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The Argument Principle: Winding Numbers and Zeros of Meromorphic Functions, Lecture notes of Complex analysis

The Argument Principle, a theorem in complex analysis that relates the value of an integral to the winding numbers of zeros and poles of a meromorphic function. The document also covers Rouche's Theorem, which provides a condition for the number of zeros of two functions to be equal. definitions, theorems, and examples.

What you will learn

  • What is the Argument Principle in complex analysis?
  • How does the Argument Principle relate the value of an integral to the winding numbers of zeros and poles?
  • What is Rouche's Theorem and how is it related to the Argument Principle?

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2021/2022

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V.3. The Argument Principle 1
V.3. The Argument Principle
Note. In this section, we concentrate on zeros and poles of a function. In the
Argument Principle we relate the value of an integral to winding numbers of zeros
or poles. In Rouche’s Theorem, a quantity related to the number of zeros and the
number of poles is given which is preserved between functions satisfying a certain
(inequality) relationship. The specific class of functions of concern is defined in the
following.
Definition V.3.3. If Gis open and fis a function defined and analytic on G
except for poles, then fis a meromorphic function on G.
Note. If fis meromorphic on G, then we can define f:GCby setting
f(z) = at each pole of f. By Exercise V.3.4 fis then a continuous mapping
where we treat Cas a metric space with the metric given in section I.6.
Note. If fis analytic at z=aand fhas a zero of order mat z=a, then
f(z) = (za)mg(z) where g(a)6= 0 be Definition IV.3.1. Hence
f0(z)
f(z)=m(za)m1g(z) + (za)mg0(z)
(za)mg(z)=m
za+g0(z)
g(z).(3.1)
Since g(a)6= 0, then g0/g is analytic “near” z=a.
pf3
pf4
pf5

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V.3. The Argument Principle

Note. In this section, we concentrate on zeros and poles of a function. In the Argument Principle we relate the value of an integral to winding numbers of zeros or poles. In Rouche’s Theorem, a quantity related to the number of zeros and the number of poles is given which is preserved between functions satisfying a certain (inequality) relationship. The specific class of functions of concern is defined in the following.

Definition V.3.3. If G is open and f is a function defined and analytic on G except for poles, then f is a meromorphic function on G.

Note. If f is meromorphic on G, then we can define f : G → C∞ by setting f (z) = ∞ at each pole of f. By Exercise V.3.4 f is then a continuous mapping where we treat C∞ as a metric space with the metric given in section I.6.

Note. If f is analytic at z = a and f has a zero of order m at z = a, then f (z) = (z − a)mg(z) where g(a) 6 = 0 be Definition IV.3.1. Hence

f ′(z) f (z) =^

m(z − a)m−^1 g(z) + (z − a)mg′(z) (z − a)mg(z) =^

m z − a+

g′(z) g(z).^ (3.1)

Since g(a) 6 = 0, then g′/g is analytic “near” z = a.

Note. If f has a pole of order m at z = a, then f (z) = (z − a)−mg(z) where g is analytic at z = a and g(a) 6 = 0 by the definition of pole of order m and Proposition V.1.6. Then

f ′(z) f (z) =^

−m(z − a)−m−^1 g(z) + (z − a)−mg′(z) (z − a)−mg(z) =^

−m z − a +^

g′(z) g(z).^ (3.2)

Again, since g(a) 6 = 0, then g′/g is analytic “near” z = a.

Theorem V.3.4. Argument Principle. Let f be meromorphic in G with poles p 1 , p 2 ,... , pm and zeros z 1 , z 2 ,... , zn repeated according to multiplicity. If γ is a closed rectifiable curve in G where γ ≈ 0 and not passing through p 1 , p 2 ,... , pm, z 1 , z 2 ,... , zn, then

1 2 πi

γ

f ′(z) f (z) dz^ =

∑^ n k=

n(γ; zk) − ∑^ m j=

n(γ; pj ).

Note. Given the representation of f ′/f given in the proof, we see that winding numbers naturally arise here. Also, we would expect a primitive of f ′/f to be log(f ), which of course does not exist on {γ} (unless the winding numbers are 0), but again this hints at multiples of 2πi.

Theorem V.3.6. Let f be meromorphic on region G with zeros z 1 , z 2 ,... , zn and poles p 1 , p 2 ,... , pm repeated according to multiplicity. If g is analytic on G and γ is a closed rectifiable curve in G where γ ≈ 0 and γ does not pass through any zero or pole of f , then

1 2 πi

γ^ g(z)

f ′(z) f (z) dz^ =

∑^ n k=

g(zk)n(γ; zk ) −

∑^ m j=

g(pj )n(γ; pj ).

Note. Ahlfors in his Complex Analysis (McGraw Hill, 1979, page 153) state Rouche’s Theorem as: Let γ ≈ 0 in region G where n(γ; z) is either 0 or 1 for any point z 6 = {γ}. Let f and g be analytic in G and for all z ∈ {γ} suppose |f (z) − g(z)| < |f (z)|. Then f and g have the same number of zeros enclosed by γ.

Note. Another statement of Rouche’s Theorem (see page 119 of Murray Spiegel, Complex Variables with an Introduction to Conformal Mapping and Its Applications in the Schaum’s Outline Series, NY: McGraw-Hill, 1964): If f and g are analytic inside and on a simple closed curve C and if |g(z)| < |f (z)| on C, then f (z) + g(z) and f (z) have the same number of zeros inside C. This version follows from Ahlfors’ version by replacing Ahlfors’ g(z) with f (z) + g(z).

Note. Rouche’s Theorem can be used to give another easy (analytic) proof of the Fundamental Theorem of Algebra.

Theorem. Fundamental Theorem of Algebra. If p(z) = zn^ + an− 1 zn−^1 + · · · + a 2 z^2 + a 1 z + a 0 is a (complex) polynomial of degree n, then p has n zeros (counting multiplicities).

Proof. We have

p(z) zn^ = 1 +^

an− 1 z +^ · · ·^ +^

a 2 zn−^2 +^

a 1 zn−^1 +^

a 0 zn

for z 6 = 0, and (^) zlim→∞^ p z(zn) = 1. So with ε = 1, we have that there exists R > 0

such that for all |z| > R we have

∣∣^ p z(zn )− 1

∣∣ < ε = 1. That is, for |z| > R,

|p(z) − zn| < |zn|. With f (z) = zn^ and g(z) = p(z), we have by Ahlfor’s version of Rouche’s Theorem (of course, this also follows from Conway’s version as well) that, since f (z) = zn^ has n zeros, then g(z) = p(z) has the same number of zeros.

Revised: 5/1/