Chapter 02-Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 2 Complex Functions and Mappings Page |1

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
Chapter 2
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

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 2.1
2. f ( z) =− z3 + 2z + z
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
2.a f (i ) =−i 3 + 2i + i =i + 2i − i =2i


2.b
© Jones & Bartlett f (2 − i ) =−(2
Learning, LLC− i )3 + 2(2 − i ) + (2 − i ) =
©−(2 − 11i ) &
Jones − 2i + 2 +Learning,
+ 4Bartlett i LLC
NOT FOR SALE OR DISTRIBUTION = 4 + 10i NOT FOR SALE OR DISTRIBUTION


2.c f (1 + 2i ) =−(1 + 2i )3 + 2(1 + 2i ) + 1 + 2i ) =−(−11 − 2i ) + 2 + 4i + 1 − 2i
= 14 + 4i & Bartlett Learning, LLC
© Jones © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
3.a For z = 1 we have |z| = 1 and Arg(z) = 0. Therefore,

f (1) = log e 1 + i (0) = 0.
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
3.b For z = 4i we have |z| = 4 and Arg(z) = 2 . Therefore,
π



π
f (4i )LLC
© Jones & Bartlett Learning, = log e 4 + i ≈ 1.38629 +©1.5708
Jones 0i.& Bartlett Learning, LLC
2
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION

3.c For z = 1 + i we have |z| = 2 and Arg(z) = π
4 . Therefore,

© Jones & Bartlett π Learning,
1 π
LLC © Jones & Bartlett Learning,
fNOT+ i )FOR
(1= logSALE
e iOR DISTRIBUTION
2 += log e 2 + i ≈ 0.34657 + 0.78540i.NOT FOR SALE OR DISTRIB
4 2 4

2
4. f ( z) =
z − 2 Re(iz ) + z
© 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 2 Complex Functions and Mappings Page |2

© Jones & Bartlett Learning, LLC 2 © Jones & Bartlett Learning, LLC
4.a f (3 − 4i
NOT FOR SALE OR DISTRIBUTION ) = 3 − 4i − 2 Re ( i (3 − 4i ) ) + 3 − 4i FOR SALE OR DISTRIBUTION
NOT
= 32 + 42 − 2 Re(3i + 4) + 3 − 4i
= 25 − 8 + 3 − 4i
= 20 − 4i & Bartlett Learning, LLC
© Jones © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
f (2 − i ) = 2 − i − Re ( i (2 − i ) ) + 2 − i
2
4.b
= 22 + 12 − 2 Re(2i + 1) + 2 − i
© Jones & Bartlett
= 5 − 2 +Learning,
2−i LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
= 5−i


f (1 + 2i ) = 1 + 2i − 2 Re ( i (1 + 2i ) ) + 1 + 2i
2
4.c
© Jones & Bartlett Learning, 2LLC2 © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION = 1 + 2 − 2 Re(i − 2) + 1 + 2i NOT FOR SALE OR DISTRIBUTION
= 5 + 4 + 1 + 2i
= 10 + 2i

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
z
6. f ( z ) = eNOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB


6.a f (2 − π i ) = e 2 cos(−π ) + ie 2 sin(−π ) = −e 2 ≈ −7.389
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
π  0 π  π  π  π 
6.b f  i = 0
e cos   + ie sin   =cos   + i sin  
3  3 3 3 3
≈ 0.5 + 0.866i
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION
 5π  loge2  5π  NOT 2
FOR
 5πSALE
 OR DISTRIBUTION
6.c f  log eˆ 2 − = i  e cos  −  + ieloge sin  − 
 6   6   6 
 5π   5π 
= 2 cos  −  + 2i sin  − 
 6 Learning,
© Jones & Bartlett  6 
 LLC © Jones & Bartlett Learning,
NOT FOR ≈ −SALE
1.732 −OR
i DISTRIBUTION NOT FOR SALE OR DISTRIB


7.a For z = 3 we have r = |z| = 3 and θ = Arg(z) = 0. Therefore,
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
f (3)OR
NOT FOR SALE i cos 2 − = 3 + i (1) 2 = 3 + i.
= 3 +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 2 Complex Functions and Mappings Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
7.b For z = –2i we have r = |z| = 2 and θ = Arg(z)FOR 2 . Therefore,
= − SALE
π
NOT FOR SALE OR DISTRIBUTION NOT OR DISTRIBUTION

