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for
Chapter 3 Analytic and Harmonic Functions
Overview
Does the notion of a
derivative of a complex function make sense? If so, how
should it be defined and what does it represent?
These and similar questions are the focus of this
chapter. As you might guess, complex derivatives have a
meaningful definition, and
many of the standard derivative theorems from from calculus (such as
the product rule and chain rule) carry over for complex
functions.
There are also some interesting applications. But not
everything is symmetric. You will learn in this chapter
that the mean value theorem
for derivatives does not extend to complex functions. In
later chapters you will see that differentiable complex functions
are, in some sense,
much more "differentiable" than differentiable real functions.
Section 3.1 Differentiable and Analytic Functions
Using our
imagination, we take our lead from elementary calculus and define the
derivative of
at
, written
, by
(3-1)
,
provided that the limit exists. If it does, we say that
the function
is
differentiable at
. If
we write
,
then we can express Equation (3-1) in
the form
(3-2)
.
If we let
and
, then
we can use the Leibniz's
notation
for the derivative:
(3-3)
.
Example 3.1. Use
the limit definition to find the derivative
of
.
Solution. Using Equation
(3-1), we have
We can drop the subscript on
to
obtain
as
a general formula.
Alternative Solution. Using Equation
(3-2), we have
We can drop the subscript on
to
obtain
as
a general formula.
Pay careful
attention to the complex value
in
Equation (3-3); the value of
the limit must be independent of
the manner in which
. If
we can find two curves that end at
along
which
approaches
two distinct values,
then
does
not have a limit
as
and
does not have a derivative
at
.
The same observation applies to the limits in Equations
(3-2) and
(3-1).
Example 3.2. Show
that the function
is
nowhere
differentiable.
Solution. We choose two approaches to the
point
and
compute limits of the two difference quotients.
We shall use formulas similar to (3-1),
for calculating the directional derivatives along horizontal and
vertical lines.
First, we approach
along
a line parallel to the
-axis
by forcing
to be of the form
.
Next, we approach
along a line parallel to the
-axis
by forcing
to be of the form
.
The limits along
the two paths are different, so there is no
possible value for the right side of Equation
(3-1).
Therefore
is
not differentiable at the point
,
and since
was arbitrary,
is
nowhere differentiable.
Remark 3.1. In
Section
2.3 we showed that
is
continuous for all
. Thus,
we have a simple example of a
function that is continuous everywhere
but differentiable
nowhere. Such functions are
hard to construct in real variables.
In some sense, the complex case has made pathological constructions
simpler!
We are seldom
interested in studying functions that aren't differentiable, or even
differentiable at only a single point.
Complex functions that have a derivative at all points in a
neighborhood of
deserve
further study. In Section
7.2,
we will prove that, if the complex function
can
be represented by a Taylor series at
, then
it must be
differentiable in some neighborhood of
. Functions
that are differentiable in neighborhoods of points are pillars
of the complex analysis edifice; we give them a special name, as
indicated in the following definition.
Definition 3.1 (Analytic
Function). The complex
function
is
analytic at the
point
provided
there
is some
such
that
exists
for all
. In
other words,
must
be differentiable
not only at
, but
also at all points in some
-neighborhood
of
.
If
is
analytic at each point in the region
, then
we say that
is
an analytic function
on
.
Again, we have a special term if
is
analytic on the whole complex plane.
Definition 3.2 (Entire
Function). If
is
analytic on the whole complex plane then
is
said to be an entire function.
Points of non-analyticity for a function are called singular points. They are important for applications in physics and engineering.
Our definition of
the derivative for complex functions is formally the same as for real
functions and is the natural extension from
real variables to complex variables. The basic
differentiation formulas are identical to those for real functions,
and we obtain the same
rules for differentiating powers, sums, products, quotients, and
compositions of functions. We can easily establish the
proof of the
differentiation formulas by using the limit theorems.
The Rules for Differentiation.
Suppose that
f(z) and g(z)
are differentiable. From Equation
(3-2) and the technique exhibited in the
solution to Example 3.1
we can establish the following rules, which are virtually identical
to those for real-valued functions.
