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Chapter 6Complex Integration
Overview
Of the two main topics studied in calculus - differentiation and integration - we have so far only studied derivatives of complex functions. We now turn to the problem of integrating complex functions. The theory you will learn is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics. For example, even though the ideas presented here belong to the general area of mathematics known as analysis, you will see as an application of them one of the simplest proofs of the fundamental theorem of algebra.
6.1 Complex Integrals
In Section 3.1 we saw how the derivative of a complex function is defined. We now turn our attention to the problem of integrating complex functions. We will find that integrals of analytic functions are well behaved and that many properties from calculus carry over to the complex case.
We introduce the integral of a complex function by defining the integral of a complex-valued function of a real variable
Definition 6.1 (Definite Integral of a Complex Integrand). Let where u(t) and v(t) are real-valued functions of the real variable t for . Then
(6-1) .
We generally evaluate integrals of this type by finding the antiderivatives of u(t) and v(t) and evaluating the definite integrals on the right side of Equation (6-1). That is, if and , we have
(6-2) .
Example 6.1. Show that .
Solution. We write the integrand in terms of its real and imaginary parts, i.e., . Here, and . The integrals of u(t) and v(t) are
, and
.
Hence, by Definition (6-1),
Explore Solution 6.1.
Example 6.2. Show that .
Solution. We use the method suggested by Definitions (6-1) and (6-2).
We can evaluate each of the integrals via integration by parts. For example,
Adding to both sides of this equation and then dividing by 2
gives . Likewise, . Therefore,
.
Explore Solution 6.2.
Complex integrals have properties that are similar to those of real integrals. We now trace through several commonalities. Let and be continuous on .
Using Definition (6-1), we can easily show that the integral of their sum is the sum of their integrals, that is
(6-3) .
If we divide the interval into and and integrate f(t) over these subintervals by using (6-1), then we get
(6-4) .
Similarly, if denotes a complex constant, then
(6-5) .
If the limits of integration are reversed, then
(6-6) .
The integral of the product f(t)g(t) becomes
(6-7)
Example 6.3. Let us verify property (6-5). We start by writing
Using Definition (6-1), we write the left side of Equation (6-5) as
which is equivalent to
Therefore, .
Explore Solution 6.3.
It is worthwhile to point out the similarity between equation (6-2) and its counterpart in calculus. Suppose that U and V are differentiable on and . Since , equation (6-2) takes on the familiar form
(6-8) . where . We can view Equation (6-8) as an extension of the fundamental theorem of calculus. In Section 6.4 we show how to generalize this extension to analytic functions of a complex variable. For now, we simply note an important case of Equation (6-8):
(6-9) .
Example 6.4. Use Equation (6-8) to show that .
Solution. We seek a function F with the property that . We note
that satisfies this requirement, so
which is the same result we obtained in Example 6.2, but with a lot less work.
Explore Solution 6.4.
Remark 6.1 Example 6.4 illustrates the potential computational advantage we have when we lift our sights to the complex domain. Using ordinary calculus techniques to evaluate
, for example, required a lengthy integration by parts procedure (Example
6.2). When we recognize this expression as the real part of , however, the solution comes quickly. This is just one of the many reasons why good physicists and engineers, in addition to mathematicians, benefit from a thorough working knowledge of complex analysis.
Extra Example 1. Show that .
Explore Solution for Extra Example 1.
Exercises for Section 6.1. Complex Integrals
Formulas. Let where u(t) and v(t) are real-valued functions of the real variable t for . Then
(6-1) .
We generally evaluate integrals of this type by finding the antiderivatives of u(t) and v(t) and evaluating the definite integrals on the right side of Equation (6-1). That is, if and , we have
(6-2) .
Exercise 1. Use Equations (6-1) and (6-2) to find
1 (a). . Solution 1 (a).
See text and/or instructor's solution manual.
Answer. .
Solution. Given the real functions u(t) and v(t) with and , we use the formula
.
First, expand the integrand into it's real and imaginary parts
.
Here we have
, and
.
Now integrate u(t) and v(t) and obtain
, and
.
Compute values for U(t) and V(t)
Compute the real definite integrals
Evaluate the complex definite integral
We are done.
Aside. We can let Mathematica double check our work.
The details for this computation are:
Some points in the interval of integration and their images , for .
In the interval the right endpoint is and the last image point
is .
We are really done.
Aside. After we have developed the topics of contour integrals (in Section 6.2), the independence of path for integration of an analytic function (in Section 6.3),
and established the Fundamental Theorem of Calculus (in Section 6.4) then we will be able to revisit this integral and use the more straightforward computation:
Remark. It is important to understand whether you are permitted to use only real variables in computing an integral,
or whether you are permitted to use a complex variable ( ) and complex functions in computing the integral.
The three popular software packages Maple, Matlab and Mathematica use complex variable based computations.
1 (b). .
1 (c). . Solution 1 (c).
Answer .
Solution. Given the real functions u(t) and v(t) with and , we use the formula
.
First, use the identity to expand the integrand
.
Here we have
, and
.
Now integrate u(t) and v(t) and obtain
, and
.
Compute values for U(t) and V(t)
Compute the real definite integrals
Evaluate the complex definite integral
We are done.
Aside. We can let Mathematica double check our work.
The details for this computation are:
Some points in the interval of integration and their images , for .
