a Linear Approximation Theorem, which asserts that if is a real-valued function of two real variables
and
, then for
near
, the value of
can be approximated by the value of the linear function
defined by

Now, the graph of is a plane passing through the point
on the graph of
(Figure 14). Moreover, the partial
- and
-derivatives of
coincide with the partial
- and
-derivatives of
at
. This means that both have the same gradient in the
- and
-directions, so you can think of the plane as the tangent plane to the graph of
at
.



The accuracy with which this tangent plane approximates the graph of depends on the smoothness of the graph of
. If the graph exhibits the kind of kink shown in Figure 12, then the approximation is not as good as for a function with continuous partial derivatives.
Theorem 7 Linear Approximation Theorem (
to
)
Let be a real-valued function of two real variables, defined on a region
in
containing
. If the partial
- and
-derivatives of
exist on
and are continuous at
, then there is an ‘error function’
such that

where as
.
Since is the distance from
to
, the theorem asserts that the error function tends to zero ‘faster’ than this distance. Theorem 7 is the real-valued function analogue of Theorem 2.
Proof
We have to show that the function defined by

satisfies

Since the partial derivatives exist on , they must be defined on some disc centred at
. Let us begin by finding an expression for
on this disc. If we apply the Mean Value Theorem to the real functions
(where
is kept constant) and
, then we obtain

where is between
and
, and

where is between
and
(see Figure 15). Hence




Substituting this expression for into the definition of
, we obtain

Dividing both sides by , and noting that

(because both and
do not exceed
), we see that

Figure 15 illustrates that as tends to
, so do
and
. So, by the continuity of the partial
- and
-derivatives at
, the two terms on the right of the inequality above must both tend to 0 as
tends to
. It follows that
tends to 0 as
tends to
.
We are now in a position to prove the Cauchy–Riemann Converse Theorem.
Theorem 5 Cauchy–Riemann Converse Theorem (revisited)
Let be defined on a region
containing
. If the partial derivatives
,
,
,
exist at
for each
are continuous at
satisfy the Cauchy–Riemann equations at
,
then is differentiable at
and

Proof
We need to show that the limit of the difference quotient for at
exists and has the value indicated in the theorem. In order to calculate the difference quotient for
at
, we find an expression for
. Since
and
fulfil the conditions of Theorem 7, it follows that

where and
are the error functions associated with
and
, respectively.
Collecting together terms, we see that

Since and
satisfy the Cauchy–Riemann equations, both expressions in the large brackets must be equal, so

Dividing by gives

The limit of this difference quotient exists, and has the required value

provided that we can show that the expression involving the error functions and
tends to 0 as
tends to
. To this end, notice that
is equal to
and so, by the Triangle Inequality,

By Theorem 7, both expressions on the right tend to 0 as tends to
. Consequently, the expression on the left must also tend to 0, and the theorem follows.
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