Chap 8 Mapping by Elementary Functions
68. Linear Transformations
w  Az ,
where A is a nonzero constant and z  0
Let A  aei ,
z  rei
w  (ar )ei (  )
rotate by   arg A.
expand or contract radius by a  A
w zB
Let w  u  iv
z  x  iy
B  b1  ib2
then (u, v)  ( x  b1 , y  b2 )
位移 translation
1
The general linear transformation
w  Az  B (A  0)
is a composition of Z  Az
Ex: w  (1  i ) z  2
i
and w  Z  B

= 2e 4  z  2
y
y
v
B
0
A
x
x
0
2
u
2
1
69. The Transformation
z
mapping between
1
w
z
nonzero points of z and w planes.
Since z z  z ,
2
1
Z
z
2
z,
w
1
is the composite of
z
wZ
a reflection in the real axis
An inversion with respect
to unit circle
Z 
Z
1
z
Z
1
arg Z  arg z
Because
lim
z 0
w
1

z
and
lim
z 
1
0
z
3
To define a one to one transformation w  T ( z ) from the extended
z plane onto the extended w plane by writing
1
T (0)  , T ()  0 and T ( z ) 
for z  0, 
z
T is contiuons throughout the extended z plane.
1
Since w  u  iv is the image of z  x  iy under w 
z
z
x  iy
w 2  2
x  y2
z
x
y
,
v

(3)
2
2
2
2
x y
x y
1
u
-v
Similarly, z  , x  2 2 y  2 2
w
u v
u v
u 
4
Let A, B, C , D be real numbers, and B 2  C 2  4 AD
The equation
A( x 2  y 2 )  Bx  Cy  D  0
(5)
represents an arbitrary circle or line,
where A  0 for a circle and A  0 for a line
A0
B
C
D
x y 0
A
A
A
B 2
C 2 4 AD  B 2  C 2
(x 
)  (y 
) 
0
2
2A
2A
4A
x2  y 2 
B 2
C 2
B 2  C 2  4 AD 2
(x 
) (y 
) (
)
2A
2A
2A
a circle when B 2  C 2  4 AD  0
5
A0
B 2  C 2  0, which means B and C are not both zero.
Bx  Cy  D  0 is a line.
u
v
,
y
by
u 2  v2
u 2  v2
we get D(u 2  v 2 )  Bu  Cv  A  0
represents a circle or line
substituting x by
in (5),
1
 The mapping w  transforms circles and lines into circles
z
and lines.
(a)A  0, D  0, A circle not passing through the origin z  0
is tranformed to a circle not passing through the origin w  0.
T
(b)A  0, D  0, A circle thru z  0 
 line not passing w  0.
T
(c)A  0, D  0, a line not passing z  0 
 a circle through w  0.
T
(d)A  0, D  0, a line thru z  0 
 a line thru w  0.
6
Ex1.
v
vertical line x  c1 , c1  0
C2  0
C1  0
1
w
z
1 2
1
(u 
)  v 2  ( )2
2c1
2c1
C1  0
u
C2  0
a point z  (c1 , y )
by eq.(3)
 (u , v)  (
c1
y
,
)
2
2
2
2
c1  y c1  y
x
,
2
2
x y
y
x2  y 2
y
C1
C1
C2  0
x
C2  0
7
Ex2.
y  c2
w
1
z
u 2  (v 
1 2
1 2
) (
)
2c2
2c2
Ex3.
1 2 2
1 2
x  c1  (u 
) v ( )
2c1
2c1
C1
C
1
2C1
1
2C1
8
70. Linear Fractional Transformation
(1)
az  b
(ad  bc  0) , a, b, c, d , complex constants
cz  d
is called a linear fractional transformation or Mobius
transformation.
w
Eq.(1) wcz  dw  az  b  0
(bc  (ad )  0)
(~ A zw  Bz  Cw  D  0,
AD - BC  0)
bilinear transformation
linear in z
linear in w
bilinear in z and w
when c  0, the condition becomes ad  0.
(1) 
w
a
b
z
d
d
a linear function
9
when c  0,
ad
a
b
(cz  d ) 
c
c
(1)  w 
cz  d
1
a bc  ad

