Chapter Ⅴ
SERIES
Exercises
1.
Show in two ways that the sequence
z n  2  i
2.
(1) n
n2
(n  1, 2,)
converges to –2.
Let rn denote the moduli and  n the principal values of the arguments of the complex
numbers z n in Exercise 1. Show that the sequence rn (n  1, 2,) converges but that the
3.
sequence  n (n  1, 2,) does not.
Show that
lim z n  z , then
if
n 
4.
lim zn  z .
n
i
Write z  re , where 0  r  1 , in the summation formula that was derived in the
example in Sec. 5.2. Then, with the aid of the theorem in Sec. 5.2, show that

r sin 
r cos   r 2
r n sin n 
and
r cos n 


2
1  2r cos   r 2
1  2r cos   r
n 1
n1
when 0  r  1 . (Note that these formulas are also valid when r  0 .)

n
5.
6.

Show that a limit of a convergent sequence of complex numbers is unique by appealing to the
corresponding result for a sequence of real numbers.
Show that

if
z
n 1
7.
n
S,

z
then

 zn  S ,

 cz
then
n 1
n 1
n
 cS .
By recalling the corresponding result for series of real numbers and referring to the theorem
in Sec. 5.2, show that

if
 zn  S

and
n 1
9.
S .
Let c denote any complex number and show that
if
8.
n
n 1
 n  T ,

then
n 1
 (z
n 1
n
n )  S  T .
Let a sequence {z n } converge to a number z . Show that there exists a positive number
M such that the inequality z n  M holds for all n , i.e., {z n } is bounded. Do this in
each of the ways indicated below.
(a) Note that there is a positive integer n0 such that
zn  z  ( zn  z)  zn  1
whenever n  n0 .
(b) Write z n  xn  iy n and recall from the theory of sequences of real numbers that the
convergence of {x n } and { y n } implies that xn  M 1 and
some positive numbers M 1 and M 2 .
yn  M 2 (n  1,2, ) for
10. Let two sequences {z n } and {wn } converge, show that the sequence {z n wn } converge
and
lim z n wn   lim z n lim wn .

n 
n 


n 
Exercises
1. Obtain the Maclaurin series representation
z 4 n1
n 0 (2n)!

z cosh( z 2 )  
( z  ) .
2. Obtain the Taylor series
( z  1) n
n!
n 0

e z  e
for the function f ( z )  e
z
( z 1  )
by
(a) using f (1) (n  0,1,2) ; (b) writing e  e
3. Find the Maclaurin series expansion of the function
z
(n )
f ( z) 
z 1
e
z
z
1
.
 
z  9 9 1  ( z 4 / 9)
4
(1) n 4 n1
( z  3) .
z

2 n2
n 0 3
4. Show that if f ( z )  sin z , then
f ( 2 n ) (0)  0 and
f ( 2n1) (0)  (1) n

Ans.
(n  0,1,2) .
Thus give an alternative derivation of the Maclaurin series (5.4.2) for sin z in Sec. 5.4.
5. Rederive the Maclaurin series (5.4.3) in Sec. 5.4 for the function f ( z )  cos z by
(a) using the definition
cos z 
e iz  e iz
2
z
in Sec. 3.6 and appealing to the Maclaurin series (5.4.1) for e in Sec. 5.4;
(b) showing that
(n  0,1,2) .
f ( 2n ) (0)  (1) n and f ( 2 n1) (0)  0
6. Write the Maclaurin series representation of the function f ( z )  sin( z ) , and point out how
it follows that
(n  0,1,2) .
f ( 2 n1) (0)  0
f ( 4 n ) (0)  0 and
7. Derive the Taylor series representation
2

1
( z  i) n

1  z n0 (1  i) n1
z i  2 .
Suggestion: Start by writing
1
1
1
1



.
1  z (1  i )  ( z  i) 1  i 1  ( z  i ) /(1  i)
8. With the aid of the identity (see Sec. 3.6)
 
cos z   sin  z   ,
2

expand cos z into a Taylor series about the point z0   / 2 .
9. Use the identity sinh( z   i )   sinh z , verified in Exercise 7 (a), Sec, 3.7, and the fact
that sinh z is periodic with period 2 i to find the Taylor series for sinh z about the
point z 0   i .
Ans. 
( z   i) 2 n1

n 0 (2n  1)!

