Chapter 06 -Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 6 Series and Residues Page |1

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Chapter 6
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Note: Solutions not appearing in this complete solutions manual can be found in the student
study guide. © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
Exercises 6.1
{2 +&(−Bartlett
2.© Jones i)n } Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
=n 1: 2 − i
)2 1
n 2 : 2 + (−i=
=
n = 3 : 2 + (−i )3 = 2 + i
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
= 4 : 2 + (−i ) 4 = 2 + 1 = 3
NOT FOR SALE ORn DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
n = 5 : 2 + (−i )5 = 2 − i


3. Using (1)
© of Section
Jones & 4.1 we have
Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
{1 + enπ i } =
1 + eπ i , 1 + e 2π i , 1 + e3π i , 1 + e 4π i , 1 + e5π i , 
=−1 1, 1 + 1, 1 − 1, 1 + 1, 1 − 1, 
= 0, 2, 0, 2, 0, 
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
4. { 2n e
{(1 + i)n } =
in
π
4 }
= n 1: 1 + i
© Jones & Bartlett Learning, LLC π © Jones & Bartlett Learning, LLC
i ⋅ 2⋅
NOT FOR = 2=: 22 e 4 2i
SALE ORn DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
π
( 2)
3 i3
n =3 : e 4
=−2 + 2i
π
n = 4 : (© 2Jones
) e 4 &= −Bartlett
4 i4
4 Learning, LLC © Jones & Bartlett Learning,
NOT5 FOR π SALE  OR2 DISTRIBUTION
2 NOT FOR SALE OR DISTRIB
n =5 : ( 2 ) e 4 =4 2  −
i5
−i  =−4 − 4i
 2 2 

 ni&+Bartlett
© Jones 2n  Learning, LLC © Jones & Bartlett Learning, LLC
6.  SALE n  OR DISTRIBUTION
NOT FOR
 3ni + 5  NOT FOR SALE OR DISTRIBUTION

ni + 2n (2n + ni )(5n − 3ni ) 10n − 3 × 2n ni + 5n ni + 3n 2
= zn = =
3ni + 5n 52 n + 4 n 2 52 n + 4 n 2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 6 Series and Residues Page |2

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
n 2 n n
NOT FOR SALE 10 + 3
ORz DISTRIBUTION n n (5 − 3 × 2 ) NOT FOR SALE OR DISTRIBUTION
= n 2n 2
+ i 2n 2
5 + 4n 5 + 4n
n 3n 2   3n 2 
10  1 +  n 1 + 
10n + 3n 2  10n   2   10n 
= Re( zn ) © =Jones2 & Bartlett = Learning,
  LLC →0 © Jones & Bartlett Learning,
52 n +FOR
NOT 4n SALE 2 n  OR
2
  5   4n 2 
4nDISTRIBUTION NOT FOR SALE OR DISTRIB
5 1 + 2 n  1 + 2 n 
 5   5 

n 2 
n
 2 
n

n5  1 − 3 ×    n  1 − 3   
n (5 n
− 3 × 2 n
)   5     
©5Jones
©=
Jones
Im(&zn )Bartlett
= Learning, LLC =  & Bartlett Learning, LLC
2n
NOT FOR SALE5OR n2
+ 4DISTRIBUTION  4 n 2
  4 n
NOT 2
 FOR SALE OR DISTRIBUTION
52 n  1 + 2 n  5n  1 + 2 n 
 5   5 
as n → ∞, zn converges to 0
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE
7. OR WeDISTRIBUTION
apply Theorem 6.1.1. From NOT FOR SALE OR DISTRIBUTION

(ni + 2) 2 4 − n 2 + 4ni 4 n 2 − 4
zn= = = +i 2
n 2i n 2i n n
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION
we see that NOT FOR SALE OR DISTRIB

4 n2 − 4
lim Re(
= zn ) lim
= 0 and lim Im( = zn ) lim = 1.
n →∞ n →∞ n n →∞ n →∞ n2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR
Therefore, by Theorem 6.1.1, the sequence convergesNOT
SALE OR DISTRIBUTION to i. FOR SALE OR DISTRIBUTION


 n(1 + i n ) 
8.
© Jones & Bartlett Learning, 
n + 1  LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
n(1 + i n )
zn =
n +1
n
→ 1©asJonesn → ∞& Bartlett Learning, LLC © Jones & Bartlett Learning,
n +1
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
But 1 + i n alternates between 0, 1 – i, 2, 1 + i which is why the sequence is divergent


© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 6 Series and Residues Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
10. OR
NOT FOR SALE {eDISTRIBUTION
1/ n
+ 2(tan −1 n)i} NOT FOR SALE OR DISTRIBUTION

zn e1/ n + 2(tan −1 n)i
=
Re( z=
n) e1/ n → 1
© Jones & Bartlett π Learning, LLC © Jones & Bartlett Learning,
Im(
= zn ) NOT −1
2(tanFOR n) →=2 × ORπ DISTRIBUTION
SALE NOT FOR SALE OR DISTRIB
2
as n → ∞
Thus zn is convergent and converges to 1 + iπ
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
11. We apply Theorem 6.1.1. From

4n + 3ni 4n + 3ni 2n − i 8n 2 + 3n 6n 2 − 4n
zn= = ⋅ = +i
2n + i
© Jones & Bartlett Learning, LLC 4n 2 + 1& Bartlett
2n + i 2n − i© Jones 4n 2 + 1Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
we see that

8n 2 + 3n 6n 2 − 4n 3
=lim Re( zn ) lim
= 2 and =lim Im( z n ) lim
= .
→∞ 4n 2 + 1 n →∞ 4n 2 + 1 2
© Jones & nBartlett
n →∞
Learning, LLC
n →∞
© Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
Therefore, by Theorem 6.1.1, the sequence converges to 2 + 32 i.


n π n n π
1 + i  Learning,
© Jones & Bartlett 1 1 4  LLC
 1  in 4 © Jones & Bartlett Learning, LLC
12. =zn =   = e    e
NOT FOR SALE4  OR 2DISTRIBUTION
2  2 2 NOT FOR SALE OR DISTRIBUTION
n
 1  π  1  π
= zn   cos n 4 + i   sin n 4
2 2 2 2
© Jones & Bartlett Learning, LLCn © Jones & Bartlett
n Learning, LLC
NOT FOR SALE OR  1
DISTRIBUTION  π  1 
= Re( zn )   cos n 4 limit
= Re(NOT
zn ) 0FOR

SALE
 →0
OR DISTRIBUTION
2 2 n →∞
2 2
n
 1  π
Im( zn ) =  sin n 4 limit Im( zn ) 0
 2 2  n →∞
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
Thus {znNOT
} converges
FOR SALEto L = OR
0 DISTRIBUTION NOT FOR SALE OR DISTRIB



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION



© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.

, Chapter 6 Series and Residues Page |4

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
∞
i
14. OR∑DISTRIBUTION
NOT FOR SALE NOT FOR SALE OR DISTRIBUTION
k =1 k ( k + 1)

∞
i ∞
1 1   1 1 1 1 
=
Sn = ∑
k 1= k (Jones
©
= i ∑  −
k + 1) &k Bartlett
1 k
 = i 1 − + − +  −
k +Learning,
1   2LLC 2 3

n +1 © Jones & Bartlett Learning,
 NOT FOR
1  SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
S=n i  1 − 
 n +1
i
→ 0 as n → ∞, so sn → i as n → ∞
n + 1
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
ThusSALE
NOT FOR the sum converges
OR to i
DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

15. The infinite series
© Jones & Bartlett Learning,
∞ LLC © Jones & Bartlett Learning, LLC
∑ (1
NOT FOR SALE OR DISTRIBUTION − i ) k
=1 + (1 − i ) + (1
k =0
− i ) 2
+NOT
3
(1 − i )FOR
+  SALE OR DISTRIBUTION


is a geometric series. It has the form given in (2) of Section 6.1 with a = 1 and z = 1 − i.
z © Jones
Since = 2 > 1, & Bartlett
this series isLearning,
divergent. LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB

k −1
∞
1
16. ∑ 4i  
3
k =1
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR
1 SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
< 1 ⇒ the series is convergent
3
k −1
1 4i 4i ∞

∑ 4i   =
 3  LLC
© Jones & Bartlett Learning,
= = 6i
1 2 © Jones & Bartlett Learning, LLC
k =1
1−
NOT FOR SALE OR DISTRIBUTION 3 3 NOT FOR SALE OR DISTRIBUTION

∞
1 k ∞  1  k −1
∑
18.= i ∑ ×i
2 ©k Jones
1  2  & Bartlett Learning, LLC
=k 0= © Jones & Bartlett Learning,
i = 1 ⇒ NOT
the series
FORdiverges
SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB


19. The infinite series
© Jones & Bartlett Learning,k
LLC 2
© Jones3
& Bartlett Learning, LLC
∞
NOT FOR SALE OR 2 DISTRIBUTION
  2   2   NOT 2  FOR SALE OR DISTRIBUTION
∑   =
k = 0  1 + 2i 
3 + 3  + 3  +  +
 1 + 2i   1 + 2i   1 + 2i 


© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


© Jones & Bartlett Learning, LLC. NOT FOR SALE OR DISTRIBUTION.