= 
cz  d
c
c
is a composition of
a bc  ad
1
W
Z  cz  d , W  , w= 
c
c
Z
 a linear fractional transformation always transforms
circles and lines into circles and lines.
1
( az  b, 皆是)
z
10
solving (1) for z,
 dw  b
z
(ad - bc  0)
cw  a
d
b
If c  0, z  w 
one-to-one mapping
a
a
a
If c  0, z & w has one-to-one mapping except at w  .
c
Denominator=0
11
define a linear fractional tranformation
T on the extended z plane such that w  a .
c
in the image of z   when c  0.
az  b
cz  d
T ( )  
a
T ( ) 
c
d
T ( )  
c
T ( z) 
(ad  bc  0)
(5)
if c  0
if c  0
if c  0
This makes T continuous on the extended z plane (Ex10, sec14).
We enlarge the domain of definition,
(5) is a one-to-one mapping of the extended z plane onto the
extended w plane.
12
i.e., T ( z1 )  T ( z2 ) whenever z1  z2
and for each point w in w-plane, there is a point z in the z -plane,
such that T ( z )  w.
There is an inverse transformation T -1
T -1 ( w)  z iff
dw  b
T 1 ( w) 
cw  a
T 1 ()  
a
T 1 ( )  
c
T 1 ()  
T ( z)  w
(ad  bc  0)
A linear fractional transformation
if c  0
if c  0
d
c
13
There is always a linear fractional transformation that maps three
given distinct points, z1, z2 and z3 onto three specified distinct
points w1, w2 and w3.
Ex1.
az  b
find w 
that
cz  d
map z1  1 z2  0
onte w1  i
z3  1
b
w2  1 w3  i
d
b0
az  b
w 
(b(a  c)  0)
cz  b
ac
a  b
ab
i 
i
c  b
cb
ic  ib  a  b
ic  ib  a  b
2ic  2b,
bd
c  -ib
a  ib
w 
1
( b  0)
i bz  b
iz  1 i  z


i bz  b iz  1 i  z
14
Ex2:
z1  1,
z2  0,
z3  1
w1  i,
w2  ,
w3  1

d 0
az  b
cz
ab
i
c
ic  a  b
w
(bc  0)
a  b
c
 c  a  b
1
i 1
c
2
i 1
i 1
a  ic  b  (i 
)c
c
2
2
(i  1) z  (i  1)
w
2z
(i  1)c  2b,
b
15
71. An Implicit Form
The equation
( w  w1 )( w2  w3 ) ( z  z1 )( z2  z3 )

( w  w3 )( w2  w1 ) ( z  z3 )( z2  z1 )
(1)
defines (implicitly) a linear fractional transformation that maps
distinct points z1 , z2 , z3 onto distinct w1 , w2 , w3 , respectively.
Rewrite (1) as
( z  z3 )( w  w1 )( z2  z1 )( w2  w3 )
(2)
 ( z  z1 )( w  w3 )( z2  z3 )( w2  w1 )
If z  z1 ,
right-hand side=0
 w  w1
If z  z3 ,
left-hand side=0
w  w3
16
If z  z2
( w  w1 )( w2  w3 )  ( w  w3 )( w2  w1 )
 w  w2
Expanding (2)  get A zw  Bz  Cw  D  0
a linear fractional transformation.
Ex1.
z1  1
w1  i
z2  0
w2  1
z3  1
w3  i
( w  i )(1  i ) ( z  1)(0  1)

( w  i )(1  i ) ( z  1)(0  1)
( w  i )( z  1)(1  i )  ( w  i )( z  1)(1  i )
( wz  iz  w  i )(1  i )  ( wz  iz  w  i )(1  i )
( wz  iz  w  i  iwz  z  iw  1)  ( wz  iz  w  i  iwz  z  iw 1)
i-z
2 wz  2iw  (- z  i )2
w( z  i )  (i - z ) w 
iz
17
equation (1) can be modified for point at infinity.
suppose z1  
replace z1 by
1
, and let z1  0
z1
1
)( z2  z3 )
( z z  1)( z2  z3 )
z1
lim
 lim 1
z1 0
z1 0 ( z  z )( z z  1)
1
3
2 1
( z  z3 )( z2  )
z1
(z 

