Chapter 07 -Solution Manual for Complex Analysis 3th Edition Dennis Zill - Patrick Shanahan

Chapter 7 Conformal Mappings Page |1

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
Chapter 7
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 7.1
2. f ( z ) = z 2 + 2iz − 3
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
f ′( z=
NOT FOR 2 z + OR
)SALE 2i DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
The entire function f ( z ) = z 2 + 2iz − 3 is conformal at all points z , z ≠ −i, since
f ′( z ) ≠ 0 for z ≠ −i
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
3. From Theorem 7.1.1, the entire function f ( z ) = z − e − z + 1 − i is a conformal mapping at

all z such that f ′( z ) ≠ 0. Now f ′( z ) =
1 + e− z =0 if
©z Jones =z
& Bartlett Learning, LLC © Jones & Bartlett Learning,
eNOT
(1 + eFOR )=e z (0) OR DISTRIBUTION
SALE NOT FOR SALE OR DISTRIB
ez + 1 = 0
e z = −1
z ln(−1) LLC
=
© Jones & Bartlett Learning, © Jones & Bartlett Learning, LLC
zDISTRIBUTION
NOT FOR SALE OR= log e −1 + i arg(−1) NOT FOR SALE OR DISTRIBUTION
z=0 + i (π + 2π n)
z (2n + 1)π i,
=
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
for n = 0, ± 1, ± 2, . Therefore, we see that f ′ is conformal for all z except z =
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
(2n + 1)π i, n = 0, ± 1, ± 2, .

2
4. ©z Jones
f ( z ) = ze −2
& Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT
2
FOR SALE 2
OR DISTRIBUTION
2
NOT FOR SALE OR DISTRIB
f ′( z=) e z − 2 + z ⋅ 2 ze z −=
2
(2 z 2 + 1)e z − 2
i 2
f ′( z ) ≠ 0 for 2 z 2 + 1 ≠ 0, i.e., for z ≠ ±
2
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION z2 −2
By Theorem 7.1., the entire function f ( z ) = ze isNOT FOR at
conformal SALE OR DISTRIBUTION
all points z,

i 2
z≠± .
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 7 Conformal Mappings Page |2

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
6. ORf (DISTRIBUTION
NOT FOR SALE z ) =−
z ln( z + i ) NOT FOR SALE OR DISTRIBUTION
1 z + i −1
f ′( z ) =
1− =
z +i z +i
f ′( z ) ≠ © Jones &i Bartlett Learning, LLC
0 for z ≠ 1 − © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION w > 0 NOT FOR SALE OR DISTRIB
ln( z + i ) is analytic at all points w= z + i, such that and −π < arg( w) < π .

z ln( z + i ) is thus analytic at all points z not on the ray emanating from z = −i
f ( z ) =−
and &
© Jones containing z =−i − 1. LLC
Bartlett Learning, © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION
By Theorem 7.1, f ( z ) =− NOT
z ln( z + i ) is conformal at all FOR
points notSALE
on theOR
ray DISTRIBUTION
specified
above, except at the point z =−i + 1.

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
7. Let z1 (t ) = i + t and z2 (t ) = i + teπ i /2 for 0 ≤ t < ∞. Then the curves C1 and C2 are rays
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
at i z=
intersecting = 1 (0) z2 (0). Furthermore, the angle θ between C1 and C2 at i is the
value of
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR π
π i /2 DISTRIBUTION (4n + 1)π NOT
arg z2′ − arg z1′ =arg e − arg1 = − 0 + 2nπ = , n=±1, ±FOR
2,  SALE OR DISTRIB
2 2

that lies in the interval [0, π ]. Hence θ = π / 2. On the other hand,
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
w1 (t )= ( z1 (t ) − i ) = t and w2 (t )= ( z2 (t ) − iNOT
) = t 3FOR
3 3
NOT FOR SALE OR DISTRIBUTION 3
e3π i /2 . SALE OR DISTRIBUTION

By changing our parameter to s = t 3 we see that C1′ and C2′ are given by w1 ( s ) = s and

© Jones & Bartlett w 2 ( s ) = se
Learning, 3π i /2
LLC, 0 ≤ s < ∞, respectively.
© Therefore, C1′ and CLearning,
Jones & Bartlett ′ LLC
2 are rays intersecting at
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
f (i ) = 0 and the angle φ between C1′ and C2′ at 0 is the value of

3π (4n + 3)π
arg w2′ − arg w1′ =arg e3π i /2 − arg1 = − 0 + 2nπ = , n =±1, ± 2, 
© Jones & Bartlett Learning, LLC 2 2 © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
that lies in the interval [0, π ]. Hence φ = 3π / 2. Since θ ≠ φ , the mapping w= ( z − i )3 is
not conformal at z = i.
© 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 7 Conformal Mappings Page |3

