intzero.htm Cauchy's Integral Theorem
Cauchy's Integral Theorem
The fundamental result - Cauchy's Integral Theorem - says roughly:
If C is a simple closed path and w = f(z) is analytic inside and on C, then
ó
(ç)
õ


C 
 f(z)  dz = 0.
There are two common approaches to this result. The first approachuses the Cauchy-Riemann equations and Green's Theorem. The second approach uses less assumptions about the regularity of the derivative f¢ and builds up the proof by first considering C to be a simple closed triangle and then approximating the general simple closed path by a simple closed polygonal path.


Cauchy's Integral Theorem using Green's Formula

Theorem. Let C be a simple closed path enclosing a region D. Suppose that on D ÈC, w = f(z) is analytic and that f¢ is continuous. Then
ó
(ç)
õ


C 
 f(z)  dz = 0.
 Proof: By the Cauchy-Riemann Equations,
f
-
z
 
 = 1
2
 æ
è 		f
x
 + i f
y
 ö
ø 		=0.
Then by Green's Formula
0 = ó
õ 		ó
õ

D 
 æ
è 		f
x
 + i f
y
 ö
ø 		 dx  dy
= ó
(ç)
õ


C 
 f  dy - i ó
(ç)
õ


C 
 f dx
= -i ó
(ç)
õ


C 
 f  (dx + i dy)
= -i ó
(ç)
õ


C 
 f(z)  dz
= 0.


Consequences of Cauchy's Integral Theorem
1.
In Simply Connected Regions Integrals of Analytic functions are Independent of the Path
A region D is simply connected if for every simple closed path C in D, all of the points inside C are also in D. The most important examples of simply connected regions are
·
Circles: {z | |z-z| < R}
·
Half Planes: {z| Âz > 0}
·
The whole complex plane C
·
Convex regions
Let w = f(z) be analytic in a simply connected region D. Let Z1 and Z2 be two points in D, and take two paths C1 and C2 in D which go from Z1 (initial point) to Z2 (terminal point). Then C1 - C2 can be broken into simple closed paths so that
0 = ó
õ

C1 - C2 
 f(z)  dz
= ó
õ

C1 
 f(z)  dz - ó
õ

C2 
 f(z)  dz,
ó
õ

C1 
 f(z)  dz
= ó
õ

C2 
 f(z)  dz.
Thus we define
ó
õ 		Z2

Z1 
 f(z)  dz = ó
õ

C 
 f(z)  dz,
where C is any path in D from Z1 to Z2.


2.
Two Circles Theorem
Let Ce be a circle inside a circle CR.

intzpic1.gif

Suppose that f(z) is analytic on the two circles and the region in between the two circles. Then
ó
(ç)
õ


Ce 
 f(z)  dz = ó
(ç)
õ


CR 
 f(z)   dz.
. The proof uses a cut C from the outer circle to the inner circle.

intzpic2.gif

Then
ó
(ç)
õ


CR 
 f(z)  dz - ó
(ç)
õ


Ce 
 f(z)  dz
= ó
(ç)
õ


CR + C - Ce - C  
 f(z)  dz
= 0.


3.
Fundamental Theorem of Calculus Version II
Let w = f(z) be analytic in a simply connected region D. For z Î D, define
F(z) = ó
õ 		z

Z0 
 f(z)  dz.
Then F(z) in analytic in D and F¢(z) = f(z).




File translated from TEX by TTH, version 3.76.
On 26 Apr 2007, 15:34.