Page 58 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 6: Cauchy’s Integral Formula

     2 Notes
3 z(s z) 2( z) 
=   2 3  g(s)ds


C  (s z   z) (s z) 
Hence,
3 z(s z) 2( z) 

G'(z   z) G'(z)  2 g(s) ds =      2 g(s)ds
 
2
 z  C (s z) 3 C  (s z   z) (s z) 3  




  z ( 3m  2 z )M ,
2
(d   z) d 3
where m = max {|s – z| : s  C}. It should be clear then that

lim G'(z   z) G'(z)  2  g(s) ds  0,

 z 0  z C (s z) 3
or in other words,
g(s)
G''(z)  2  3 ds.

C (s z)
Suppose f is analytic in a region D and suppose C is a positively oriented simple closed curve in
D. Suppose also the inside of C is in D. Then from the Cauchy Integral formula, we know that
f(s)

2 if(z)   ds

C s z
and so with g = f in the formulas just derived, we have
1 f(s) 2 f(s)
f'(z)   ds, and f''(z)   ds
2pi (s z) 2 2 i (s z) 3



C
C
for all z inside the closed curve C. Meditate on these results. They say that the derivative of an
analytic function is also analytic. Now suppose f is continuous on a domain D in which every
0
point of D is an interior point and suppose that f(z)dz  for every closed curve in D. Then we

C
know that f has an antiderivative in D—in other words f is the derivative of an analytic function.
We now know this means that f is itself analytic. We thus have the celebrated Morera’s Theorem:
0
If f : D  C is continuous and such that f(z)dz  for every closed curve in D, then f is analytic

C
in D.
Example:
Let’s evaluate the integral

e z dz,
z 3 
C
where C is any positively oriented closed curve around the origin. We simply use the equation

2 f(s)
f''(z)   ds
2 i (s z) 3


C

LOVELY PROFESSIONAL UNIVERSITY 51