In the interval the right endpoint is and the last image point
is .
We are really done.
Aside. After we have developed the topics of contour integrals (in Section 6.2), the independence of path for integration of an analytic function (in Section 6.3),
and established the Fundamental Theorem of Calculus (in Section 6.4) then we will be able to revisit this integral and use the more straightforward computation:
Remark. It is important to understand whether you are permitted to use only real variables in computing an integral,
or whether you are permitted to use a complex variable ( ) and complex functions in computing the integral.
The three popular software packages Maple, Matlab and Mathematica use complex variable based computations.
1 (d). .
1 (e). .Solution 1 (e).
Exercise 2. Let be integers. Show that .
Exercise 3. Show that provided . Solution 3.
Exercise 4. Given that and are continuous on . Establish the following:
4 (a). Identity (6-3) .
4 (b). Identity (6-4) .
4 (c). Identity (6-6) .
4 (d). Identity (6-7)
.
Exercise 5. Let , where u and v are differentiable. Show that
. Solution 5.
6.2 Contours and Contour Integrals
In Section 6.1 we learned how to evaluate integrals of the form , where f(t) was complex-valued and was an interval on the real axis (so that t was real, with
). In this section, we define and evaluate integrals of the form , where f(t) is complex-valued and C is a contour in the plane (so that z is complex, with ). Our main result is Theorem 6.1, which shows how to transform the latter type of integral into the kind we investigated in Section 6.1 .
We will use concepts first introduced in Section 1.6. Recall that to represent a curve C we used the parametric notation
(6-10) for ,
where x(t) and y(t) are continuous functions. We now place a few more restrictions on the type of curve to be described. The following discussion leads to the concept of a contour, which is a type of curve that is adequate for the study of integration.
Recall that C is simple if it does not cross itself, which means that whenever , except possibly when and . A curve C with the property that
is a closed curve. If is the only point of intersection, then we say that C is a simple closed curve. As the parameter t increases from the value to the value
, the point starts at the initial point , moves along the curve C , and ends up at the terminal point . If C is simple, then moves continuously from to as t increases and the curve is given an orientation, which we indicate by drawing arrows along the curve. Figure 6.1 illustrates how the terms simple and closed describe a curve.
Figure 6.1 The terms simple and closed used to describe curves.
The complex-valued function is said to be differentiable on if both and are differentiable for . Here we require the one-sided derivatives of
and to exist at the endpoints of the interval. As in Section 6.1, the derivative is
for .
The curve C defined by Equation (6-10) is said to be a smooth curve if is continuous and nonzero on the interval. If C is a smooth curve, then C has a nonzero tangent vector at each point , which is given by the vector . If , then the tangent
vector is vertical. If , then the slope of the tangent line to C
at the point is given by . Hence for a smooth curve the angle of inclination of its tangent vector is defined for all values of and is continuous. Thus a smooth curve has no corners or cusps. Figure 6.2 illustrates this concept.
Figure 6.2 The term smooth used to describe curves.
If C is a smooth curve, then , the differential of arc length, is given by
.
The function is continuous, as and are continuous functions, so the length of the curve C is
(6-11) .
Now consider C to be a curve with parameterization
for .
The opposite curve traces out the same set of points in the plane, but in the reverse order, and has the parametrization
for .
Since , is merely C traversed in the opposite sense, as illustrated in Figure 6.3.
Figure 6.3 The curve and its opposite curve .
A curve C that is constructed by joining finitely many smooth curves end to end is called a contour. Let denote n smooth curves such that the terminal point of the curve coincides with the initial point of for . We express the contour C by the equation
.
A synonym for contour is path.
Example 6.5. Find a parameterization of the polygonal path from shown in Figure 6.4.Here is the line from , is the line from , and is the line from .
Figure 6.4 The polygonal path from .
Solution. We express C as three smooth curves, or . If we set and , we can use Equation (1-48) to get a formula for the straight-line segment joining two points:
,
for . When simplified, this formula becomes
, for .
Similarly, the segments are given by
, for , and
, for .
Explore Solution 6.5.
We are now ready to define the integral of a complex function along a contour C in the plane with initial point A and terminal point B. Our approach is to mimic what is done in calculus. We create a partition of points that proceed along C from A to B and form the differences for . Between each pair of partition points we select a point on C, as shown in Figure 6.5, and evaluate the function . These values are used to make a Riemann Sum for the partition:
(6-12) .
Figure 6.5 Partition points and function evaluation points for a Riemann sum along the contour C from .
Assume now that there exists a unique complex number L that is the limit of every sequence of Riemann sums given in Equation (6-12), where the maximum of tends toward
0 for the sequence of partitions. We define the number L as the value of the integral of the function f(z) taken along the contour C.
Definition 6.2 (Complex Integral). Let C be a contour. Then
,
provided that the limit exists in the sense previously discussed.
Note that in Definition 6.2, the value of the integral depends on the contour. In Section 6.3 the Cauchy - Goursat theorem will establish the remarkable fact that, if f(z) is analytic, then
is independent of the contour.
Example 6.6. Use a Riemann sum to get an approximation for the integral where C is
a the line segment joining the point .
Solution. Set n=8 in Equation (6-12) and form the
partition . For this situation, we have a uniform
increment .
For convenience we select for . Figure 6.6 shows the points and .
