**for
the**

**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 of Theorem 8.8 is in the book.
**

**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.

**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.

The points on the contour that winds around .Figure 8.10

The points on the contour that winds around .Figure 8.11

**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 .

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 of Theorem 8.9 is in the book.
**

**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 of Theorem 8.10 is in the book.
**

**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 .

**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 all 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 .

**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 .

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

**The Next Module
is**

**Introduction
to the z-transform**

**Return to the Complex
Analysis Modules**

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell