Weiqi Luo (骆伟祺)
School of Software
Sun Yat-Sen University
Email
： weiqi.luo@yahoo.com
Office
：
# A313


Convergence of Sequences; Convergence of Series
 Taylor Series; Proof of Taylor's Theorem; Examples;

Laurent Series; Proof of Laurent's Theorem; Examples

Absolute and Uniform Covergence of Power Series

Continuity of Sums of Power Series

Integration and Differentiation of Power Series

Uniqueness of Series Representations
 Multiplication and Division of Power Series
2 School of Software

55. Convergence of Sequences

The limit of Sequences
An infinite sequence z
1
, z
2
, …, z n
, … of complex number has a limit z if, for each positive number ε, there exists a positive integer n
0 n>n
0 | z n
   such that when
Denoted as n lim
 z n
 z
Note that the limit must be unique if it exists;
Otherwise it diverges
3 School of Software

55. Convergence of Sequences

Theorem
Suppose that z n
Then
= x n
+ iy n
(n = 1, 2, . . .) and z = x + iy. n lim
 z n
 z
If and only if n lim
 x n
 x & lim n
 y n
 y
Proof:
If n lim
 x n
 x & lim n
 y n
 y then, for each positive number ε, there exists n
1 such that
| x n x

2
, n
 n
1
| y n y

2
, n
 n
2 and n
2
,
4 School of Software

55. Convergence of Sequences
Let n
0
=max(n
1
,n
2
), then when n>n
0
| x n x

2
& | y y n

2
| z n
  x n
 iy n
) ( x iy
 x n x ) ( n
 y ) |

| x n
 x |

| y n y |
  
2 2 n lim
 z
 exists a positive integer n
0 z such that, when n>n
0
| z n z | | ( x n
 iy n
) ( x iy ) | | ( x n x ) i y n
 y ) |
 
|
| x n
  x n x ) ( n
 y ) |
 y n
  x n x ) ( n
 y ) |
 
 n lim
 x n
 x & lim n
 y n
 y
5 School of Software

55. Convergence of Sequences

Example 1
The sequence z n

1
3
  n
, ( 1, 2,...) converges to i since
1 n lim(
 n
3
1
) lim n
 n
3
 i n lim1

0 i 1 i
6 School of Software

55. Convergence of Sequences

Example 2
When z n i n
2 n
, ( n

1, 2,...)
The theorem tells us that n lim
 z n
 lim( 2) n

  i n lim(
 n
2 n
)
    
2
If using polar coordinates, we write r n

| z n
| &
  n
Argz , ( n 1, 2,...) n
 n lim
 r n
 n lim 4


1
4
 n
2
 n
 n lim

Argz
2 n
  n lim

Argz
2 n

1
  
Why?
Evidently, the limit of Θ n tends to infinity.
does not exist as n
7 School of Software

56. Convergence of Series

Convergence of Series
An infinite series n



1 z n z
1 z
2
...
z n
...
Series of complex number converges to the sum S if the sequence
S
N

N  n

1 z n z
1 z
2
...
z
N
, N

(1, 2,...) Sequence of partial sums converges to S; we then write n



1 z n

S The series has at most one limit, otherwise it diverges
8 School of Software

56. Convergence of Series

Theorem
Suppose that z n
Then
If and only if
= x n n



1

 n

1 x n

X
+ iy n z n

S
& n



1
(n = 1, 2, . . .) and S = X + iY. y n

Y n



1 z n

S
N lim

S
N

S S
N
 n
N  

1 x n
 i
N y n

X
N
 iY
N n

1
N lim

X
N

X & lim
N

Y
N

Y n



1 x n

X & n



1 y n

Y
9 School of Software

56. Convergence of Series

Corollary 1
If a series of complex numbers converges, the nth term converges to zero as n tends to infinity.

 z n z
1 z
2
...
z n n

1 theorem, both the two following real series converse. n



1 x n
& n



1 y n
Then we get that x n and y n
( why?
), and thus n lim
 z n
 n lim
 converge to zero as n tends to infinity x n
 i lim n
 y n
  
0
10 School of Software

56. Convergence of Series

Absolutely convergent
If the series n



1
| z n
|
 n



1 x n
2  y n
2 , ( z n
 x n
 iy n
) of real number converges, n
 y n then the series is said to be absolutely convergent.
11 School of Software

56. Convergence of Series

Corollary 2
The absolute convergence of a series of complex numbers implies the convergence of that series.
| x n
|
 x
2 n
 y
2 n
| y n
|
 x
2 n
 y
2 n n