Figure 6.6 Partition and evaluation points for the Riemann sum .
One possible Riemann sum, then, is
By rounding the terms in this Riemann sum to five decimal digits, we obtain an approximation for the integral:
This result compares favorably with the precise value of the integral, which you will soon see equals
.
Explore Solution 6.6.
In general, obtaining an exact value for an integral given by Definition 6.2 is a daunting task. Fortunately, there is a beautiful theory that allows for an easy computation of many contour integrals. Suppose that we have a parametrization of the contour C given by the function for . That is, C is the range of the function over the interval , as Figure 6.7 shows.
Figure 6.7 A parametrization of the contour C by for .
It follows that
where and are the points contained in the interval with the property that and , as is also shown in Figure 6.7. If for all k we multiply the kth term in the last
sum by , then we get
The quotient inside the last summation looks suspiciously like a derivative, and the entire quantity looks like a Riemann sum. Assuming no difficulties, this last expression should
equal , as defined in Section 6.1. Of course, if we're to have any hope of this happening, we would have to get the same limit regardless of how we parametrize the contour C. As Theorem 6.1 states, this is indeed the case.
Theorem 6.1. Suppose that is a continuous complex-valued function defined on a set containing the contour C. Let be any parametrization of C for . Then
.
Proof.
Two important facets of Theorem 6.1 are worth mentioning. First, Theorem 6.1 makes the problem of evaluating complex-valued functions along contours easy, as it reduces the task to the evaluation of complex - valued functions over real intervals - a procedure that you studied in Section 6.1. Second, according to Theorem 6.1, this transformation yields the same answer regardless of the parametrization we choose for C.
Example 6.7. Give an exact calculation of the integral in Example 6.6: where C is a
the line segment joining the point .
Solution. We must compute , where C is the line segment
joining . According to Equation (1-48), we can parametrize C
by , for . As , Theorem 6.1 guarantees that
Each integral in the last expression can be done using integration by parts. (There is a simpler way-see Remark 6.1.) We leave as an exercise to show that the final answer simplifies
to , as we claimed in Example 6.6.
Explore Solution 6.7.
Example 6.8. Evaluate the contour integral where C is a the upper semicircle with radius 1 centered at .
Solution. The function , for is a parametrization for C. We apply
Theorem 6.1 with . (Note: ), and .) Hence
.
Explore Solution 6.8.
To help convince yourself that the value of the integral is independent of the parametrization chosen for the given contour, try working through Example 6.8 with , for .
A convenient bookkeeping device can help you remember how to apply Theorem 6.1.
Because , you can symbolically equate z with z(t) and with .
These identities should be easy to remember because z is supposed to be a point on the contour
C parametrized by z(t), and , according to the Leibniz notation for the derivative.
If , then by the preceding paragraph we have
(6-13)
where are the differentials for , respectively (i.e., is equated with , etc.). The expression is often called the complex differential of z. Just as
are intuitively considered to be small segments along the x and y axes in real variables, we can think of dz as representing a tiny piece of the contour C. Moreover, if we write
we can put Equation (6-11) into the form
(6-14) .
so we can think of as representing the length of .
Suppose , and is a parametrization for the contour C. Then
(6-15)
where we are equating u(t) with u(z(t)), x' with x'(t), and so on.
If we use the differentials given in Equation (6-13), then we can write Equation (6-15) in terms of line integrals of the real-valued functions u(t) and v(t) , giving
(6-16) ,
which is easy to remember if we recall that symbolically
.
We emphasize that Equation (6-16) is merely a notational device for applying Theorem 6.1. You should carefully apply Theorem 6.1as illustrated in Examples 6.7 and 6.8 before using any shortcuts suggested by the latter.
Example 6.9. Show that , where is the line segment from , and is the portion of the parabola joining , as indicated in Figure 6.8.
Figure 6.8 The two contours and joining .
Solution. The line segment joining is given by the slope intercept
formula , which can be written as . If we choose the parametrization and , we can write segment as
and
for .
Along we have . Applying Theorem 6.1 gives
.
We now multiply out the integrand and put it into its real and imaginary parts:
Similarly, we can parametrize the portion of the parabola joining by and and so that
and
for .
Along we have . Theorem 6.1 now gives
Explore Solution 6.9 (a)
Explore Solution 6.9 (b)
Extra Example 1. Evaluate the contour integrals of starting at the points . (a) Use the line segment joining the points. (b) Use a portion of a parabola joining the points.Remark. The example in the text used the function which is difficult for hand computations but it is not a challenge for Mathematica, hence we choose to use the function
.
Explore Solution for Extra Example 1 (a)
Explore Solution for Extra Example 1 (b)
Extra Example 2. Evaluate the contour integrals of starting at the points . (a) Use the line segment joining the points. (b) Use a portion of a parabola joining the points.Remark. This example illustrates the situation when f(z) is not analytic.
Explore Solution for Extra Example 2 (a)
Explore Solution for Extra Example 2 (b)
In Example 6.9, the value of the two integrals is the same. This outcome doesn't hold in general, as Example 6.10 shows.
Example 6.10. (a) Show that but that ,
where is the semicircular path from , in the upper half plane, and is the polygonal path from , respectively, shown in Figure 6.9.
Figure 6.9 The two contours and joining .
Solution. We parametrize the semicircle as
and ,
for .