( z   i  ) .
10. What is the largest circle within which the Maclaurin series for the function tanh z
converges to tanh z ? Write the first two nonzero terms of that series.
11. Show that when z  0 ,
ez
1 1 1 1 z2

   
 ;
z 2 z 2 z 2! 3! 4!
sin( z 2 ) 1 z 2 z 6 z10
(b)
 2 


 .
z2
z
3! 5! 7!
(a)
12. Derive the expansions
sinh z 1  z 2 n1
(0  z  ) ;
 
z2
z n0 (2n  3)!

1
1
1 z
3
3
 2 n1
(b) z cosh     z  
(0  z  ) .
2 2
n 1 ( 2n  2)! z
13. Show that when 0  z  4 ,

1
1
zn
.


4 z  z 2 4 z n 0 4 n  2
(a)
Exercises
1. Find the Laurent series that represents the function
 1 
f ( z )  z 2 sin  2 
z 
in the domain 0  z   .
Ans. 1 

(1) n
1
 (2n  1)!  z
n 1
4n
.
2. Derive the Laurent series representation
ez
1   ( z  1)n
1
1 





2
( z  1)
e  n  0 (n  2)! z  1 ( z  1)2 
(0  z  1  ) .
3. Find a representation for the function
1
1
1
 
1  z z 1  (1/ z )
in negative powers of z that is valid when 1  z   .
f ( z) 
(1) n1
.

zn
n 1

Ans.
4. Give two Laurent series expansions in powers of z for the function
f ( z) 
1
,
z (1  z )
2
and specify the regions in which those expansions are valid.

Ans.
1 1
z n   2 (0  z  1) ;

z z
n 0
5. Represent the function


n 3
1
zn
(1  z  ) .
f ( z) 
z 1
z 1
(a) by its Maclaurin series, and state where the representation is valid;
(b) by it Laurent series in the domain (1  z  ) .
Ans. (a)  1  2


1
.
zn
( z  1) ; (b) 1  2
 zn
n 1
n 1
6. Show that when 0  z  1  2 ,

z
( z  1)n
1
.
 3 n  2 
( z  1)( z  3)
2( z  1)
n 0 2
7. Write two Laurent series in powers of z that represent the function
f ( z) 
1
z (1  z 2 )
in certain domains and specify those domains.
(1) n1
(1  z  ) .

2 n 1
n 1 z
n 0
(a) Let a denote a real number, where 1  a  1 , and derive the Laurent series

Ans.
8.
 (1) n1 z 2n1 
1
z

(0  z  1) ;
representation

a
an

z  a n1 z n
( a  z  )
i
(b) Write z  e in the equation obtained in part (a) and then equate real parts and
imaginary parts on each side of the result to derive the summation formulas

 a n cos n 
n 1
a cos  a 2
1  2a cos  a 2

and
a
n 1
n
sin n 
a sin 
,
1  2a cos  a 2
where 1  a  1. (Compare Exercise 4, Sec. 5.2.)
9. Suppose that a series

 x[n]z
n
n  
converges to an analytic function X (z ) in some annulus R1  z  R2 . That sum X (z )
is called the z  transform of x[n]( n  0,1,2,) . Use expression (5.5.5), Sec. 5.5,
for the coefficients in a Laurent series to show that if the annulus contains the unit circle
z  1 , then the inverse z  transform of X (z ) can be written
1 
X (e i )e in d (n  0,1,2,) .



