Chapter 01 -Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 1 Complex Numbers and the Complex Plane Page |1

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Chapter 1
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Note: Solutions not appearing in this complete solutions manual can be found in the student
study guide.
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NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
Exercises 1.1
2.a 2i 3 − 3i 2 + 5i
© Jones
=2i&2 ⋅Bartlett
i − 3i 2 + 5Learning,
i =−2i − 3( −LLC
1) + 5i © Jones & Bartlett Learning, LLC
NOT FOR SALE OR
=−2i + 3 + 5i =3 + 3iDISTRIBUTION NOT FOR SALE OR DISTRIBUTION

2.b 3i 5 − i 4 + 7i 3 − 10i 2 − 9
3i 4 ⋅ i − i 4 +LLC
© Jones & Bartlett= Learning, 7i 2 ⋅ i − 10i 2 − 9 © Jones & Bartlett Learning, LLC
3i − 1 − 7i + 10 − 9
NOT FOR SALE OR= DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
= −4i

5 2 20
2.c + − 18
i i3 © i Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
5i NOT
2 FOR
20 SALE2OR DISTRIBUTION NOT FOR SALE OR DISTRIB
=2 + 2 − 2 9 =−5i − + 20
i i ⋅ i (i ) i
2i
=−5i − 2 + 20 =−5i + 2i + 20 =20 − 3i
i
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE3 OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
 2 
2.d 2i 6 +   + 5i −5 − 12i
 −i 
8 5 8 5
=2(i 2 )3 + + 5 − 12i =−2 − 2 + 4 − 12i
© Jones & Bartlett Learning, LLC3
( −i ) i © Jones
i ⋅i i ⋅i & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
8 5 8i 5i
=−2 + + − 12i =−2 + 2 + 2 − 12i
i i i i
=−2 − 8i − 5i − 12i =−2 − 25i
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
3. (5 − 9i ) + (2 − 4i ) = (5 + 2) + ( −9 − 4)i = 7 − 13i
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
4. 3(4 − i ) − 3(5 + 2i )
=12 − 3i − 15 − 6i =−3 − 9i
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6. i (4 − i ) + 4i (1 + 2i )
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
= 4i − i 2 + 4i + 8i 2 = ri + 1 + 4i − 8 =−7 + 8i

7. (2 − i )(4 + i ) =8 − 12i + 2i − 3i 2 =8 − 10i + 3 =11 − 10i
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NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


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

, Chapter 1 Complex Numbers and the Complex Plane Page |2

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NOT FOR SALE  1 1   2 5  NOT FOR SALE OR DISTRIBUTION
8. OR DISTRIBUTION
− i + i
 2 4  3 3 
1 5 1 5 1 4 5 4+5 2
= + i − i − i2 = + i + = + i
3 6 6 12 3 6 12 12 3
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
9 2 3 2
= + NOT i = FOR
+ i SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
12 3 4 3

i i 1− i i = i2 i +1 1 i
10. = ⋅ =2 2
= =+
1 + i& Bartlett
© Jones 1 + i 1 −Learning,
i (1) − (i LLC) 2 2 2 © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
2 − 4i 2 − 4i 3 − 5i 6 − 10i − 12i + 20i 2 −14 − 22i 7 11
11. = = = 2
= − − i
3 + 5i 3 + 5i 3 − 5i 9 − 15i + 15i − 25i 34 17 17

© Jones & Bartlett 10 − 5i 5(2LLC
Learning, − i ) 5(2 − i )(3 − i ) 5(6 2i − 3i +&i 2)Bartlett Learning, LLC
©−Jones
12. = = =
+ 2i 2(3 + i ) 2(3 + i )(3 − i )
NOT FOR SALE OR6DISTRIBUTION 2(9 −
NOT i 2 ) SALE OR DISTRIBUTION
FOR