Applying Theorem 6.1, we have , so
and
We parametrize in three parts, one for each line segment:
where in each case. Integrating over these three line segments we obtain
,
,
.
We get our answer by adding the three integrals along the three segments:
Note that the value of the contour integral along isn't the same as the value of the contour integral along , although both integrals have the same initial and terminal points.
Explore Solution 6.10 (a)
Explore Solution 6.10 (b)
Contour integrals have properties that are similar to those of integrals of a complex function of a real variable, which you studied in Section 6.1. If C is given by Equation (6-10), then the integral for the opposite contour -C is
Using the change of variable in this last equation and the property
that , we obtain
(6-17) .
If two functions f and g can be integrated over the same path of integration C, then their sum can be integrated over C, and we have the familiar result
.
Constant multiples also behave as we would expect:
.
If two contours and are placed end to end so that the terminal point of coincides with the initial point of , then the contour is a continuation of , and
(6-18) .
If the contour C has two parametrizations
for , and
for ,
and there exists a differentiable function such that
(6-19) , , and for ,
then we say that is a reparametrization of the contour C. If f is continuous on C, then we have
(6-20) .
Equation (6-20) shows that the value of a contour integral is invariant under a change in the parametric representation of its contour if the reparametrization satisfies Equations (6-19).
We now give two important inequalities relating to complex integrals.
Theorem 6.2 (Absolute Value Inequality). If is a continuous function of the real parameter t, then
(6-21) .
Proof.
Theorem 6.3 (ML - Inequality). If is continuous on the contour C, then
(6-23) .
Proof.
Example 6.11. Use Inequality (6-23) to show that ,where C is the straight-line segment from .
Solution. Here , and the terms represent the distance from the point z to the points , respectively. Referring to Figure 6.10 and using a geometric argument, we get
, for z on C.
Figure 6.10 The distances for z on C.
Thus we have
.
Because L, the length of C, equals 1, Inequality (6-23) implies that
.
Exercises for Section 6.2. Contours and Contour Integrals
Exercise 1. Give a parametrization of each contour.
1 (a). , as indicated in Figure 6.11.
Figure 6.11. The contour . Where
for , and for .Solution 1 (a).
1 (b). , as indicated in Figure 6.12.
Figure 6.12. The contour . Where
for , and for , and for .
Exercise 2. Sketch the following curves.
2 (a). , for .
2 (b). , for .
2 (c). , for . Solution 2 (c).
Exercise 3. Consider the integral , where C is the positively oriented upper semicircle of radius 1, centered at 0.
3 (a). Give a Riemann sum approximation for the integral by selecting and using the
points and . Solution 3 (a).
3 (b). Compute the integral exactly by selecting a parametrization for C and applying Theorem 6.1.Solution 3 (b).
Exercise 4. Show that the integral where C is a the line segment joining the
point
of Example 6.7 simplifies to .
Exercise 5. Evaluate from along the following contours, as shown in Figures 6.13 (a) and 6.13 (b).
5 (a). The polygonal path with vertices , as shown in Figures 6.13 (a).
Figure 6.13 (a). The contour . Solution 5 (a).
5 (b). The contour C that is the upper half of the circle , oriented clockwise, as shown in Figures 6.13 (b).
Figure 6.13 (a). The contour . Solution 5 (b).
Exercise 6. Evaluate from along the following contours, as shown in Figures 6.14 (a) and 6.14 (b).
6 (a). The polygonal path with vertices , as shown in Figure 6.14 (a).
Figure 6.14 (a). The contour .
6 (b). The contour C that is the left half of the circle , oriented clockwise. clockwise, as shown in Figure 6.14 (b).
Figure 6.14 (b). The contour that is oriented clockwise, as shown.
Exercise 7. Recall is the circle of radius r centered at a, oriented counter-clockwise.
7 (a). Evaluate . Solution 7 (a).
7 (b). Evaluate .
7 (c). Evaluate . (The minus sign in means the clockwise orientation.)Solution 7 (c).
7 (d). Evaluate . (The minus sign in means the clockwise orientation.)
7 (e). Evaluate , where C is the portion of in the first quadrant.Solution 7 (e).
7 (f). Evaluate , where C s the upper half of .
7 (g). Evaluate , where C is the upper half of . Solution 7 (g).
Exercise 8. Let be a continuous function on the circle .
Show that .
Exercise 9. Use the results of Exercise 8 with to evaluate
9 (a). .Solution 9 (a).
9 (b). , where is an integer.Solution 9 (b).
Exercise 10. Use the techniques of Example 6.11 to show that
10 (a). , where C is the first quadrant portion of .
10 (b). .
Exercise 11. Evaluate , where C is the line segment from . Solution 11.
Exercise 12. Evaluate , where C given by .
Exercise 13. Evaluate , where C is the straight-line segment joining .Solution 13.
Exercise 14. Evaluate , where C is the square with vertices taken with the counterclockwise orientation.
Exercise 15. Evaluate , where C is the straight-line segment joining . Solution 15.
Exercise 16. Let , be a smooth curve. Give a meaning for each of the following expressions.
16 (a). .
16 (b). .
16 (c). .
16 (d). .
Exercise 17. Evaluate , where C is the polygonal path from that consists of theline segments from and . Solution 17.
Exercise 18. Let be defined on , where .
