1.5 A geometric interpretation of derivatives
As we mentioned in this session’s introduction, the derivative of a real function is often pictured geometrically as the gradient of the graph of the function. This interpretation is useful in real analysis, but it is of little use in complex analysis, since the graph of a complex function is not two-dimensional.
Fortunately, there is another way of interpreting derivatives that works for complex functions.
If a complex function is differentiable at a point
, then any point
close to
is mapped by
to a point
close to
.
Indeed, by the Linear Approximation Theorem,

where as
. So if
, then, to a close approximation,

Multiplication of by
has the effect of scaling
by the factor
and rotating it about 0 through the angle
; see Figure 6. We refer to
as a complex scale factor, because it causes both a scaling and a rotation.


We can rewrite the equation above as

From this we see that is obtained by scaling and rotating the vector
based at
by the complex scale factor
, as illustrated in Figure 7.

Another useful way to picture how behaves geometrically is to consider the effect it has on a small disc centred at
(still assuming that
). From equation 2 (above), we see that, to a close approximation, a small disc centred at
is mapped to a small disc centred at
. In the process, the disc is rotated through the angle
, and it is scaled by the factor
(see Figure 8). As usual, the rotation is anticlockwise if
is positive, and clockwise if it is negative.



The geometric interpretation of derivatives is more complicated if , and we do not discuss it here.
Example 7
Using the notion of a complex scale factor, describe what happens to points close to under the function
.
Solution
To a close approximation, a small disc centred at is mapped by
to a small disc centred at

In the process, the disc is scaled by the factor and rotated through the angle
.
Now , so

which has modulus and principal argument
.
So scales the disc by the factor
and rotates it anticlockwise through the angle
.
Exercise 10
Using the notion of a complex scale factor, describe what happens to points close to under the function

Answer
To a close approximation, a small disc centred at is mapped by
to a small disc centred at

In the process the disc is scaled by the factor and rotated through the angle
.
By the Quotient Rule,

So

This has modulus and principal argument
.
So scales the disc by the factor
and rotates it clockwise through the angle
.
It is important to bear in mind that the complex scale factor interpretation of a derivative is only an approximation, and that it is unlikely to be reliable far from the point under consideration.
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