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