8.7  The Argument Principle and Rouché's Theorem

    We now derive two results based on Cauchy's residue theorem. They have important practical applications and pertain only to functions all of whose isolated singularities are poles.

Definition 8.3 (Meromorphic Function).      A function f(z) is said to be meromorphic in a domain D provided the only singularities of f(z) are isolated poles and removable singularities.

    We make three important observations relating to this definition.

(i).       Analytic functions are a special case of meromorphic functions.

(ii).      Rational functions , where P(z) and Q(z) are polynomials, are meromorphic in the entire complex plane.

(iii).     By definition, meromorphic functions have no essential singularities.

    Suppose that f(z) is analytic at each point on a simple closed contour C and f(z) is meromorphic in the domain that is the interior of C.  We assert without proof that Theorem 7.13 can be extended to meromorphic functions so that f(z) has at most a finite number of zeros that lie inside C.   Since the function    is also meromorphic, it can have only a finite number of zeros inside C, and so f(z) can have at most a finite number of poles that lie inside C.

    Theorem 8.8, known as the argument principle, is useful in determining the number of zeros and poles that a function has.

Theorem 8.8 (Argument Principle).  Suppose that f(z) be meromorphic in the simply connected domain D.  Suppose that f(z) is meromorphic in the simply connected domain D and that C is a simple closed positively oriented contour in D such that f(z) has no zeros or poles  ()  for  .  Then

(8-34)            ,  

where    is the number of zeros of f(z) that lie inside C and    is the number of poles that lie inside C.

Proof.

Proof of Theorem 8.8 is in the book.
Complex Analysis for Mathematics and Engineering

Remark about Theorem 8.8 

Corollary 8.1.  Suppose that f(z) is analytic in the simply connected domain D.  Let C be a simple closed positively oriented contour in D such that for .  Then  

            ,  

where    is the number of zeros of f(z) that lie inside C.

Proof.

Remark 8.3.  Certain feedback control systems in engineering must be stable.  A test for stability involves the function , where F(z) is a rational function.  If G(z) does not have any zeros in the region , then the system is stable.  We determine the number of zeros of G(z)  by writing , where P(z) and Q(z) are polynomials with no common zero.  Then , and we can check for the zeros of by using Theorem 8.8.  We select a value R so that for and then integrate along the contour consisting of the right half of the circle and the line segment between . This method is known as the Nyquist stability criterion.

    Why do we label Theorem 8.8 as the argument principle?  The answer lies with a fascinating application known as the winding number.
Recall that a branch of the logarithm function, , is defined by

            ,  


where    and  .  Loosely speaking, suppose that for some branch of the logarithm, the composite function were analytic in a simply connected domain D containing the contour C.  This would imply that is an antiderivative of the function for all .  Theorems 6.9 and 8.8 would then tell us that, as z winds around the curve C, the quantity would change by .  Since is purely imaginary, this result tells us that would change by radians.  In other words, as z winds around C, the integral would count how many times the curve winds around the origin.

    Unfortunately, we can't always claim that is an antiderivative of the function for all .  If it were, the Cauchy-Goursat theorem would imply that .  Nevertheless, the heuristics that we gave - indicating that counts how many times the curve winds around the origin - still hold true, as shown in the book.

Figure 8.10  The points on the contour that winds around .

Figure 8.11  The points on the contour that winds around .

Example 8.25.  The image of the circle under is the curve shown in Figure 8.12.  Note that the image curve    winds twice around the origin. We check this by computing

            .

The residues of the integrand are at 0 and -1. Thus

              

            Figure 8.12  The image curve    under  .

Explore Solution 8.25.

    Finally, we note that if , then , and thus we can generalize what we've just said to compute how many times the curve winds around the point a. Theorem 8.9 summarizes our discussion.

Theorem 8.9 (Winding Number).  Suppose f(z) is meromorphic in the simply connected domain D.  If C is a simple closed, positively oriented contour in D such that for   and , then  

            ,  

known as the winding number of about a, counts the number of times the curve winds around the point a.  If , the integral counts the number of times the curve winds around the origin.

Proof.

Proof of Theorem 8.9 is in the book.
Complex Analysis for Mathematics and Engineering

Remark 8.4.  Letting in Theorem 8.9 gives

              

which counts the number of times the curve C winds around the point a.  If C is not a simple closed curve, but crosses itself perhaps several times, we can show (but omit the proof) that still gives the number of times the curve C winds around the point a. Thus winding number is indeed an appropriate term.

    We close this section with a result that will help us gain information about the location of the zeros and poles of meromorphic functions.

Theorem 8.10 (Rouche's Theorem).  Suppose that f(z) and g(z) are meromorphic functions defined in the simply connected domain D, that C is a simply closed contour in D, and that f(z) and g(z) have no zeros or poles for .  If the strict inequality   holds for all , then  .

Proof.

Proof of Theorem 8.10 is in the book.
Complex Analysis for Mathematics and Engineering

Corollary 8.2.  Suppose that f(z) and g(z) are analytic functions defined in the simply connected domain D, that C is a simple closed contour in D, and that f(z) and g(z) have no zeros for .  If the strict inequality    holds for all , then .  

Proof.

Remark 8.5.  Theorem 8.10 is usually stated with the requirement that f(z) and g(z) satisfy the condition  ,  for  .  The improved theorem that we gave was proved by T. Estermann (in the book "Complex Numbers and Functions," London: Athlone -Oxford University Press, 1962, section 18.2, p.156.), and also by Irving Glicksberg (see the American Mathematical Monthly, 83 (1976), pp. 186-187). The weaker version is adequate for most purposes, however, as the following examples illustrate.

Example 8.26.  Show that al four zeros of the polynomial    lie in the disk  .

Solution.  Let  .  Then  ,  and at points on the circle    we have the relation  

                

Of course, if  ,  then as we indicated in Remark 8.5 we certainly have  ,  so that the conditions for applying Corollary 8.2 are satisfied on the circle .  The function f(z) has a zero of order 4 at the origin, so g(z) must have four zeros inside .

Explore Solution 8.26.

Example 8.27.  Show that the polynomial    has one zero in the disk  .  

Solution.  Let ,  then  .  At points on the circle    we have the relation  

              

The function f(z) has one zero at the point in the disk , and the hypotheses of Corollary 8.2 hold on the circle .  Therefore g(z) has one zero inside .

Explore Solution 8.27.

Exercises for Section 8.7.  The Argument Principle and Rouché's Theorem