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE (iz − 3) 2 , z0 =
8. ORf (DISTRIBUTION
z) = −3i NOT FOR SALE OR DISTRIBUTION
′( z ) 2i (iz − 3) since f ′(−3i ) =
First note that f is an entire function and that f = 0, for

any two smooth curves C1 and C2 parameterized by z1 (t ) and z2 (t ) that intersect at z0 ,
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
if we find parametrizations
NOT FOR SALE OR of the images C1′ and C2′ of the 2 curves,NOT
DISTRIBUTION w1 (t )FOR w2 (t ), we
and SALE OR DISTRIB

will not be able to find the angle between the 2 curves using arg( w2′ ) − arg( w1′)1 because:

w
=
© Jones & Bartlett
1 t=
0) f ′ ( z1 (tLLC
′ w1′(Learning, 0 ) ) ⋅ z1 (t=
′ 0 ) f ′( z0 ) ⋅=
z1′ 0 © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
and

′ w2′ (t=
w=
2 0) f ′ ( z2 (t0 ) ) ⋅ z2′ (t=
0) f ′( z0 ) ⋅ =
z2′ 0
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
For example, if C1 is represented by z1 (t ) = t − 1 − 3i, z1 (1) = −3i. C1′ is given by

w1 (t ) =( i (t − 1 − 3i ) − 3) =−(t − 1) 2
2


w1′ = w1′(1)
©= −2(1 −&
Jones 1) Bartlett
=0 Learning, LLC © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIB
f ( z) = (iz − 3) 2 , z0 = −3i is not conformal at z0 = −3i because f ′( z0 ) = 0


10. z0 = 0 does not belong to the domain of f.
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR
f is SALE ORnot
therefore DISTRIBUTION
conformal at z0 = 0 NOT FOR SALE OR DISTRIBUTION


11. By mapping H-4 of Appendix III with the identification a = 2 we see that
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
NOT FOR SALE OR DISTRIBUTION πz  NOT FOR SALE OR DISTRIBUTION
w = cos  .
 2 


12. Entry C-3 in Appendix III gives a conformal mapping of region R onto region R′.
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT
The desired FOR SALE
mapping is thusOR
w =DISTRIBUTION
e z
NOT FOR SALE OR DISTRIB
From C-3, we also have that the image of the curve from A to B is the interval [−1, 0) on
the u-axis. We can verify this using parametrizations. The curve from A to B is
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
parametrized
NOT FOR SALE OR z (t ) = t + π i, t ≤ 0. Its image is given
by DISTRIBUTION NOTby FOR et +π i = OR
w(t ) = SALE −et , DISTRIBUTION
t ≤ 0. The
result mentioned above is thus verified.


© 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 7 Conformal Mappings Page |4

© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
14. Appendix
NOT FOR SALE OR DISTRIBUTIONIII does not have an entry that
NOTmaps
FORtheSALE
given region R onto the given region
OR DISTRIBUTION
R′. Therefore, we construct a conformal mapping that does this by composing two
mappings in the table. Entry H-1 gives a conformal mapping of R onto the upper half-
© Jones & z
1 −Bartlett Learning, LLC © Jones & Bartlett Learning,
plane by f ( z ) = i ⋅ . In addition, from entry M-4 we see that the upper half-plane is
NOT FOR1 + SALE
z OR DISTRIBUTION NOT FOR SALE OR DISTRIB

z ) ( z 2 − 1)1/2 .
mapped onto R′ by g (=
1
© Jones & Bartlett Learning, LLC   1&
© Jones 2
− zBartlett
  2 Learning, LLC
The first composition of
NOT FOR SALE OR DISTRIBUTION these 2 functions w g
= ( NOT
f ( z ) )  FOR
=  i ⋅ 
SALE − 1 . DISTRIBUTION
OR
  1 + z  
1
  1 − z 2  2
= w i  + 1 therefore maps R′©onto R′.

© Jones & Bartlett Learning, LLC Jones & Bartlett Learning, LLC
  1+ Z  
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
The first of these mappings maps the arc Ab onto the interval [–1, 1] on the real axis (see
entry H-1). Inspection of entry M-4 reveals that [–1, 1] on the real axis is mapped onto
the interval [0, 1] on
© Jones & the imaginary
Bartlett axis. LLC
Learning, © Jones & Bartlett Learning,
NOT FOR SALE OR DISTRIBUTION NOT
Thus we conclude that the ac AB is mapped onto the interval [0, 1] on the FOR
v-axis.SALE OR DISTRIB

15. We construct a conformal mapping by composing two mappings from Appendix III.
© Jones & entry
From Bartlett
H-6Learning,
the region RLLC © Jones
is mapped on to the upper & Bartlett
half-plane Im( w) > Learning,
0 by LLC
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
eπ / z + e −π / z
f ( z) = .
eπ / z − e −π / z
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning, LLC
Now from E-4 with the identifications θ 0 = π and α = 1/ 2, the upper half-plane is
NOT FOR SALE OR DISTRIBUTION NOT FOR SALE OR DISTRIBUTION
′
mapped onto the region R by

g ( z ) = z1/2 .
© Jones & Bartlett Learning, LLC © Jones & Bartlett Learning,
NOT
Therefore, the FOR
desiredSALE OR is
mapping DISTRIBUTION
given by NOT FOR SALE OR DISTRIB

1/2
 eπ / z + e −π / z 
= w g= ( f ( z ) )  π / z −π / z  .
© Jones & Bartlett Learning,  e LLC −e  © 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.