1
| x n
|
 n



1 n

 

1
| y n
|



1 n x
2 n
 y
2 n x 2 y n
 2 n n



1 n



1
| x n
|
| y n
|
Converge n



1 z n 1 2 z n
Converge
...
12 n



1 x n n



1 y n
Converge
School of Software

56. Convergence of Series

The remainder ρ
N after N terms
S
 n



1 z n z
1 z
2 z
N
 z
N

1
 z
N

2

...
S
N
ρ
N
  
N
S S
N
|
   
N
S S
N
|
Therefore, a series converges to a number S if and only if the sequence of remainders tends to zero.
13 School of Software

56. Convergence of Series

Example
With the aid of remainders, it is easy to verify that when |z| <1,
Note that 1 n



0 z n z z
2
...
z n 

1
1

1
 z n

1 z
, z

1
1
 z
The partial sums
S ( )
N

N n

1 

0 z n
1 z z ...
z
N

1 
1

1
 z
N z
, z

1
If then
1
1 z
, z

1 
N z

S z

S
N z
 z
1

N z
, z

1
N
|

N z

|1
 z |
When |z|<1 ρ
N tends to zero, but not when |z|>1
14 School of Software

56. Homework
 pp.188-189
Ex. 2, Ex. 3, Ex. 5, Ex. 9
15 School of Software

57. Taylor Series

Theorem
Suppose that a function f is analytic throughout a disk
|z − z
0
| < R
0
, centered at z
0 and with radius R
0
. Then f (z) has the power series representation
 n



0 n
(
 z
0 n
) , (| z
 z
0
|

R
0
) a n
 f z
0 , ( n

0,1, 2,...) n !
That is, series converges to f (z) when z lies in the stated open disk.
a n

2
1
 i

C
( z
 z
0
) n

1 Refer to pp.167
16 School of Software

57. Taylor Series

Maclaurin Series
When z
0
=0 in the Taylor Series become the Maclauin Series y=e x
 n



0 f n !
(0) z n
,(| z
 z
0
|

R
0
)
In the following Section, we first prove the Maclaurin Series, in which case f is assumed to be assumed to be analytic throughout a disk |z|<R
0
17 School of Software

58. Proof the Taylor’s Theorem


 n

0 f n !
(0) z n
,(| z
 z
0
|

R
0
)
Proof:
Let C
0 denote and positively oriented circle |z|=r
0
, where r<r
0
<R
0
Since f is analytic inside and on the circle C
0 and since the point z is interior to C
0
, the Cauchy integral formula holds

1
2
 i

C
0 s
 z
,

,| |

R
0 s
1
 z

1 s

1 z s

1 s 1

1 w
, w
 z s w

Refer to pp.187
18 School of Software

58. Proof the Taylor’s Theorem
1 s
 z

N n


0

1 1 s n

1 z n  z
N
1
( s

)
N

1
2
 i

C
0 s
 z

N n



1
0
1
2
 i

C
0 s n

1 z n 
1
2
 i z
N
C
0

( s

) N
Refer to pp.167

N n

1 

0 f f n !
(0) n !
(0) z n  z
N
2
 i

C
0
( s

)
N
ρ
N
19 School of Software

58. Proof the Taylor’s Theorem
When
N lim


N

N lim
 z
N
2
 i

C
0
( s

)
N

0 f z

N lim (
 n

1 N 

0 f n !
(0) z n  
N
)
 n



0 f n !
(0) z n n



0 f n !
(0) z n
|

N

2 z

N i

C
0
( s
( )

)
N
|

| |

N M
2 ( r
0
 r r
0
N
2
 r
0
Where M denotes the maximum value of |f(s)| on C
0
|

N
|
 r
0
Mr r

0 ( ) r r
0
N r
( )

1 r
0
N lim


N

0
20 School of Software

59. Examples

Example 1
Since the function f (z) = e z is entire, it has a Maclaurin series representation which is valid for all z. Here f (n) (z) = e z (n = 0, 1, 2, . . .) ; and because f (n) (0) = 1 (n = 0, 1, 2, . .
.) , it follows that e z  n



0 z n n
!
z
 
)
Note that if z=x+i0, the above expansion becomes e x 

 n

0 x n n !
, ( x )
21 School of Software

59. Examples

Example 1 (Cont’)
The entire function z 2 e 3z also has a Maclaurin series expansion, e z  n