z 2  z3
z  z3
The desired equation is
( w  w1 )( w2  w3 ) z2  z3

( w  w3 )( w2  w1 ) z  z3
18
Ex2.
z1  1
w1  i
z2  0
w2  
z3  1
w3  1
w  w1 ( z  z1 )( z2  z3 )

w  w3 ( z  z3 )( z2  z1 )
w  i ( z  1)(0  1)

w  1 ( z  1)(0  1)
( w  i )( z  1)  ( w  1)( z  1)
 wz  iz  w  i  wz  z  w  1
2 wz  iz  i  z  1
 (i  1) z  (i -1)
(i  1) z  (i  1)
w
2z
19
72. Mapping of the upper Half Plane
Determine all 1inear fractional transformation T that
T
Im z  0


w 1
T
Im z  0


w 1
Choose three points z  0, 1, 
that will be mapped to
by
z0
z
0
1
1

w 1
az  b
(ad - bc  0)
cz  d
b
w
 1,
b  d 0
d
a
w
only if c  0
(c  0時w  不在圓內)
c
a
w   1,
a  c 0
c
b
a z a
w 
c zd
c
w
20
Since
z  1,
a
b
d
 1 and

0
c
a
c
z  z0
w  ei
, z1  z0  0
z  z1
w  ei
(5)
1  z0
1
1  z1
1  z1  1  z0
or
(1  z1 )(1  z1 )  (1  z0 )(1  z0 )
but
z1 z1  z0 z0
 z1  z1  z0  z0
i.e. Re z1  Re z0
 z1  z0 , or z1  z0
if z1  z0 , (5) is a constant transformation
 z1  z0 ,
z1  z0
21
z  z0
we
z  z0
i
when z  z0 ,
(6)
w0
since w  0 is inside
w 1

z0 is above the x axis.
or
Im z0  0
w
Z0
Z
z  z0
Z0
z  z0
if z is above the x-axis
z - z0
z - z0
1
if z is on the x-axis
if z is below the x-axis
z - z0
z - z0
1
z  z0
z  z0
1
 (6) is what we want
22
Ex1.
iz
i z  i
w
e
iz
z i
Ex2.
w
z 1
z 1
maps
has the above mapping property
y  0 onto v  0
y0
onto v  0
(1) z real  w real
Since the image of y  0 is either a line or a circle.
 it must be the real axis v  0.
(2) v  Im w  Im
( z -1)( z  1)
 Im
( z  1)( z  1)
z 1 z  z
2
z 1
2

2y
z 1
2
y  0, v  0
y  0, v  0
also linear fractional transformation is onto.
 Q.E.D.
23
73. Exponential and Logarithmic Transformations
The transformation w  e z
  ei  e x eiy
Thus   e x ,
or   e x ,
(1)
w   ei , z  x  iy
  y  2n ,
n any integer
  y transformation from z plane to w plane
x  c1 vertical line
z  (c1 , y ), its image   ec1 ,   y
(c1 , 2 )
ec1
(c1 , 0)
many-to-1 mapping
24
(2)
y  c2 horizontal line
C2
C2
1-to-1 mapping
Ex1
w  ez
a  x  b, c  y  d maps onto e a    eb , c    d
C'
y
d
c
D
C
D'
B'
A
d
B
x
a
A'
 c
b
25
Ex2.
 b
ib
ic
 c
 a
ia
w  log z  ln r  i
(r  0,       2 )
any branch of log z , maps onto a strip
v
y
i (  2 )
0
i 0

i
x
0
u
26
Ex3.
w  log
z 1
z 1

principal branch
z 1
and w  log Z
z 1

maps upper half plane y  0 onto
(0     )
upper half plane v  0
maps upper half plane
is a composition of Z 
onto the strip 0  v  
27
w  sin z
74. The transformation
Since sin z  sin x cosh y  i cos x sinh y
w  sin z
 u  sin x cosh y, v  cos x sinh y
Ex1. w  sin z


