Page 103 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes We have, thus, produced a power series having the given analytic function as a limit:


j

f(z)   c (z z ) , z z  r,

j
0
0
j 0

where,
1 f(s)
c =  j 1 ds


j 2 i (s z ) 
C
0
9.4 Keywords
Taylor series: f is analytic on the open disk |z – z | < r. Let z be any point in this disk and choose
0
C to be the positively oriented circle of radius , where |z – z | <  < r. Then for s  C we have
0
 
1  1  1   1     (z z ) j

0
s z (s z ) (z z ) (s z ) 1  z z 0   0 j 1





j 0 (s z )


0 
0

0
  s z 

0 
Cauchy Integral Formula
n! f(s)
(n)

f (z )  2 i (s z ) n 1 ds,for n  0,1,2,.....
0



0
C
This is the famous Generalized Cauchy Integral Formula. Recall that we previously derived this
formula for n = 0 and 1.
9.5 Self Assessment
1. f is analytic on the open disk |z – z | < r. Let z be any point in this disk and choose C to
0
be the positively oriented circle of radius , where |z – z | <  < r. Then for s  C we
0
have
 
1 1 1  1  .................

s z  (s z ) (z z )  (s z )  1  z z 0   





0 
0
0
  s z 

0 
2. The circle of ................. is the largest circle centered at z inside of which the limit f is
0
analytic.
3. Suppose f is analytic in the region ................., and let C be a positively oriented simple
closed curve around z in this region.
0
1 
j
4. Laurent series for f, the one valid for the region 1 < |z| < . When, f(z)     z is
z j 0

equal to .................
96 LOVELY PROFESSIONAL UNIVERSITY