argument.htm Argument Principle
Argument Principle


Zeroes and Poles
For the moment, we shall consider a function f(z) analytic in the punctured disk
×
D
 
z0,R 
 = {z |0 < |z-z0| £ R}.
Then
f(z)
= ¥
å
n = -¥ 
 an (z-z0)n,
an
= 1
2pi
 ó
(ç)
õ


Cz0,r 
 f(z) (z-z0)-n - 1  dz.
·
If f(z) = ån = 0¥ an (z-z0)n, f(z) may be extended by defining f(z0) = a0, and the resulting function is analytic in |z-z0| £ R.

·
If f(z) = ån = N¥ an (z-z0)n, N ³ 0, aN ¹ 0, f(z) is said to have a zero of order N at z=z0. Near z=z0,
f(z) = (z-z0)N ·g(z),
where g(z) is analytic in |z-z0| £ R, g(z0) ¹ 0.

·
If f(z) = ån = - M¥ an (z-z0)n, M ³ 0, a-M ¹ 0, f(z) is said to have a pole of order M at z=z0. Near z=z0,
f(z) = (z-z0)-M ·g(z),
where g(z) is analytic in |z-z0| £ R, g(z0) ¹ 0.

·
If f(z) = ån = - ¥¥ an (z-z0)n, an ¹ 0 for infinitely many negative n, then f(z) is said to have an essential singularity at z=z0.

·
The coefficient of (z-z0)-1 is called the residue of f(z) at z=z0, and is written
Res(f,z=z0) = Resf(z)|z=z0 = 1
2pi
 ó
(ç)
õ


Cz0,r 
 f(z)  dz.

·
Let f(z) be analytic in the punctured disk
×
D
 
z0,R 
 = {z |0 < |z-z0| £ R}.
Then for r small and positive,
ó
(ç)
õ


Cz0,r 
 f(z)  dz = 2 pi Resf(z)|z=z0.

·
Let f(z) be analytic in the punctured disk
×
D
 
z0,R 
 = {z |0 < |z-z0| £ R}.
Suppose that f(z) has a zero of order N > 0, at z=z0.
For z near z0, f(z) = (z-z0)N g(z), g(z) analytic, and g(z0) ¹ 0. It follows that
f¢(z)
f(z)
= N(z - z0)N-1 g(z) + (z-z0)N g¢(z)
(z-z0)N g(z)
=N (z - z0)-1 + analytic,
so that
Res æ
è 		f¢
f
 ,z = z0 ö
ø
= N
= order of zero at z=z0
Then for r small and positive,
1
2pi
 ó
(ç)
õ


Cz0,r 
 f¢(z)
f(z)
  dz = N.

There is another interpretation of the number N. For the moment let fN(z) = (z - z0)N. Follow the argfN(z) as Cz0,r is traversed in the counterclockwise direction. The change in argument of (z - z0)N, denoted by DCz0,r argfN(z) is exactly 2p N. This is the first statement of the Argument Principle :
1
2 p
 DCz0,r argfN(z)
= N
= order of zero.
In the case f(z) has a zero of order N at z=z0, we expect that an antiderivative of the function [(f¢(z))/f(z)] is log(f(z)). This is the case locally, at least if we are near enough to a point z1 on Cz0,r. As the path Cz0,r is traversed counterclockwise, the logarithm of f(z) may be defined locally in a continuous manner, but when we make one full revolution around the circle returning to z1, the argument of f(z) may have changed by a multiple of 2p. We have that
ó
(ç)
õ


Cz0,r 
 f¢(z)
f(z)
  dz
= i ·DCz0,r argf(z)
= 2 pi ·N.
= 2 pi ·order of zero at z=z0.
Thus for the small circle Cz0,r,
N
= 1
2p
 DCz0,r arg f(z)
= 1
2pi
 ó
(ç)
õ


Cz0,r 
 f¢(z)
f(z)
  dz

·
Let f(z) be analytic in the punctured disk
×
D
 
z0,R 
 = {z |0 < |z-z0| £ R}.
Suppose that f(z) has a pole of order M > 0, at z=z0.
Then for r small and positive,
1
2pi
 ó
(ç)
õ


Cz0,r 
 f¢(z)
f(z)
  dz = - M.
Mimicking the discussion above for zeroes, we obtain for the small circle Cz0,r
- M
= 1
2p
 DCz0,r arg f(z)
= 1
2pi
 ó
(ç)
õ


Cz0,r 
 f¢(z)
f(z)
  dz


We are now ready to state the Argument Principle .

Theorem. Let C be a simple closed path. Suppose that f(z) is analytic and nonzero on C and meromorphic inside C.
·
List the zeroes of f inside C as z1, ¼,zk with multiplicities N1, ¼,Nk, an let
ZC = N1 + ¼+ Nk.
·
List the poles of f inside C as w1, ¼,wj with orders N1, ¼,Nk, an let
PC = M1 + ¼+ Mj.
Then
ZC - PC
= 1
2p
 DC arg f(z)
= 1
2pi
 ó
(ç)
õ


C 
 f¢(z)
f(z)
  dz.

Proof. calculate the integral two ways. First take a local antiderivative log(f(z)) to obtain
1
2pi
 ó
(ç)
õ


C 
 f¢(z)
f(z)
  dz
= 1
2p
 DC arg f(z).
Second take small circles around each zi and wi and the usual cuts from C to to the circles. In this way, obtain
1
2pi
 ó
(ç)
õ


C 
 f¢(z)
f(z)
  dz
= j
å
i=1 
 1
2pi
 ó
(ç)
õ


Czi,r 
 f¢(z)
f(z)
  dz + k
å
i=1 
 1
2pi
 ó
(ç)
õ


Cwi,r 
 f¢(z)
f(z)
  dz
= j
å
i=1 
 Ni - k
å
i=1 
 Mi
= ZC - PC.



Corollary. Let C be a simple closed path. Suppose that f(z) is analytic and nonzero on C and analytic inside C.
·
List the zeroes of f inside C as z1, ¼,zk with multiplicities N1, ¼,Nk, an let
ZC = N1 + ¼+ Nk.
Then
ZC
= 1
2p
 DC arg f(z)
= 1
2pi
 ó
(ç)
õ


C 
 f¢(z)
f(z)
  dz.
Briefly stated: Let C be a simple closed path. Suppose that f(z) is analytic and nonzero on C and analytic inside C. Then
1
2p
 DC arg f(z)
= number of zeroes inside C - counting multiplicities.


Indices and Winding Numbers


Let C be a simple closed path. Suppose that f(z) is analytic and nonzero on C and meromorphic inside C. Then w = f(z) = f(z(t)) is a closed path (not necessarily simple). call this path f(C). As w traverses f(c), the number of times the argument of w changes by a multiple of 2p is called the index or winding number of the path f(C). The Argument Principle says that the winding number of f(C) is ZC - PC.


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On 26 Apr 2007, 16:06.