Show that there is no number such that .
In other words, the mean value theorem for definite integrals that you learned in calculus does not hold for complex functions.
Exercise 19. Use the ML inequality to show that , where is the Legendre polynomial defined on by
Solution 19.
Exercise 20. Explain how contour integrals in complex analysis and line integrals in calculus are different.
How are contour integrals in complex analysis and line integrals in calculus similar ?
6.3 The Cauchy-Goursat Theorem
The Cauchy-Goursat theorem states that within certain domains the integral of an analytic function over a simple closed contour is zero. An extension of this theorem allows us to replace integrals over certain complicated contours with integrals over contours that are easy to
evaluate. We demonstrate how to use the technique of partial fractions with the Cauchy - Goursat theorem to evaluate certain integrals. In Section 6.4 we will see that the Cauchy-Goursat theorem implies that an analytic function has an antiderivative. To begin, we need to introduce some new concepts.
Recall from Section 1.6 that each simple closed contour C divides the plane into two domains. One domain is bounded and is called the interior of C; the other domain is unbounded and is called the exterior of C. Figure 6.15 illustrates this concept, which is known as the Jordan curve theorem.
Figure 6.15 The interior and exterior of simple closed contours.
Recall also that a domain D is a connected open set. In particular, if are any pair of points in D, then they can be joined by a curve that lies entirely in D. A domain D is said to be a simply connected domain if the interior of any simple closed contour C contained in D is contained in D. In other words, there are no "holes" in a simply connected domain. A domain that is not simply connected is said to be a multiply connected domain. Figure 6.16 illustrates uses of the terms simply connected and multiply connected.
Figure 6.16 Simply connected and multiply connected domains.
Let the simple closed contour C have the parametrization for . Recall that if C is parametrized so that the interior of C is kept on the left as z(t)
moves around C, then we say that C is oriented positively (counterclockwise); otherwise, C is oriented negatively (clockwise). If C is positively oriented, then -C is negatively oriented. Figure 6.17 illustrates the concept of positive and negative orientation.
Figure 6.17 Simple closed contours that are positively and negatively oriented.
Green's theorem is an important result from the calculus of real variables. It tells you how to evaluate the line integral of real-valued functions.
Theorem 6.4 (Greens Theorem). Let C be a simple closed contour with positive orientation and let R be the domain that forms the interior of C. If P and Q are continuous and have
continuous partial derivatives at all points on C and R, then
.
Proof.
Proof of Theorem 6.4 is in the book.Complex Analysis for Mathematics and Engineering
We are now ready to state the main result of this section.
Theorem 6.5 (Cauchy-Goursat Theorem). Let f(z) be analytic in a simply connected domain D. If C is a simple closed contour that lies in D, then
.
Proof.
Proof of Theorem 6.5 is in the book.Complex Analysis for Mathematics and Engineering
Example 6.12. Let us recall that (where n is a positive integer) are all entire functions and have continuous derivatives. The Cauchy-Goursat theorem implies that, for any simple closed contour,
(a) ,
(b) , and
(c) .
Explore Solution 6.12 (a)
Explore Solution 6.12 (b)
Explore Solution 6.12 (c)
Example 6.13. If C is a simple closed contour such that the origin does not lie interior to C, then
there is a simply connected domain D that contains C in which is analytic, as is
indicated in Figure 6.22. The Cauchy-Goursat theorem implies that .
Figure 6.22 A simple connected domain D containing the simple closed contour C that does not contain the origin.
We want to be able to replace integrals over certain complicated contours with integrals that are easy to evaluate. If is a simple closed contour that can be "continuously deformed'' into another simple closed contour without passing through a point where f is not analytic, then the value of the contour integral of f over is the same as the value of the integral of f over . To be precise, we state the following result.
Theorem 6.6 (Deformation of Contour). Let and be two simple closed positively oriented contours such that lies interior to . If f(z) is analytic in a domain D that contains both and and the region between them, as shown in Figure 6.23, then
.
Figure 6.23 The domain D that contains the simple closed contours and and the region between them.
Proof.
Proof of Theorem 6.6 is in the book.Complex Analysis for Mathematics and Engineering
We now state as a corollary an important result that is implied by the deformation of contour theorem. This result occurs several times in the theory to be developed and is an important tool for computations. You may want to compare the proof of Corollary 6.1 with your solution to Exercise 23 from Section 6.2.
Corollary 6.1. Let denote a fixed complex value. If C is a simple closed contour with positive orientation such that lies interior to C, then
(i) , and
(ii) , where n is any integer except .
Proof.
Demonstration for (i).
Demonstration for (ii).
The deformation of contour theorem is an extension of the Cauchy-Goursat theorem to a doubly connected domain in the following sense. Let D be a domain that contains and and the region between them, as shown in Figure 6.23. Then the contour is a parametrization of the boundary of the region R that lies between so that the points of R lie to the left of C as a point z(t) moves around C. Hence C is a positive orientation of the
boundary of R, and Theorem 6.6 implies that .
We can extend Theorem 6.6 to multiply connected domains with more than one "hole.'' The proof, which is left for the reader, involves the introduction of several cuts and is similar to the proof of Theorem 6.6.
Theorem 6.7 (Extended Cauchy-Goursat Theorem). Let be simple closed positively oriented contours with the property that lies interior to C for and
the set of interior to has no points in common with the set interior to if . Let f(z) be analytic on a domain D that contains all the contours and the region between C and , as shown in Figure 6.26. Then
.