0 z n n
!
z
 
)
Replace z by 3z
2 3 z e z  n



0
3 n n !
z n

2
, (| |
 
)
If replace n by n-2, we have
2 3 z e z  n



2
3 n

2
( n

2)!
z n
, (| |
 
)
22 School of Software

59. Example2

Example 2
Trigonometric Functions sin z
 e iz  e
 iz
2 i


( 1) n


0 n
(2 z n
2 n

1

1)!
, (| Z |
 
) cos z
 e iz  e
 iz
2


 n

0
( 1) n z
2 n n
(2 )!
, (| Z |
 
)
23 School of Software

59. Examples

Example 4
Another Maclaurin series representation is
1
1
 z
 n



0 z n  since the derivative of the function f(z)=1/(1-z), which fails to be analytic at z=1, are f ( )
 n !
(1
 z ) n

1
, ( n

0,1, 2,...)
In particular, f (0)
 n !, ( n

0,1, 2,...)
24 School of Software

59. Examples
 Example 4 (Cont’) substitute –z for z
1
1
 z


 n n z z n


0

1
1
 z


 n

0 z n  z
1


 n
( 1) ( z
 n
1) , (| z
  n


0 replace z by 1-z
25 School of Software

59. Examples

Example 5
 z
2

 z
3 z
5 z
1 2(1
3
1

 z z
2
2

 z
1
3
(2

1

1 z
2
) expand f(z) into a series involving powers of z.
We can not find a Maclaurin series for f(z) since it is not analytic at z=0. But we do know that expansion
1

1 z
2
1 z
2  z
4  z
6   
Hence, when 0<|z|<1 Negative powers

1 z
3
(2 1 z
2
 z
4
 z
6 z ...)

1 1 z
3 z z z
3 z
26 School of Software
...

59. Homework
 pp. 195-197
Ex. 2, Ex. 3, Ex. 7, Ex. 11
27 School of Software

60. Laurent Series

Theorem
Suppose that a function f is analytic throughout an annular domain
R
1
< |z − z
0
| < R
2
, centered at z
0
, and let C denote any positively oriented simple closed contour around z
0 and lying in that domain.
Then, at each point in the domain, f (z) has the series representation


 n

0 n
(
 z
0
) n 

 n

1 b n
( z
 z
0
) n
, ( R
1
  z
0
|

R
2
) a n

2
1
 i

C
( z
 z
0
) n

1
, ( n

0,1, 2,...) b n

2
1
 i

C
( z
 z
0
) n 1
, ( n

1, 2,...)
28 School of Software

60. Laurent Series

Theorem (Cont’)
 n



0 n
(
 z
0
) n 

 n

1
( b n z
 z
0
) n
, ( R
1
  z
0
|

R
2
) a n

2
1
 i

C
( z
 z
0
) n

1
, ( n

0,1, 2,...) b n

2
1
 i

C
( z
 z
0
) n 1
, ( n

1, 2,...)

1  n

( b
 n z
 z
0
)
 n


1  n
 b
 n
( z
 z
0
) n
 n


 n
(
 z
0 n
) , ( R
1
  z
0
|

R
2
) c n
 
 b
 n
, n n
,

 
0
1 c n

2
1
 i

C
( z
 z
0
) n

1
, ( n
  
29 School of Software

60. Laurent Series
 Laurent’s Theorem
If f is analytic throughout the disk |z-z
0
|<R
2
,


 n

0 n
(
 z
0
) n  n



1
( b n z
 z
0
) n
, ( R
1
  z
0
|

R
2
) reduces to Taylor
Series about z
0 b n

2
1
 i

C
( z
 z
0
) n 1

1
2
 i

C
( z
 z
0
) n

1 f z dz n

1, 2,...)
Analytic in the region |z-z
0
|<R
2 b n

0, ( n

1, 2,...)
 n



0 n
(
 z
0
) n a n

2
1
 i

C
( z
 z
0
) n

1
 f z
0 , ( n

0,1, 2,...) n !
30 School of Software

62. Examples

Example 1
Replacing z by 1/z in the Maclaurin series expansion e z 

 n

0 z n n !
1
2 z z z
3
    
1!
2!
3!
...(| |
 
)
We have the Laurent series representation e
1/ z  n



0
1 n
1
1

1
 z z
2
1!
2!
3!
1 z
3
   
)
There is no positive powers of z, and all coefficients of the positive powers are zeros.
b n

2
1
 i

C
( z

0) n 1
, ( n

1, 2,...)
1
 
1 z
2
1
 i

C
( z

0) where c is any positively oriented simple closed contours around the origin

1
2
 i

C
1/ z e dz

C
1/ z e dz

2
 i
31 School of Software

62. Examples

Example 2
The function f(z)=1/(z-i) 2 is already in the form of a
Laurent series, where z
0
=i,. That is
1
(

)
2
 n


 c z i n
 n
( ) , (0 | | ) where c
-2
=1 and all of the other coefficients are zero.

