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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/