(1 + i )(1 − 2i ) 1 − 2i + i − 2i 2 3−i (3 − i )(11 + 2i )
14. = = 2
=
(2 + i )(4 − 3i ) 8 − 6i + 4i − 3i 11 − 2i 112 − (2i ) 2
© Jones2 & Bartlett Learning, LLC © Jones & Bartlett Learning,
33 + 6i −NOT
11i −FOR
2i SALE
35 − 5i OR35DISTRIBUTION
5i 7 i NOT FOR SALE OR DISTRIB
= = − = −
121 + 4 125 125 125 25 25

(5 − 4i ) − (3 + 7i ) (5 − 3) + ( −4 − 7)i
15. =
(4 + 2i ) + (2 − 3i
© Jones & Bartlett Learning,) + (2 − 3)i
(4 + 2)LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION 2 − 11i NOT FOR SALE OR DISTRIBUTION
=
6−i
2 − 11i 6 + i
=
© Jones & Bartlett Learning, LLC 6−i 6+i © Jones & Bartlett Learning, LLC
2
NOT FOR SALE OR DISTRIBUTION 12 + 2 i − 66i − 11 i NOT FOR SALE OR DISTRIBUTION
=
36 + 6i − 6i − i 2
23 − 64i
=
37
© Jones &23 Bartlett
64 Learning, LLC © Jones & Bartlett Learning,
= − i
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
37 37

(4 + 5i ) + 2i 3 4 + 5i + 2i ⋅ i 2 4 + 3i (4 + 3i )(4i − 3)
16. = 2 2
= =
(2 + i ) 4
© Jones & Bartlett Learning, + i + 4i LLC4i + 3 (4i ) 2 − 9 © Jones & Bartlett Learning, LLC
NOT FOR 16iSALE
− 12 + OR2i 2 −DISTRIBUTION
9i 7i − 14 14 7i NOT FOR SALE OR DISTRIBUTION
= = = −
−16 − 9 −25 25 25


© 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 1 Complex Numbers and the Complex Plane Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
18. (1 + i ) 2
NOT FOR SALE OR DISTRIBUTION (1 − i ) 3
NOT FOR SALE OR DISTRIBUTION
= (1 + i 2 + 2i )(1 − i ) 2 (1 − i )
= (2i )(1 − 2i + i 2 )(1 − i )= (2i )( −2i )(1 − i )
= −4i 2 (1©− Jones
i) = &i Bartlett
4(1 − )= 4 − 4i Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
1 1 2+i
19. (3 + 6i ) + (4 − i )(3 + 5i ) + =(3 + 6i ) + (12 + 20i − 3i − 5i 2 ) +
2−i 2−i 2+i
2+i
=(3 + 6i ) + (17 + 17i ) + © Jones & Bartlett
© Jones & Bartlett Learning, LLC Learning, LLC
NOT FOR SALE OR DISTRIBUTION 4 + 2i FOR
NOT i2
− 2i − SALE OR DISTRIBUTION
5 5 2+i
=(3 + 6i ) + (17 + 17i ) +
5 5 5
15 + 30i 85 + 85i 2 + i
= + +
© Jones & Bartlett Learning, LLC 5 © Jones 5 & Bartlett
5 Learning, LLC
NOT FOR SALE OR DISTRIBUTION 102 116NOT FOR SALE OR DISTRIBUTION
= + i
5 5
2
 2−i 
20. (2 + 3i ) © Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
1 + 2iFOR
NOT  SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
2−i 2 − i 1 − 2i 2 − 4i − i + 2i 2 −5i
=⋅ = == −i
1 + 2i 1 + 2i 1 − 2i 12 − (2i ) 2 5
2
 2 − i Learning, LLC 2
(2 +&3iBartlett
© Jones )  =(2 + 3i )( −i ) =−2 − 3i © Jones & Bartlett Learning, LLC
NOT FOR SALE 2i  DISTRIBUTION
 1 + OR NOT FOR SALE OR DISTRIBUTION
3
 1 
22. 1 − i 
 2 
© Jones & Bartlett Learning, 3 LLC 1  3(2) 1  1 © Jones
2 &3 Bartlett Learning, LLC
3 2   1 
= (1) +
NOT FOR SALE OR DISTRIBUTION (1) ⋅  − i  + (1)  − i
NOT + − i
 FORSALE OR DISTRIBUTION
1!  2  2!  2   2 
3 1 1 3 3 1
=−1 i + 3 i 2 − i 3 =− 1 i + ( −1) + i
2 4 8 2 4 8
3 ©3Jones 1 &1 Bartlett
11 Learning, LLC © Jones & Bartlett Learning,
=−1 − i+ i= − i
4 NOT 2 FOR 8 4SALE 8 OR DISTRIBUTION NOT FOR SALE OR DISTRIB