Figure 6.26 The multiply connected domain D and the contours in the statement of the extended Cauchy-Goursat theorem.
Proof.
Example 6.14. Show that , where C is the circle taken with positive orientation.
Solution. Using partial fraction decomposition gives
, so
(6-38) .
The points lie interior to C, so Corollary 6.1 implies that
.
Substituting these values into Equation (6-38) yields
Explore Solution 6.14.
Example 6.15. Show that , where C is the circle taken with positive orientation.
Figure 6.27 The circle and the points .
Solution. Using partial fractions again, we have
In this case, lies interior to C but does not, as shown in Figure 6.27. By Corollary 6.1, the second integral on the right side of the above equation has the value . The first integral equals zero by the Cauchy-Goursat theorem because the
function is analytic on a simply connected domain that contains C. Thus
Explore Solution 6.15.
Example 6.16. Show that , where C is the "figure eight" contour shown in Figure 6.2
Figure 6.28 The contour .
Solution. Again, we use partial fractions to express the integral:
(6-39) .
Using the Cauchy-Goursat theorem, Property (6-17), and Corollary 6.1 (with ), we compute the value of the first integral on the right side of Equation (6-39):
Similarly, we find that
If we substitute the results of the last two equations into Equation (6-39) we get
Explore Solution 6.16.
Exercises for Section 6.3. The Cauchy-Goursat Theorem
6.4 The Fundamental Theorems of Integration
Let f be analytic in the simply connected domain D. The theorems in this section show that an antiderivative F can be constructed by contour integration. A consequence will be the fact that in a simply connected domain, the integral of an analytic function f along any contour joining is the same, and its value is given by . As a result, we can use the antiderivative formulas from calculus to compute the value of definite integrals. The next two theorems are generalizations of the Fundamental Theorems of Calculus.
Theorem 6.8 (Indefinite Integrals or Antiderivatives). Let f(z) be analytic in the simply connected domain D. If is a fixed value in D and if C is any contour in D with initial point and terminal point z, then the function
is well-defined and analytic in D, with its derivative given by .
Proof.
Proof of Theorem 6.8 is in the book.Complex Analysis for Mathematics and Engineering
Remark 6.2. It is important to stress that the line integral of an analytic function is independent
of path. In Example 6.9 we showed that , where and were different contours joining . Because the integrand is an analytic function, Theorem 6.8 lets us know ahead of time that the value of the two integrals is the same; hence one calculation would have sufficed. If you ever have to compute a line integral of an analytic function over a difficult contour, change the contour to something easier. You are guaranteed to get the same answer. Of course, you must be sure that the function you're dealing with is analytic in a simply connected domain containing your original and new contours.
Exploration.
If we set in Theorem 6.8, then we obtain the following familiar result for evaluating a definite integral of an analytic function.
Theorem 6.9 (Definite Integrals). Let f(z) be analytic in a simply connected domain D. If and are two points in D joined by a contour C, then
,
where F(z) is any antiderivative of f(z) in D.
Proof.
Proof of Theorem 6.9 is in the book.Complex Analysis for Mathematics and Engineering
Theorem 6.9 gives an important method for evaluating definite integrals when the integrand is an analytic function in a simply connected domain. In essence, it permits you to use all the rules of integration that you learned in calculus. When the conditions of Theorem 6.9 are met, applying it is generally much easier than parametrizing a contour.
Example 6.17. Show that where is the principal branch of the square root function and C is the line segment joining .
Remark. Sometimes we write this as .
Solution. We showed in Example 3.10 (see Section 3.2) that if
, then , where the principal branch of the square root function is used in both the formulas for F(z) and F'(z). We note that C is contained in the simply connected domain
, which is the open disk of radius 4 centered at the midpoint of the segment
C. Since is analytic in the domain and is an anti-derivative of , Theorem 6.9 guarantees that
Explore Solution 6.17.
Example 6.18. Show that , where C is the line segment between .
Solution. An antiderivative of is . Because F(z) is entire, we use Theorem 6.9 to conclude that
Explore Solution 6.18.
Example 6.19. We let be the simply connected domain which
is the z-plane slit along the negative x-axis, shown in Figure 6.32. We know that is analytic in D, and has an antiderivative for all z in D. If C is a contour in D that joins the point to the point , then Theorem 6.9 implies that
.
Figure 6.32 The simply connected domain D shown in Examples 6.19 and 6.20.
Example 6.20. Show that , where C is the unit circle , taken with positive orientation.
Solution. We let C be that circle with the point -1 omitted, as shown in Figure 6.32(b). The contour C is contained in the simply connected domain D of Example 6.19. We know
that is analytic in D, and has an antiderivative , for all . Therefore, if we let approach -1 on C through the upper half-plane and approach -1 on C through the lower half-plane,
Explore Solution 6.20.
Extra Example 1. Show that .
Explore Solution for Extra Example 1.
Exercises for Section 6.4. The Fundamental Theorems of Integration
6.5 Integral Representations for Analytic Functions
We now present some major results in the theory of functions of a complex variable. The first result is known as Cauchy's integral formula and shows that the value of an analytic function f(z) can be represented by a certain contour integral. The derivative, , will have a similar representation. In Section 7.2, we use the Cauchy integral formulas to prove Taylor's theorem and also establish the power series representation for analytic functions. The Cauchy integral formulas are a convenient tool for evaluating certain contour integrals.