2
10. (a) Let z be any complex number, and let C denote the unit circle
w  e i
(     )
in the w plane. Then use that contour in expression (5.5.5). Sec. 5.5, for the coefficients in
a Laurent series, adapted to such series about the origin in the w -plane, to show that

z 
1 
exp   w     J n ( z ) w n
(0  w  ) ,
w  n
2 
x[n ] 
where
J n ( z) 
1 
exp[ i ( n  z sin )d
2   
(n  0,1,2,) .
(b) With the aid of Exercise 6, Sec. 4.2, regarding certain definite integrals of even and odd
complex-valued functions of a real variable, show that the coefficients in part (a) can be
written
J n ( z) 
1 
cos( n  z sin )d
 0
(n  0,1,2,) .
11. (a) Let f (z ) denote a function which is analytic in some annular domain about the origin
i
(     ) . By taking that circle as the path of
integration in expressions (5.5.2) and (5.5.3), Sec. 5.5, for the coefficients a n and bn in a
Laurent series in powers of z , show that
n
n

 e i  
1 
1  
i
i  z 
f ( z) 
f (e )d 
 f (e ) ei    z  d
2 
2 n 1 

  
when z is any point in the annular domain.
i
(b) Write u ( )  Re[ f (e )] , and show how it follows from the expansion in part (a) that
1 
1  
u( ) 
u
(

)
d


 u() cos[n(  )]d .
2   
 n 1  
This is one form of the Fourier series expansion of the real-valued function u ( ) on the
interval       . The restriction on u ( ) is more severe than is necessary in order
that includes the unit circle z  e
for it to be represented by a Fourier series.
Exercises
1.
By differentiating the Maclaurin series representation

1
  zn
1  z n 0
( z  1) ,
obtain the expansions

1

 (n  1) z n
(1  z ) 2 n0
( z  1) ,
and
2.

2

(n  1)( n  2) z n

3
(1  z )
n 0
By substituting 1 /(1  z ) for z in the expansion

1

 (n  1) z n
(1  z ) 2 n0
( z  1) .
( z  1) ,
found in Exercise 1, derive the Laurent series representation

1
(1) n (n  1)


z 2 n2 ( z  1) n
3.
(1  z  1  ) .
(Compare Example 2, Sec. 5.9.)
Find the Taylor series for the function
1
1
1
1

 
z 2  ( z  2) 2 1  ( z  2) / 2
about the point z 0  2 . Then, by differentiating that series term by term, show that
1 1 
 z 2
  (1) n (n  1)
( z  2  2) .

2
4 n 0
z
 2 
With the aid of series, prove that the function f defined by means of the equations
n
4.
(e z  1) / z, when z  0,
f ( z)  
when z  0
1,
5.
is entire,
Prove that if
 cos z
 z 2  (  / 2) 2 , when z    / 2,
f ( z)  
 1 ,
when z    / 2,
 
6.
then f is an entire function.
In the w -plane, integrate the Taylor series expansion (see Example 4, Sec. 5.4)
1 
  (1) n ( w  1) n
w n 0
( w  1  1)
along a contour interior to the circle of convergence from w  1 to w  z to obtain the
representation
(1)n 1
( z  1)n
n
n 1

log z  
7.
( z  1  1) .
Use the result in Exercise 6 to show that if
 log z
, when z  1,

f ( z)   z  1

when z  1,
1,
then f is analytic throughout the domain given by 0  z   ,    arg z   .
8.
Prove that if f is analytic at z 0 and f ( z 0 )  f ( z 0 )    f
function g defined by the equations
 f ( z)
 ( z  z ) m1 ,

0
g ( z )   ( m1)
( z0 )
f
,
 ( m  1) !
(m)
( z 0 )  0 , then the
when z  z0 ,
when z  z0 ,
is analytic at z 0 .
9.
Suppose that a function f (z ) has a power series representation

f ( z )   an ( z  z0 ) n
n 0
inside some circle z  z0  R . Use Theorem 5.9.2 in Sec. 5.9, regarding term by term
differentiation of such a series, and mathematical induction to show that
(n  k ) !
an  k ( z  z 0 ) k
k
!
k 0

f ( n) ( z)  
(n  0,1, 2,)
when z  z0  R . Then, by setting z  z0 , show that the coefficients an (n  0,1, 2, )
are the coefficients in the Taylor series for f about z 0 . Thus give an alternative proof of
Theorem 5.10.1 in Sec. 5.10.
10. Consider two series