5 20 60 120
23. ( −2 + 2i )5 = ( −2)5 + ( −2) 4 (2i )1 + ( −2)3 (2i ) 2 + ( −2) 2 (2i )3 + ( −2)1 (2i ) 4 + (2i )5
1 2 6 24
© Jones & Bartlett= −32 + (5)(16)(2i ) + (10)( −8)( −4) + (10)(4)( −8i ) + (5)( −2)(16) + Learning,
Learning, LLC © Jones & Bartlett 32i LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
= −32 + 160i + 320 − 320i − 160 + 32i
= 128 − 128i

© 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 1 Complex Numbers and the Complex Plane Page |4

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
8 8(7) 8(7)(6)
NOT FOR SALE
24. OR(1DISTRIBUTION
8 8 7
+ i ) = (1) + (1) (i ) + (1) (i ) NOT
6 2
+ FOR(1)SALE
5
(i )3 OR DISTRIBUTION
1! 2! 3!
8(7)(6)(5) 4 4 8(7)(6)(5)(4) 3 5
+ (1) (i ) + (1) (i ) +
4! 5!
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
8(7)(6)(5)(4)(3)
NOT FOR SALE OR
2 8(7)(6)(5)(4)(3)(2)
6DISTRIBUTION 7 8 NOT FOR SALE OR DISTRIB
(1) (i ) + (1)(i ) + i
6! 7!
= 1 + 8i − 28 + 56i 3 + 70i 4 + 56i 5 + 28i 6 + 8i 7 + i 8
=1 + 8i − 28 − 56i + 70 + 56i − 28 − 8i + 1
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= 16
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1
26. z =
© Jones & Bartlett Learning,
(1 + i )(1 −LLC
2i )(1 + 3i ) © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
2
(1 + i )(1 − 2i )(1 + 3i ) = (1 − 2i + i − 2i )(1 + 3i )
= (3 − i )(1 + 3i ) = 3 + 9i − i − 3i 2 = 6 + 8i
1 1 6 − 8i 6 − 8i 6 − 8i 6 8
= © Jones ⋅ & Bartlett
= Learning,
2
= LLC −
= i © Jones & Bartlett Learning,
6 + 8i NOT 6 + 8iFOR i 36 −OR
6 − 8SALE 64iDISTRIBUTION
100 100 100 NOT FOR SALE OR DISTRIB
3 4 3 2
= − i= − i
50 50 50 25
3 −2
= Re( z ) = ; Im( z )
© Jones & Bartlett
50 Learning,25 LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION


1  1 
27. Re   = Re  
z
© Jones & Bartlett Learning,  x + iy 
LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
 1 x − iy 
= Re  
 x + iy x − iy 
 x − iy 
=©Re
Jones
 x2 −& Bartlett 2Learning,
2  LLC © Jones & Bartlett Learning,
 ixy + ixy + i y 
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
 x − iy 
= Re  2 2 
x +y 
© Jones  x
& Bartlett y 
= Re  Learning,
2 2
− i 2LLC2  © Jones & Bartlett Learning, LLC
NOT FOR SALE OR  x DISTRIBUTION
+y x +y  NOT FOR SALE OR DISTRIBUTION
x
= 2
x + y2

© 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.