Theorem 6.10 (Cauchy Integral Formula). Let f(z) be analytic in the simply connected domain D, and let C be a simple closed positively oriented contour that lies in D. If is a point that lies interior to C, then
.
Proof.
Proof of Theorem 6.10 is in the book.Complex Analysis for Mathematics and Engineering
Example 6.21. Show that , where C is the circle with positive orientation.
Solution. We have and . The point lies interior to the circle, so Cauchy's integral formula implies that
,
and multiplication by establishes the desired result.
Explore Solution 6.21.
Example 6.22. Show that , where C is the circle with positive orientation.
Solution. Here we have . We manipulate the integral and use Cauchy's integral formula to obtain
Explore Solution 6.22.
Example 6.23. Show that , where C is the circle with positive orientation.
Solution. We see that . The only zero of this
expression that lies in the interior of C is .
We set and use Theorem 6.10 to conclude that
Explore Solution 6.23.
Theorem 6.11 (Leibniz's Rule). Let G be an open set, and let be an interval of real numbers. Let and its partial derivative with respect to z be continuous functions for all z in G and all t in I. Then
is analytic for z in G, and
.
Proof.
Demonstration for Theorem 6.11.
We now generalize Theorem 6.10 to give an integral representation for the derivative, . We use Leibniz's rule in the proof and note that this method of proof is a mnemonic
device for remembering Theorem 6.12.
Theorem 6.12 (Cauchy's Integral Formulae for Derivatives). Let be analytic in the simply connected domain D, and let C be a simple closed positively oriented contour that lies in D. If z is a point that lies interior to C, then for any integer , we have
.
Proof.
Proof of Theorem 6.12 is in the book.Complex Analysis for Mathematics and Engineering
Example 6.24. Let denote a fixed complex value. Show that, if C is a simple closed positively oriented contour such that lies interior to C, then
, and(6-50)
, for any integer .
Solution. We let . Then for . Theorem 6.10 implies that the value of the first integral in Equations (6-50) is
,
and Theorem 6.12 further implies that
.
This result is the same as that proven earlier in Corollary 6.1. Obviously, though, the technique of using Theorems 6.10 and 6.12 is easier.
Explore Solution 6.24 (a).
Explore Solution 6.24 (b).
Example 6.25. Show that , where C is the circle with positive orientation.
Solution. If we set , then a straightforward calculation shows
that . Using Cauchy's integral formulas with , we conclude that
Explore Solution 6.25.
We now state two important corollaries of Theorem 6.12.
Corollary 6.2. If is analytic in the domain D, then all derivatives exists for (and therefore are analytic in D).
Proof.
Proof of Corollary 6.2 is in the book.Complex Analysis for Mathematics and Engineering
Remark 6.3. This result is interesting, as it illustrates a big difference between real and complex functions. A real function can have the property that exists everywhere in a domain D, but exists nowhere. Corollary 6.2 states that if a complex function has the property that exists everywhere in a domain D, then, remarkably, all derivatives of exist in D.
Corollary 6.3. If is a harmonic function at each point in the domain D, then all
partial derivatives , , , , exists and are harmonic functions.
Proof.
Proof of Corollary 6.3 is in the book.Complex Analysis for Mathematics and Engineering
Extra Example 1. Show that the partial derivatives of are harmonic functions.
Explore Extra Solution 1.
Exercises for Section 6.5. Integral Representations for Analytic Functions
6.6 The Theorems of Morera and Liouville and Extensions
In this section we investigate some of the qualitative properties of analytic and harmonic functions. Our first result shows that the existence of an antiderivative for a continuous function is equivalent to the statement that the integral of f(z) is independent of the path of integration. This result stated in a form that will serve as a converse to the Cauchy-Goursat theorem.
Theorem 6.13 (Morera's Theorem). Let f(z) be a continuous function in a simply connected
domain D. If for every closed contour in D, then f(z) is analytic in D.
Proof.
Proof of Theorem 6.13 is in the book.Complex Analysis for Mathematics and Engineering
Cauchy's integral formula show how the value can be represented by a certain contour integral. If we choose the contour of integration C to be a circle with center , then we can show that the value is the integral average of the values of f(z) at points z on the circle C.
Theorem 6.14 (Gauss's Mean Value Theorem). If f(z) is analytic in a simply connected domain D that contains the circle , then
.
Proof.
Proof of Theorem 6.14 is in the book.Complex Analysis for Mathematics and Engineering
We now prove an important result concerning the modulus of an analytic function.
Theorem 6.15 (Maximum Modulus Principle). Let f(z) be analytic and nonconstant in the bounded domain D. Then does not attain a maximum value at any point in D.
Proof.
Proof of Theorem 6.15 is in the book.Complex Analysis for Mathematics and Engineering
We sometimes state the maximum modulus principle in the following form.
Theorem 6.16 (Maximum Modulus Principle). Let f(z) be analytic and nonconstant in the bounded domain D. If f(z) is continuous on the closed region R that consists of D and all of its boundary points B, then assumes its maximum value, and does so only at point(s) on the boundary B.
Proof.