 π
f (−2i ) =2 + i cos 2  −  =2 + i (0) 2 =2.
 2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
7.c For z = 2 – i we have = r z= 5. Moreover, from equating the Cartesian and polar

forms of the point z = 2 − i = 5(cos θ + i sin θ ) we obtain cos θ = . Therefore,
© Jones & Bartlett Learning, LLC 2
© Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION  2  4 NOT FOR SALE OR DISTRIBUTION
f (2 − i ) = 5 + i   = 5 + 5 i ≈ 2.23607 + 0.8i.
 5

θ 
© Jones & Bartlett
8. = Learning,  + i cos(2θ )
f ( z ) r sin LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION 2 NOT FOR SALE OR DISTRIBUTION

8.a −2 =2eiπ
 π  & Bartlett Learning, LLC
f (−2) =©2sin   + i cos(2π ) = 2 + i
Jones © Jones & Bartlett Learning,
NOT FOR2  SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB

π
i
8.b 1 + i = 2e 4


π 
© Jones & Bartlett Learning, π 
LLC π  © Jones & Bartlett Learning, LLC
=f (1 + i ) 2 sin  + i cos
= 2 sin   ≈ 0.5412
NOT FOR SALE OR DISTRIBUTION
8
 
2 8 NOT FOR SALE OR DISTRIBUTION

π
−i
8.c −5i =5e 2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
 −π   2
NOT FOR SALE ORf DISTRIBUTION
(−5i ) =5sin   + i cos(−π ) =−5  NOT i
 −FOR SALE OR DISTRIBUTION
 4   2 
≈ −3.5355 − i

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
10. f ( z) =−3z + 2 z − i
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
=−3( x + iy ) + 2( x + iy ) − i =
−3 x − 3iy + 2( x − iy ) − i
=−3 x − 3iy + 2 x − 2iy − i =− x + i (−5 y − 1)

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
The real
NOT FOR SALE andOR
imaginary part of u and v of f(z) as a function
DISTRIBUTION of x and
NOT FOR y areOR
SALE thus:
DISTRIBUTION

u ( x, y ) =− x and v( x, y ) =−5 y − 1.

© 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 2 Complex Functions and Mappings Page |4

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
11. OR
NOT FOR SALE we set z = x + iy, then
If DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
f ( z ) = ( x + iy )3 − 2( x + iy ) + 6
=x 3 + 3x 2 yi + 3xy 2i 2 + y 3i 3 =2 x − 2 yi + 6
© Jones
=x 3 +&3xBartlett
2
yi − 3xyLearning,
2
− y 3i =2 x LLC
=2 yi + 6 © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
= ( x 3 − 3 xy 2 − 2 x + 6) + (3x 2 y − y 3 − 2 y )i.

Therefore, Re( f ) = x 3 − 3 xy 2 − 2 x + 6 and Im( f )= 3 x 2 y − y 3 − 2 y.
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
12. ) z2 + z 2
f ( z=
= ( x + iy ) 2 + ( x + iy ) 2 = ( x + iy ) 2 + ( x − iy ) 2
= x 2 − y 2 + 2ixy + x 2 − y 2 − 2ixy = 2 x 2 − 2 y 2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
The real and imaginary parts of u and v of f(z) as a function of x and y are thus:

2 x 2 − 2 y 2 and v( x, y ) =
u ( x, y ) = 0.
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT 1 FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
14. f ( z )= z +
z
1 x − iy
= ( x + iy ) + = x + iy + 2 2
© Jones & Bartlett Learning, x + iy LLC x + y © Jones & Bartlett Learning, LLC
NOT FOR SALE OR 
x DISTRIBUTIONy  NOT FOR SALE OR DISTRIBUTION
=+x 2 2
+i y − 2 
x +y  x + y2 

The real and imaginary parts of u and v of f(z) as a function of x and y are thus:
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION y NOT FORySALE OR DISTRIBUTION
u ( x, y ) =
x+ 2 2
and v( x, y ) =
y− 2 .
x +y x + y2


15. If we set©z=Jones
x + iy,&then 2( x +Learning,
Bartlett (2 y + 1)i. So, from (3) of©Section
iy ) + i = 2 x +LLC Jones2.1 we
& Bartlett Learning,
have: NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB

e 2 z +i = e 2 x + (2 y +1)i
2x
BartletteLearning,
© Jones & = cos(2 y + 1)
LLC+ ie 2 x sin(2 y + 1). © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
=
Therefore, Re( f ) e 2 x cos(2 y + 1) and
= Im( f ) e 2 x sin(2 y + 1).

© 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.