Chapter 04 -Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 4 Elementary Functions Page |1

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Chapter 4
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Note: Solutions©not appearing
Jones in this complete
& Bartlett Learning,solutions
LLC manual can be found in the student
© Jones & Bartlett Learning,
study guide. NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB

Exercises 4.1
© Jones & Bartlett 2x
Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE 3eOR ie − z u
−DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
2. = f ( z) =
z3 −1 + i v
u ′v − v′u (6e 2 z + ie − z )( z 3 − 1 + i ) − (3z 2 )(3e 2 z − ie − z )
= ′
f ( z) =
v2 ( z 3 − 1 + i)2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
6 z 3e 2 z − 6e 2 z + 6ie 2 z + iz 3e − z − ie − z − e − z − 9 z 2 e 2 z + 3iz 2 e − z
′
NOT FOR SALE OR⇒DISTRIBUTION
f ( z) = NOT FOR SALE OR DISTRIBUTION
( z 3 − 1 + i)2
(6 z 3 − 9 z 3 + 6i − 6)e 2 z + (iz 3 + 3iz 2 − i − 1)e − z
=
( z 3 − 1 + i)2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
3. Using (3) in Section 4.1 and the rules from Section 3.2:

d [ iz − iz ] iz d d
e − e= e [iz ] − e − iz [−iz ]
dz
© Jones & Bartlett Learning,dzLLC dz © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTIONiz − iz
= ie + ie . NOT FOR SALE OR DISTRIBUTION


4. ie 4 ⇒ f ′( z ) =
ie1/2 =
f ( z) = ieu u ′
© Jones & Bartlett Learning, LLC1 i © Jones & Bartlett Learning, LLC
f ′( z ) =
ie1/2  − 2  =
− 2 e1/2
NOT FOR SALE OR DISTRIBUTION  z  z NOT FOR SALE OR DISTRIBUTION



6. ( ) z−
i
arg e z© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
x + i ( y −1) [ x + i ( y −1)][ x −iy ] x 2 −ixy + ix ( y −1) + y ( y −1)
z −1 NOT FOR SALE
x2 + y 2
OR DISTRIBUTIONx2 + y 2
NOT FOR SALE OR DISTRIB
e
= z
e= ex + iy
= e
x 2 + y ( y −1)  x 
i − 2 2 
x2 + y 2  x +y 
=e e

arg eSALE
NOT FOR z( )
© Jones & Bartlett
z −i
=
Learning, LLC
x
+ 2nπ
−OR2 DISTRIBUTIONn∈ z
© Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION
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.

, Chapter 4 Elementary Functions Page |2

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
7. OR
NOT FOR SALE ByDISTRIBUTION
(5) in Section 4.1 NOT FOR SALE OR DISTRIBUTION

 i
arg ( e z −i / z ) = Im  z −  + 2nπ , n = 0, ± 1, ± 2,  .
 z
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
If set z= NOT
x + iyFOR, thenSALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB

 i  i 
Im  z − = Im  x + iy − 
 z  x + iy 
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR = 
DISTRIBUTION i x − iy  NOT FOR SALE OR DISTRIBUTION
Im  x + iy − 
 x + iy x − iy 
 y + xi 
= Im  x + iy − 2 
 x + y2 
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION  y 
NOT FOR   OR DISTRIBUTION
x SALE
= Im  x − 2 2
+i y − 2 2 
 x +y  x + y 
x
= y− 2 2
.
x + y
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
Therefore,

x
arg ( e z −i / z ) = y − 2 + 2nπ , n = 0, ± 1, ± 2,  .
© Jones & Bartlett Learning, y2
x +LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
8. ie z + 1 =ie z + 1 =ie 2 + 1 =−ie z + 1
=−ie x −iy + 1 =−ie x e − iy =−ie 2 (cos y − i sin y ) + 1
© Jones & Bartlett =Learning,
e x (−i cos yLLC
− sin y ) + 1 = 1 − e x sin y −©ie xJones
cos y & Bartlett Learning, LLC
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10. f ( z ) = e 2 z +i = e 2 x − 2 yi +i = e 2 x ei (1− 2 y ) = e 2 x ( cos(1 − 2 y ) + i sin(1 − 2 y ) )
= e 2 x cos(1 − 2 y ) + ie 2 x sin(1 − 2 y )
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NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
11. If set z= x + iy and use (1) in Section 4.1, then

2 2
e z = e( x +iy )
© Jones & Bartlett 2Learning, LLC © Jones & Bartlett Learning, LLC
y 2 + 2 xyi
NOT FOR SALE=OR e x −DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
2
− y2 2
− y2
= ex cos 2 xy + ie x cos 2 xy.


© 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 4 Elementary Functions Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
1 1 x −iy x
NOT FOR SALE OR DISTRIBUTION 2 2 2 2  NOT y FOR SALEy OR DISTRIBUTION
12. f ( z=
) e=
z
e x +=
iy
e x + y= e x + y  cos 2 − i sin 2
 x + y2 x + y 2 
x x
y x2 + y 2
2 2 y
= e cos 2 2
− ie x + y sin 2 2
© Jonesx&+Bartlett
y x + yLLC
Learning, © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
2 2
z − y 2 + i 2 xy 2 2 2 2
14. f (=
x) e= ex = e x − y cos 2 xy + e x − y sin 2 xyi
= U + Vi
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
U andSALE
NOT FOR V are continuous and differentiable on R 2 , also
OR DISTRIBUTION theirFOR
NOT partial derivatives
SALE are
OR DISTRIBUTION
continuous 1