C dz
( z i ) n

3 c n

2
1
 i

C
( z
 dz z
0
) n

3
, ( n
  
0, n
 2

,
 
2
 
2 where c is any positively oriented simple contour around the point z
0
=i
32 School of Software

62. Examples
Consider the following function


1
( z

1)( z

2)
 z
1

1

1 z

2 which has the two singular points z=1 and z=2, is analytic in the domains
D z

1
:| | 1
D
2
D
3
33 School of Software

62. Examples

Example 3
The representation in D
1 is Maclaurin series.
 z
1

1
 z
1

 
2 1
1
 z
1

1
Refer to pp. 194 Example 4 where |z|<1 and |z/2|<1
  n



0 z n  n



0
2 z n n

1
 n



0
(2 n 1 
1) z n z

34 School of Software

62. Examples

Example 4
Because 1<|z|<2 when z is a point in D
2
, we know
 z
1

1
 z
1

2
1 z
1 1
  z
Refer to pp. 194 Example 4 where |1/z|<1 and |z/2|<1
 n



0 z n
1

1
 n



0 z
2 n n

1
 n



1 z
1 n
 n



0 z n
2 n

1
35 School of Software

62. Examples

Example 5
Because 2<|z|<∞ when z is a point in D
3
, we know
 z
1

1
 z
1

2
1 z

1
 
1 (1/ ) z

1
Refer to pp. 194 Example 4 where |1/z|<1 and |2/z|<1
 n



0 z n
1

1
 n



0 z
2 n n

1
 n



0
 n z n

1
 n



1
 n

1 z n
  
)
36 School of Software

62. Homework
 pp. 205-208
Ex. 3, Ex. 4, Ex. 6, Ex. 7
37 School of Software

63~66 Some Useful Theorems

Theorem 1 (pp.208)
If a power series n



0 n
(
 z
0
) n converges when z = z
1
(z
1
≠ z
0
), then it is absolutely convergent at each point z in the open disk |z − z
0
| < R
1 where R
1
= |z
1
− z
0
|
38 School of Software

63~66 Some Useful Theorems

Theorem 2 (pp.210)
If z
1 is a point inside the circle of convergence |z − z
0
| =
R of a power series n



0 n
(
 z
0
) n then that series must be uniformly convergent in the closed disk |z − z
0
| ≤ R
1
, where R
1
= |z
1
− z
0
|
39 School of Software

63~66 Some Useful Theorems

Theorem (pp.211)
A power series n



0 n
(
 z
0
) n represents a continuous function S(z) at each point inside its circle of convergence |z − z
0
| = R.
40 School of Software

63~66 Some Useful Theorems

Theorem 1 (pp.214)
Let C denote any contour interior to the circle of convergence of the power series S(z), and let g(z) be any function that is continuous on C. The series formed by multiplying each term of the power series by g(z) can be integrated term by term over C; that is,

C
 n



0 n
 n a ( )( z
0
) dz
C

 n



0 n
(
 z
0
) n
Corollary: The sum S(z) of power series is analytic at each point z interior to the circle of convergence of that series.
41 School of Software

63~66 Some Useful Theorems

Theorem 2 (pp.216)
The power series S(z) can be differentiated term by term. That is, at each point z interior to the circle of convergence of that series,
 n



0
( a z n
(
 z
0 n
) ) '
 n



0 n
(
 z
0
) n

1
42 School of Software

63~66 Some Useful Theorems
The uniqueness of Taylor/Laurent series representations

Theorem 1 (pp.217)


(
 z ) n If a series n 0 n

0 converges to f (z) at all points interior to some circle |z − z
0
| = R, then it is the Taylor series expansion for f in powers of z − z
0
.

Theorem 2 (pp.218)

If a series n

 n
(
 z
0
) n  n



0 n
(
 z
0
) n  n



1
( b n z
 z
0
) n converges to f (z) at all points in some annular domain about z
0
, then it is the Laurent series expansion for f in powers of z − z
0 for that domain.
43 School of Software