Proof of Theorem 6.16 is in the book.Complex Analysis for Mathematics and Engineering
Example 6.26. Let . If we set our domain D to be , then f(z) is continuous on the closed region . Prove that
,
and this value is assumed by f(z) at a point on the boundary of D.
Solution. From the triangle inequality and the fact that in D, it follows that
(6-58) .
If we choose , where , then
so the vectors and lie on the same ray through the origin. This is the requirement for the Inequality (6-58) to be an equality (see Exercise 19 in Section 1.3). Hence , and the result is established.
Explore Solution 6.26.
Extra Example 1. Let . If we set our domain D to be , then
f(z) is continuous on the closed region . Show that , and this
value is assumed by at a point on the boundary of D.
Explore Solution for Extra Example 1.
Extra Example 2. Let . If we set our domain D to be , then f(z) is continuous on the closed region . Show
that , and this value is assumed by at a point on the boundary of D.
Explore Solution for Extra Example 2.
Theorem 6.17 (Cauchy's Inequalities). Let f(z) be analytic in the simply connected domain D that contains the circle . If holds for all points
, then
for .
Proof.
Proof of Theorem 6.17 is in the book.Complex Analysis for Mathematics and Engineering
Theorem 6.18 shows that a nonconstant entire function cannot be a bounded function.
Theorem 6.18 (Liouville's Theorem). If f(z) is an entire function and is bounded for all values of z in the complex plane, then f(z) is constant.
Proof.
Proof of Theorem 6.18 is in the book.Complex Analysis for Mathematics and Engineering
Example 6.27. Show that the function sin(z) is not a bounded function.
Solution. We established this characteristic with a somewhat tedious argument in Section 5.4. All we need do now is observe that f(z) is not constant, and hence it is not bounded.
Explore Solution 6.27.
We can use Liouville's theorem to establish an important theorem of algebra.
Theorem 6.19 (Fundamental Theorem of Algebra). If P(z) is a polynomial of degree , then P(z) has at least one zero.
Proof.
Proof of Theorem 6.19 is in the book.Complex Analysis for Mathematics and Engineering
Corollary 6.4. Let P(z) be a polynomial of degree . Then P(z) can be expressed as the product of linear factors. That is,
where are the zeros of P(z) counted according to multiplicity an A is a constant.
Proof.
Extra Example 3. Find the n zeros of the equation .
Explore Solution for Extra Example 3.
Extra Example 4. Find the n zeros of the equation .
Explore Solution for Extra Example 4.
Extra Example 5. Find the roots of the Chebyshev polynomial.
Explore Solution for Extra Example 5.
Exercises for Section 6.6. The Theorems of Morera and Liouville and Extensions
6.7 The Fundamental Theorem of Algebra
This section is a supplement to the textbook.
In Section 6.6 we developed the background (Theorems 6.13 - 6.18) for the proof of the Fundamental Theorem of Algebra.
Theorem 6.13 (Morera's Theorem). Let f(z) be a continuous function in a simply connected
domain D. If for every closed contour in D, then f(z) is analytic in D.
Theorem 6.14 (Gauss's Mean Value Theorem). If f(z) is analytic in a simply connected domain D that contains the circle , then
.
Theorem 6.15 (Maximum Modulus Principle). Let f(z) be analytic and nonconstant in the bounded domain D. Then does not attain a maximum value at any point in D.
Theorem 6.16 (Maximum Modulus Principle). Let f(z) be analytic and nonconstant in the bounded domain D. If f(z) is continuous on the closed region R that consists of D and all of its boundary points B, then assumes its maximum value, and does so only at point(s) on the boundary B.
Theorem 6.17 (Cauchy's Inequalities). Let f(z) be analytic in the simply connected domain D that contains the circle . If holds for all points
, then
for .
Theorem 6.18 (Liouville's Theorem). If f(z) is an entire function and is bounded for all values of z in the complex plane, then f(z) is constant.
Theorem 6.19 (Fundamental Theorem of Algebra). If P(z) is a polynomial of degree , then P(z) has at least one zero.
Proof.
Proof of Theorem 6.19 is in the book.Complex Analysis for Mathematics and Engineering
Corollary 6.4. Let P(z) be a polynomial of degree . Then P(z) can be expressed as the product of linear factors. That is,
where are the zeros of P(z) counted according to multiplicity an A is a constant.
Proof.
In Section 1.1, we introduced the formulas of Cardano and Tartaglia. Historically, formulas have been developed for the quadratic equation, cubic equation and quartic equation. There is no general formula for polynomial equations higher than fourth degree (see Abel's Impossibility Theorem).
The solution of the cubic equations. The depressed cubic equation has roots
Exploration.
Example 1. Find the zeros of the equation .
Explore Solution for 1.
The solution of the cubic equations. The general cubic equation has roots
Exploration.
Example 2. Find the zeros of the equation .
Explore Solution 2.
The solution of the quartic equations. Mathematica can construct the solutions to the general quartic equation.
Exploration.
Example 3. Find the n zeros of the equation .
Explore Solution 3.
Example 4. Find the n zeros of the equation .
Explore Solution 4.
Example 5. Find the roots of the Chebyshev polynomial.
Explore Solution 5.
Extra Example 6. Find the n zeros of the equation .
Explore Solution 6.
Exercises for Section 6.7 The Fundamental Theorem of Algebra
