Power Series in Complex Analysis

1. -K.Anitha M.Sc., M.Phil.,
-A.Sujatha M.Sc., M.Phil.,PGDCA
Department of Mathematics ( SF )
V.V.Vanniaperumal College for Women,
Virudhunagar.
Complex Analysis
( 15UMTC62)

2. Power Series
 An infinite series of the form
where the coefficients and the z are
complex numbers is called power series.
 Every complex power series
has radius of convergence R and has a circle of
convergence defined by |z – z0| = R, 0 < R < .
0
0
( )k
k
k
a z z




3. Taylor Series
Let f be analytic within a domain D and let z0 be a point
in D. Then f has the series representation
valid for the largest circle C with center at z0 and radius
R that lies entirely within D.
  k
k
k
zz
k
zf
zf )(
!
)(
0
0
0
)(
 



4. Maclurin series
 Some important Maclurin series




0
2
!!2!1
1
k
k
z
k
zzz
e 






0
1253
)!12(
)1(
!5!3
sin
k
k
k
k
zzz
zz 




0
242
)!2(
)1(
!4!2
1cos
k
k
k
k
zzz
z 

5. Laurent Series
 Isolated Singularities
Suppose z = z0 is a singularity of a complex
function f. For example, 2i and -2i are sigularities
of The point z0 is said to be an
isolated singularity, if there exists some deleted
neighborhood or punctured open disk 0 < |z – z0|
< R throughout which is analytic.
2
( )
4
z
f z
z



6. Laurent’s Series
Let f be analytic within the annular domain D defined
by . Then f has the series
representation
valid for . The coefficients ak are given
by
where C is a simple closed curve that lies entirely within
D and has z0 in its interior.
Rzzr  || 0




k
k
k zzazf )()( 0
Rzzr  || 0
1
0
1 ( )
, 0 , 1, 2 , ,
2 ( )
k kC
f s
a ds k
i s z 
   


7. Example
Expand in a Laurent series valid
for 0 < |z| < 1.
Solution
We can write
)1(
18
)(
zz
z
zf



 
...999
1
...1)
1
8(
1
118
)1(
18
)(
2
2








zz
z
zz
zzz
z
zz
z
zf

8. Zeros and Poles
 Introduction
Suppose that z = z0 is an isolated singularity of f
and
(1)
is the Laurent series of f valid for 0 < |z – z0| < R.
The principal part of (1) is
 










1 0
0
0
0 )(
)(
)()(
k k
k
kk
k
k
k
k zza
zz
a
zzazf










1 01
0
)(
)(
k
k
k
k
k
k
zz
a
zza

9. Classification
(i) If the principal part is zero, z = z0 is called a
removable singularity.
(ii) If the principal part contains a finite number of
terms, then z = z0 is called a pole. If the last
nonzero coefficient is a-n, n  1, then we say it
is a pole of order n. A pole of order 1 is
commonly called a simple pole.
(iii) If the principal part contains infinitely many
nonzero terms, z = z0 is called an essential
singularity.

10. Example
 We form
that z = 0 is a removable singularity
 From
0 < |z|. Thus z = 0 is a simple pole.
Moreover, (sin z)/z3 has a pole of order 2.

!5!3
1
sin 42
zz
z
z
...
!5!3
1sin 3
2

zz
zz
z

11. Residues
 Residue
The coefficient a-1 of 1/(z – z0) in the Laurent series is
called the residue of the function f at the isolated
singularity. We use this notation
a-1 = Res(f(z), z0)
 Cauchy’s Residue Theorem
Let D be a simply connected domain and C a simply
closed contour lying entirely within D. If a function f is
analytic on and within C, except at a finite number of
singular points z1, z2, …, zn within C, then
 

n
k
kC
zzfsidzzf
1
),)((Re2)( 