2
maps
x

2
(1-to-1)
, y0
v0
onto
y
E
M'
A
M
L'
L
1
D

B

2
c

E'
D'
B'
A'
x
2
28
A. boundary of the strip  real axis
(1) BA segment
x

2
, y0
u  cosh y, v  0
e y  e y
cosh y 
2
e y  e y
sinh y 
2
(2) DB segment
y0
u  sin x
v0
(3) DE segment

x- , y0
2
u   cosh y, v  0
29
B. Interior of strip maps onto upper half v  0 of w plane
line x  c1
0  c1 

2
u  sin c1 cosh y, v  cos c1 sinh y
(-  y  )
u2
v2

 1 hyperbola
2
2
sin c1 cos c1
with foci at the points w   sin 2 c1  cos 2 c1  1
30
Consider a horizontal line segment y  c2 ,    x   , c2  0
its image is u  sin x cosh c2 ,
u2
v2

1
2
2
cosh c1 sinh c2
v  cos x sinh c2
an ellipse
w   cosh 2 c2  sinh 2 c2  1
with foci at
v
y
A
B
C
C'
D
E
y  C2  0
D'
B'



2
0

2

1
u
1
x
A' E '
31
Ex2.
bi
C
C'
B
D
L'
L
F
E



2
2
c2  0,
Ex3.
A
u  sin x,
v0
(


2
E'
A'
1
1
x

2
B'
)
cos z  sin( z  )
2
Z  z
Ex4.
D'

2
, w  sin Z
w  sinh z
w  i sin(iz )
Z  iz , W  sin Z
w  iW
w  cosh z
 cos(iz )
32
z
75. Mapping by Branches of
z
1
2
1
2
are the two square roots of z when z  0
if z  r exp(i )
(r  0,      )
1
i (  2k )
then z 2  r exp
( k  0,1)
2
i
principal root r exp
2
z
1
2
can also be written
z
1
2
1
 exp( log z )
2
z0
The principal branch F0 ( z ) of z
1
2
is obtained by
taking the principal branch of log z
1
F0 ( z )  exp( log z )
2
i
or F0 ( z )  r exp
2
( z  0,   Argz   )
(r  0,      )
33
Ex1
v
C
C'
B
R2
w z
R1
1
R2 '
2
R1'
2
D
D'
A
0  r  2,0   
B'

2
u
A'
0    2, 0   
2

4
w  F0 (sin z )
Ex2
( z  0,   Arg z   )
 Z  sin z, w  F0 ( Z )
y
y
v
D' '
D
A
D'
sin Z

2
C
B
x
F0 ( z )
C'
x
B'
A'
C' '
u
B' '
A' '
34
when       and the branch
log z  ln r  i(  2 ) is used,
z
1
2
i (  2 )
 F1 ( z )  r exp
2
i
  r exp
2
=  F0 ( z )
other branches of z
z
1
n
1
    2  3
2
i
f a ( z )  r exp
(r  0,       2 )
2
1
i (  2k )
 exp( log z )  n r exp
, k  0,1, 2,...n  1
n
n
35
76. Square roots of polynomials
Ex1.
Branches of ( z  z0 )
Z  z  z0
with
Each branch of Z
1
2
1
2
is a composition of
wZ
1
2
yields a branch of ( z  z0 )
1
2
1
When Z  R ei , branches of Z 2 are
1
i
Z 2  R exp
( R  0,       2 )
2
If we write
R  z  z0 ,   Arg ( z - z0 ) and   arg( z  z0 )
two branches of ( z  z0 )
i
2
i
and g 0 ( z )  R exp
2
G0 ( z )  R exp
1
2
are
( R  0,      )
( R  0,0    2 )
36
G0 ( z ) is defined at all points in the z plance except z  0
and the ray Argz   .
The transformation w  G0 ( z ) is a one-to-one
mapping of the domain
   Arg ( z  z0 )  
z - z0  0,
onto the right half Re w  0 of the w-plane
y
y
z
z0
R
v
Z

w
R
R

x
x

2
u
The transformation w  g 0 ( z ) maps the domain
0  arg( z  z0 )  2
z  z0  0,
in a ont-to-one manner onto the upper half plane Im w  0
Ex.2
( z  1)
2
1
2
1
 ( z  1) ( z  1)
2
1
2
( z  1)
37