∂u 2 2 2 2 
© Jones = 2 xe x − yLLC
& Bartlett Learning, cos 2 xy − 2 ye x − y sin 2 xy© Jones & Bartlett Learning, LLC
∂x
NOT FOR SALE OR DISTRIBUTION NOT  FOR
∂u ∂v
∂v  ⇒ SALE=OR 1 DISTRIBUTION
∂x ∂y
=−2 ye x2 − y 2
sin 2 xy + 2 xe x2 − y 2
cos 2 xy 
∂y 

∂u x2 − y 2 2
y2 
−2©
= yeJones cos& Bartlett
2 xy − 2 xe x −Learning,
sin 2 xy LLC © Jones & Bartlett Learning,
∂y NOT FOR SALE OR DISTRIBUTION  ∂u ∂v NOT FOR SALE OR DISTRIB
 ⇒ = 2
∂v ∂x ∂x
= 2 xe x2 − y 2
sin 2 xy + 2 ye x2 − y 2
cos 2 xy 
∂x 

1 and
© Jones 2 ⇒ f Learning,
& Bartlett satisfies Cauchy-Riemann
LLC equations ©BJones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
A and B ⇒ f is differentiable


15. Identifying b = –2 in part (iii) of the exponential mapping properties, we see that the
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
image is
NOT FOR SALE OR DISTRIBUTIONthe ray arg(w) = − 2. The mapping
NOT is
FORdepicted
SALE below.
OR DISTRIBUTION

x
16. w e=
= ,x 3
e3+iy =e&
w(3 + iy )©=Jones 3
(cos y + i sin
Bartlett y ) = X + iYLLC
Learning, © Jones & Bartlett Learning,
3 NOT FOR SALE
X = e cos y  X 2
YOR DISTRIBUTION
2 NOT FOR SALE OR DISTRIB
 ⇒ + = 1
Y = e3 sin y  (e3 ) 2 (e3 ) 2
⇒ X 2 +Y2 = e6
© Jones
the & Bartlett
image of (x Learning, LLC
= 3) is a circle © eJones
of center (0,0) of radius 3 & 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 4 Elementary Functions Page |4

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
18. image of square
NOT FOR SALE OR DISTRIBUTION 0, 1, 1 + i, i to find the image
NOT we have
FOR SALE to OR
findDISTRIBUTION
the image of each side
= x 0 (0 < y < 1)
−+ iy
w e=
= cos y + i sin y
⇒ x 2 +&y 2Bartlett
© Jones = 1 (0 < yLearning,
< 1) LLC © Jones & Bartlett Learning,
=y 0 NOT FOR
(0 < x < 1) SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
x +i 0
=w e= ex
=x 1 (0 < y < 1)
+ iy
w=
© Jones &e1Bartlett
e1 cosLearning,
= y + ie sin y LLC
= X + iY © Jones & Bartlett Learning, LLC
NOT FOR SALE OR 2 DISTRIBUTION
2 2 NOT FOR SALE OR DISTRIBUTION
= x + y e circle center 0 and radius e (0 < y < 1)
=y 1 (0 < x < 1)
x +i
w ==ee e x (cos1 + i sin1) =
x + iy
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
x y
NOT FOR SALE OR DISTRIBUTION
= ⇒ y= tan1x piece of thisNOT
line. FOR SALE OR DISTRIBUTION
cos1 sin1


19. We determine the image of each of the four sides of the rectangle 0 ≤ x ≤ log e 2,
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
−π / 4 ≤NOT
y ≤ π FOR
/ 2 separately.
SALE OR DISTRIBUTION
Identifying NOT FOR
a = 0 in part (ii) of the exponential SALE OR DISTRIB
mapping
properties, we see that the image of the vertical side x = 0, −π / 4 ≤ y ≤ π / 2 is a segment
of the circle w= e= 0
1. Since −π / 4 ≤ y ≤ π / 2, and y is an argument of w = ez, the
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
imageSALE
NOT FOR of thisOR is the circular segment w = 1, −π NOT
sideDISTRIBUTION / 4 ≤ arg(
FOR ≤ π / 2.OR
w) SALE In aDISTRIBUTION
similar
manner, we see that the image of the vertical side x = log e 2, −π / 4 ≤ y ≤ π / 2 is the

circular segment= w e= log e 2
2, −π / 4 ≤ arg( w) ≤ π / 2. Now identifying b = −π / 4 in
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
part (iii)
NOT FOR SALE OR DISTRIBUTION of the exponential mapping properties,
NOT FORweSALE see that
OR theDISTRIBUTION
image of the horizontal
side y = −π / 4, 0 ≤ x ≤ log e 2 lies in the ray arg(w) = −π / 4. Since 0 ≤ x ≤ log e 2 and ex

is an increasing function on its domain, we have 1 = e0 ≤ e x ≤ eloge 2 =
2. On the other
hand, w © x
= eJones
and so ≤ w ≤ 2.Learning,
& 1Bartlett Therefore,LLC size y = &−πBartlett
© Jones
the image of the horizontal / 4, Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
0 ≤ x ≤ log e 2 is the line segment arg(w) = −π / 4, 1 ≤ w ≤ 2. With the appropriate
modifications to this last procedure, we find that the image of the horizontal side y =
π /&
© Jones 2, Bartlett
0 ≤ x ≤ log LLCsegment arg(w) = π / ©
e 2 is the line
Learning, 1 ≤ w ≤&2.Bartlett
3, Jones The image of the LLC
Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
rectangle is the region bounded by the images of its four sides. Therefore, the image of
the rectangle 0 ≤ x ≤ log e 2, −π / 4 ≤ y ≤ π / 2 is the annular region
1 ≤ w ≤ 2, −π / 4 ≤ arg( w) ≤ π / 2. The mapping is depicted below.
© 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.