Page 99 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes and the radius of convergence is R. Then we know, of course, that the limit function f is analytic
for |z – z | < R. We showed that if f is analytic in |z – z | < r, then the series converges for
0
0
|z – z | > r. Thus r  R, and so f cannot be analytic at any point z for which |z – z | > R. In other
0
0
words, the circle of convergence is the largest circle centered at z inside of which the limit f is
0
analytic.
Example:
Let f(z) = exp(z) = e . Then f(0) = f’(0) = ... = ƒ (0) = ... = 1, and the Taylor series for f at z = 0 is
z
(n)
0
 1
 z j
j 0 j!

and this is valid for all values of z since f is entire. (We also showed earlier that this particular
series has an infinite radius of convergence.)
9.2 Laurent Series

Suppose f is analytic in the region R < |z – z | < R , and let C be a positively oriented simple
2
0
1
closed curve around z in this region.
0

Notes We include the possibilities that R can be 0, and R = .
1
2
We shall show that for z  C in this region

  b j
j

f(z)   a (z z )   j ,
j
0
j 0 j 1 (z z ) 0


where,
1 f(s)

a  2 i (s z ) j 1 ds,for j  0,1,2,...
j



0
C
and
1 f(s)


b  2 i (s z )   ds,for j 1,2,...
j
j 1


0
C
The sum of the limits of these two series is frequently written

j
f(z)   c (z z ) ,

j
0
j – 
where,
1 f(s)


c  2 i (s z ) j 1 ds, j  0, 1, 2,....

j



0
C
This recipe for f(z) is called a Laurent series, although it is important to keep in mind that it is
really two series.
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