S1 ( z )   an ( z  z0 ) n ,
n 0

bn
,
n
n1 ( z  z0 )
S2 ( z)  
which converge in some annular domain centered at z 0 . Let denote any contour lying in
that annulus, and let g (z ) be a function which is continuous on C . Modify the proof of
Theorem 5.9.1, Sec. 5.9, which tells us that


C
g ( z )S1 ( z )dz   a n  g ( z )( z  z0 ) n dz ,
C
n 0
to prove that

C

g ( z)
dz .
C ( z  z )n
0
g ( z )S 2 ( z )dz   bn 
n1
Conclude from these results that if


n
n 0

bn
,
n
n1 ( z  z0 )
 cn ( z  z0 )n   cn ( z  z0 ) n  
S ( z) 
then

C

c 
g ( z )S ( z )dz 
n
n  
C
g ( z )( z  z 0 ) n d z .
11. Show that the function
f 2 ( z) 
1
z 1
2
( z  i )
is the analytic continuation (Sec. 2.16) of the function

f1 ( z )   (1) n z 2 n
( z  1)
n 0
into the domain consisting of all points in the z plane except z  i .
12. Show that the function f 2 ( z )  1 / z ( z  0) is the analytic continuation (Sec. 2.16) of the
function
2

f1 ( z )   (n  1)( z  1) n
( z  1  1)
n 0
into the domain consisting of all points in the z plane except z  0 .
Exercises
1. Use multiplication of series to show that
ez
1
1
5
(0  z  1) .
 1 z  z2 
2
z ( z  1) z
2
6
2. By writing csc z  1 / sin z and then using division. Show that
 1
1 1
1
csc z   z  
  z3  
(0  z   ) .
2
z 3!
5!
 (3!)
3. Use division to obtain the Laurent series representation
1
1 1 1
1 3
   z
z 
e  1 z 2 12
720
z
(0  z  2 ) .
4. Use the expansion
1
1 1 1
7
 3  
z 
z sinh z z
6 z 360
2
(0  z   )
in Example 2, Sec. 5.11, and the method illustrated in Example 1, Sec. 5.6, to show that

C
dz
i

,
z sinh z
3
2
when C is the positively oriented unit circle z  1 .
5. Follow the steps below, which illustrate an alternative to straightforward division of series, to
obtain representation (5.11.8) in Example 2, Sec. 5.11.
(a) Write
1
 d 0  d1 z  d 2 z 2  d 3 z 3  d 4 z 4   ,
4
1  z / 3! z / 5! 
2
where the coefficients in the power series on the right are to be determined by multiplying the
two series in the equation


1
1
1  1  z 2  z 4  (d 0  d1 z  d 2 z 2  d 3 z 3  d 4 z 4  ) .
5!
 3!

Perform this multiplication to show that


1 
1 
(d 0  1)  d1 z   d 2  d 0  z 2   d 3  d1  z 3
3! 
3! 



1
1 
  d 4  d 2  d 0  z 4    0
3!
5! 

when z   .
(b) By setting the coefficients in the last series in part (a) equal to zero, find the values of
d 0 , d1 , d 2 , d 3 and d 4 . With these values, the first equation in part (a) becomes equation
(5.11.8), Sec. 5.11.
6. Use mathematical induction to verify formula (5.11.3), Sec. 5.11, for the nth derivative of the
product of two differentiable functions.
7. Let f (z ) be can entire function that is represented by a series of the form.
( z  ) .
(a) By differentiating the composite function g ( z )  f [ f ( z )] successively, find the first
three nonzero terms in the Maclaurin series for g (z ) and thus show that
f [ f ( z )]  z  2a2 z 2  2(a22  a3 ) z 3  
( z  ) .
f ( z )  z  a2 z 2  a3 z 3  
(b) Obtain the result in part (a) in formal manner by writing
f [ f ( z )]  f ( z )  a2 [ f ( z )]2  a3 [ f ( z )]3   ,
replacing f (z ) on the right-hand side here by its series representation, and then collecting
terms in like powers of z .