When you have a complex-valued function of a real or complex variable, we can talk about
differentiating it in exactly the same way as when the function is a real-valued function
of a real variable. You can form the difference quotient in exactly the same way, and if
the limit of the difference quotient exsists at a particular point we call that the value
of the derivative at that point.
Although we shall not study analytic functions in this course, the class of analytic
functions is extremely important in analysis, and throughout mathematics. There is a
very beautiful and complete theory of analytic function which is developed in
the area of mathematics called complex analysis.
Just to whet your appetite, here are some of the amazing results which can be
proved about analytic functions:
If a function is analytic then its derivative is also analytic (this is is stark contrast
to the case of real-valued functions)
Every analytic function can be expressed in terms of a power series
Analytic functions are determined by their "boundary values": if you have an analytic
function on the unit disk, then its values on the unit circle determine its behavior
inside the disk as well (and the values inside the disk can be expressed in terms of
the values on the boundary by a beautiful formula involving integration).
is a function which is defined on a region of the complex plane which
includes points of the real line, then one can talk about both the complex and the
real derivative of f
. The complex and the real derivative both use the same formula
for the difference quotient.
The difference between them is that for the complex derivative, z
through both real and complex values. For the real derivative, only real
values of z
are considered. Thus, if the complex derivative of a function exists
at a point in the real line, then the real derivative also exists, and the two are equal. The
converse is not true.
For our purposes, this will mean that anything we say about complex derivatives of power series
when z is real will equally well hold for the real derivative.
There are versions of
differentiation that look formally exactly like their real analogues.
In fact the proofs are the same too; look back at the proofs of those results and you'll
see that there wasn't anywhere we did anything that wouldn't have worked just as well in
the complex plane.