(3-4)
,
(3-5)
,
(3-6)
,
(3-7)
,
(3-8)
,
(3-9)
,
(3-10)
.
Important particular cases of Equations
(3-9) and
(3-10), respectively,
are
(3-11)
,
(3-12)
.
Exploration for the Rules for Differentiation.
Example 3.3. Use
Formula (3-12) to
calculate
.
Hint. Use
,
, and
.
Solution. An easy computation yields
The proofs of the
rules given in Equations through (3-10)
depend on the validity of extending theorems for real functions to
their
complex companions. Equation
(3-8), for example, relies on Theorem
3.1.
Theorem
3.1. If
is
differentiable at
then
is
continuous at
.
Proof. From Equation (3-1), we obtain
.
Using the multiplicative property of limits given in Theorem 2.3 in
Section
2.3, we get
This result implies that
, which
in turn implies that
.
Therefore,
is
continuous at
.
The Derivative
of ![]()
We can establish
Equation (3-8)
, from
Theorem 3.1.
Letting
and
using Definition 3.1, we write
.
If we subtract and add the term
in
the numerator, we get
Using the definition of the derivative given by Equation
(3-1) and the continuity
of
, we
obtain
,
which is what we wanted to establish.
We leave the proofs of the other differentiation rules as exercises.
The rule for
differentiating polynomials carries over to the complex case as
well.
If we let
be
a polynomial of degree
,
so that
,
then mathematical induction, along with Equations
(3-5) and
(3-7), gives
.
Again, we leave the details of this proof for the reader to finish,
as an exercise.
We shall use the
differentiation rules as aids in determining when functions are
analytic. For example, Equation
(3-9) tells us
that if
are
polynomials, then their quotient
is
analytic at all points where
. This
condition
implies that the function
is
analytic for all
.
The square root
function is more
complicated. If ![]()
, then
is
analytic at
all points except
(because
is undefined) and at points that lie along the negative
-axis. Recall
from Exercise 17,
in Section
2.3, that the argument
function is not continuous along the negative
-axis. Therefore
the function
,
is not continuous at points that lie along the negative
-axis.
We close this
section with a complex extension of a famous theorem, which is
attributed to Guillaume
de l'Hôpital (1661-1704),
the proof of will be given in Section
7.5.
Theorem 3.2 (L'Hôpital's
Rule). Assume that
and
are
both analytic at
.
If
,
, and
, then
.
Exploration for L'Hôpital's Rule.
Extra Example
1. Use L'Hôpital's rule to
find
.
Optional Material for the Internet
Real Concepts in Complex Analysis.
Many of the
calculus concepts about derivatives are easy to extended to complex
functions. For example, in calculus
we learned that the derivative is the limit of the difference
quotients
as
goes
to zero. We can compare
our calculus experience with some new and interesting graphs in the
complex plane.
Graphical explorations of difference quotients.
Extra Example
2. Consider the real
function
,
which is differentiable, and it's derivative is the limit of the
real difference
quotients
.
![[Graphics:Images/AnalyticFunctionMod_gr_454.gif]](analyticfunction/AnalyticFunctionMod/Images/AnalyticFunctionMod_gr_454.gif)
We can illustrate
convergence of the real difference
quotients
by
comparing graphs for decreasing
values of
. For
illustration purposes we plot the real
graphs
for
.
![]()
![]()
The
graph of
.
Figure
E.E.3. The graphs
of
for
.
where
and
the graph of
.
By looking at the
above graphs we should get a good feeling about visualizing
limits of functions
over an interval. In
particular,
we hope that this gives you a good feeling about the the
formula
, where
we have used
the function
in
this illustration to get
.
The real function
can be extended into the complex plane by replacing the real variable
with the complex variable
.
The same algebraic computations are involved in finding the limit of
the complex difference
quotients.
Extra Example
3. Consider the complex
function
,
which is differentiable, and it's derivative is the limit of the
complex difference
quotients
.
