Section 4.5
0.
Cauchy Integral Formula (Version #0) Let G be an open subset of  and let
f  A(G ) . Let a  G and let R satisfy B(a, R )  G . Then , for z such that
| z  a | r  R we have
f ( z) 
1
f ( w)
dw
2 i |w a| r w  z
Cauchy’s Theorem (Version #0) Let G be an open subset of  and let f  A(G ) .
Let a  G and let R satisfy
B(a, R )  G . Let  be a closed rectifiable curve such
{ }  B( a, R) . Then,
 f
1.
Two approaches to extending the Cauchy Integral Formula and Cauchy’s Theorem:
  0 wrt G, i.e., n( , a)  0 for all a   \ G
Homotopy condition:   0 (Section 4.6)
a.
Homology condition:
b.
2.
Let
a.
b.
c.
3.
0

a  \{ } . Define
1
dw
n( , a ) 

2 i  w  a
n( , a ) is continuous on components of  \{ }
n( , a ) is integer valued
n( , a ) is 0 on the unbounded component of  \{ }
be a closed rectifiable curve in  and let
Leibnitz’s Rules
a.
Let  :[a, b]  [c, d ]   such that
 is continuous on [a, b]  [c, d ] .
b
Define g (t ) 
  (s, t )ds . Then, g is continuous on [c ,d]. If
a
 2 ( s, t ) 

 ( s, t ) is continuous on [a, b]  [c, d ] , then g  is
t
b
continuously differentiable on [c ,d] and is given by g '(t ) 
  (s, t )ds .
2
a
b.
(Exercise 4.2.2) Let G be open and let

be a closed rectifiable curve. Let
 :{ }  G   be continuous on { }  G . Define
g ( z )    ( w, z )dw . Then, g is continuous on G. If


 ( w, z ) is continuous on { }  G , then g is analytic on G and is
z
given by g '( z )    2 ( w, z )dw
 2 ( w, z ) 

4.
Lemma (5.1) Let
Fm ( z )  


be a rectifiable curve and let
 ( w)
(w  z )m
 be continuous on { } .
Define
dw for z   \{ } . Then, Fm  A( \{ }) and
Fm  mFm 1 .
5.
Cauchy Integral Formula (Version #1) Let G be an open subset of  and let
f  A(G ) . Let  be a closed rectifiable curve in G such that   0 wrt G. Then, for
z  G \{ }
n( , z ) f ( z ) 
6.
f ( w)
1
dw

2 i  w  z
Cauchy’s Theorem (Version #1) Let G be an open subset of  and let f  A(G ) . Let

be a closed rectifiable curve in G such that
 f
7.
wrt G. Then,
0
Cauchy Integral Formula (Version #2) Let G be an open subset of  and let
f  A(G ) . Let  1 ,  2 ,  ,  m be closed rectifiable curves in G such that
 1   2     m  0 wrt G.
m
Then, for z  G \
f ( z ) n( k , z) 
k 1
8.
 0
m
{
k
},
k 1
f (w)
1 m
dw

2 i k 1  k w  z
Cauchy’s Theorem(Version #2) Let G be an open subset of  and let f  A(G ) . Let
 1 ,  2 ,  ,  m be closed rectifiable curves in G such that  1   2     m  0 wrt G.
Then,
m


k 1
9.
f 0
k
Corollary to CIF #1. Let G be an open subset of  and let f  A(G ) . Let

be a
  0 wrt G. Then, for z  G \{ } and k > 0
k!
f ( w)
n( , z ) f ( k ) ( z ) 
dw
2 i  ( w  z )k 1
closed rectifiable curve in G such that
10.
Corollary to CIF #2. ) Let G be an open subset of  and let f  A(G ) . Let
 1 ,  2 ,  ,  m be closed rectifiable curves in G such that  1   2     m  0 wrt G.
Then, for z  G \
m
{
k
} and k > 0
k 1
m
m
f (w)
f (k ) ( z) n( k , z )  k !  
dw
 k ( w  z )k 1
2

i
k 1
k 1
11.
Morera’s Theorem. Let G be a region. Let f : G   be continuous on G such
that
f
T
 0 for every triangular path T in G. Then, f is analytic on G.