![[Graphics:Images/AnalyticFunctionMod_gr_481.gif]](analyticfunction/AnalyticFunctionMod/Images/AnalyticFunctionMod_gr_481.gif)
We can illustrate
convergence of the complex difference
quotients
by
comparing graphs for decreasing values
of
. For
illustration purposes we plot the graphs
for
.
We cannot draw a graph of
-dimensional
space into
-dimensional
space, it is necessary to choose a domain
in the
-plane
for our graphs.
The
domain
in the
-plane
for the following graphs.
![]()
![]()
The
graph of
.
Figure
E.E.4. The unit square in the
-plane,
and it's images under the mappings
for
,
where
and
the graph of
.
By looking at the
above graphs we should get a good feeling about visualizing
limits of functions in the
complex plane.
In particular, we hope that this gives you a good feeling about the
the formula
, where
we
have used the function
in
this illustration to get
.
Remark. The final resting
place of the points
are
,
,
, and
.
Graphical Explorations of Polynomial Approximations.
Many concepts from
calculus will be extended to complex
functions, including the approximation of
functions. Derivatives will
play an important role, just as they did in the calculus of
real functions. Let us give a
preview of some things we will be studying.
The following three polynomial approximations are usually discussed
in calculus.
is
an approximation to
.
is
an approximation to
.
is
an approximation to
.
In the above
graphs, is easy to visualize the real
functions and their polynomial
approximations
,
,
.
We assume that the reader is familiar with the details for
constructing these approximations, or can easily find them.
However, when we
extend these real functions to
complex functions, we must select an
appropriate
domain
for each function in the
-plane
in order to construct it's graph. The following
complex function
examples give illustrations similar to the above
real approximations, but extended into
the complex plane.
Extra Example
4. Given
, from
calculus we know that
,
.
The Maclaurin polynomial of degree
is
.
Hence, the mapping
, has
the "linear approximation"
.
The
domain
in the
-plane
for the following graphs.
![]()
Figure
E.E.5. The domain
is
a square in the
-plane,
and
it's images under the mappings
and
.
In
the last two graphs, one can visualize the
complex function approximation
.
Remark 1. This is a
trivial example of a "linear transformation" that was
studied in Section
2.1.
Also,
is
a "linear approximation" to
.
Remark 2. Complex
Taylor polynomials and approximations will be introduced in Section
7.2.
The function
is
the familiar Maclaurin polynomial approximation of
degree
.
Remark 3. Analytic
functions that satisfy
are
conformal mappings and will be studied in Section
10.1.
Extra Example
5. Given
, from
calculus we know that
,
,
.
The Maclaurin polynomial of degree
is
.
Hence, the mapping
, has
the "quadratic approximation"
.
The
domain
in the
-plane
for the following graphs.
![]()
Figure
E.E.6. The domain
is
a rectangle in the
-plane,
and
it's images under the mappings
and
.
In
the last two graphs, one can visualize the
complex function approximation
.
Remark 4. The
mapping
is
similar to the mapping
that
was studied in Section
2.2.
Also,
is
a "quadratic approximation" to
.
Remark 5. Complex
Taylor polynomials and approximations will be introduced in Section
7.2.
The function
is
the familiar Maclaurin polynomial approximation of
degree
.
Remark 6. Analytic
functions that satisfy
are
conformal mappings and will be studied in Section
10.1.
We will see that the mapping
is
not conformal at the origin.
Extra Example
6. Given
, from
calculus we know that the first few derivatives are
,
,
,
.
The Maclaurin polynomial of degree
is
.
Hence, the mapping
, has
the "cubic approximation"
.
The
domain
in the
-plane
for the following graphs.
![]()
Figure
E.E.7. The domain
is
a square in the
-plane,
and
it's images under the mappings
and
.
In
the last two graphs, one can visualize the
complex function approximation
.
Remark 7. Complex
Taylor polynomials and approximations will be introduced in Section
7.2.
The function
is
the familiar Maclaurin polynomial approximation of
degree
.
Remark 8. Analytic
functions that satisfy
are
conformal mappings and will be studied in Section
10.1.
Exercises for Section 3.1. Differentiable and Analytic Functions
Mean Value Theorem and Rolle's Theorem
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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell