Chap 8 Mapping by Elementary Functions
68. Linear Transformations
w = Az ,
where A is a nonzero constant and z ≠ 0
Let A = aeiα ,
z = reiθ
w = (ar )ei (α +θ )
rotate by α = 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
π
= 2e 4 ⋅ z + 2
y
y
v
B
0
A
x
x
0
2
u
2
1
1
z
mapping between
69. The Transformation
w=
1
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
+ y2
x
z
−y
x
, v= 2
(3)
x2 + y2
x + y2
u
1
-v
Similarly, z = , x = 2 2 y = 2 2
w
u +v
u +v
∴u =
4
2
Let A, B, C , D be real numbers, and B 2 + C 2 > 4 AD
The equation
(5)
A( x 2 + y 2 ) + Bx + Cy + D = 0
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
=0
(x +
) + (y +
) +
2A
2A
4 A2
x2 + y2 +
(x +
B 2
C 2
B 2 + C 2 − 4 AD 2
) + (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.
−v
u
substituting x by 2 2 , y by 2 2 in (5),
u +v
u +v
we get D (u 2 + v 2 ) + Bu − Cv + A = 0
represents a circle or line
∴ The mapping w =
1
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
→ a circle through w = 0.
(c)A = 0, D ≠ 0, a line not passing z = 0 
T
→ a line thru w = 0.
(d)A = 0, D = 0, a line thru z = 0 
6
3
Ex1.
v
vertical line x = c1 , c1 ≠ 0
C2 < 0
C1 < 0
1
↓w=
z
1 2 2
1
(u −
) + v = ( )2
2c1
2c1
C1 > 0
u
C2 > 0
a point z = (c1 , y )
by eq.(3) → (u , v) = (
c1
−y
, 2
)
2
c + y c1 + y 2
2
1
x
,
2
x + y2
y
C1
C1
−y
2
x + y2
C2 > 0
x
C2 < 0
7
Ex2.
y = c2
↓w=
1
z
u 2 + (v +
1 2
1 2
) +(
)
2c2
2c2
Ex3.
x ≥ c1 → (u −
1 2 2
1
) + v ≤ ( )2
2c1
2c1
C1
C
1
2C1
1
2C1
8
4
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.
a
b
(1) → w = z +
a linear function
d
d
9
when c ≠ 0,
a
ad
+b
(cz + d ) −
c
c
(1) → w =
cz + d
a bc − ad
1
⋅
= +
c
c
cz + d
is a composition of
1
a bc − ad
Z = cz + d , W = , w= +
W
Z
c
c
∴ a linear fractional transformation always transforms
circles and lines into circles and lines.
1
( ∵ az + b, 皆是)
z
10
5
solving (1) for z ,
− dw + b
(ad - bc ≠ 0)
cw − a
d
b
If c = 0, z = w −
one-to-one mapping
a
a
z=
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 (∞ ) = ∞
T (z) =
T (∞ ) =
a
c
d
T (− ) = ∞
c
(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
6
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
that
cz + d
map z1 = −1 z2 = 0
find w =
onte w1 = −i
z3 = 1
b
w2 = 1 w3 = i
d
b≠0
az + b
(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,
c = -ib
a = ib
(∴ b ≠ 0)
i bz + b
iz + 1 i − z
∴w =
=
=
−i bz + b −iz + 1 i + z
=1
b=d
∴w =
14
7
Ex2:
z1 = 1,
z2 = 0,
w1 = i,
w2 = ∞,
z 3 = −1
w3 = 1
↓
d =0
az + b
w=
(bc ≠ 0)
cz
a+b
−a + b
i=
1=
c
−c
ic = a + b
− c = −a + b
i −1
b=
c
(i − 1)c = 2b,
2
i −1
i +1
c
)c=
a = ic − b = (i −
2
2
(i + 1) z + (i − 1)
w=
2z
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
8
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 1
lim
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
9
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
a = c ≠0
w = = 1,
c
b
a z+ a
∴w =
c z+d
c
w=
20
10
Since
a
b
d
= 1 and
=
≠0
c
a
c
z − z0
, z1 = z0 ≠ 0
w = eiα
z − z1
w = eiα
z = 1,
(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
w = eiα
when z = z0 ,
z − z0
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
11
Ex1.
w=
i−z
z −i
= eiπ
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.
z −1+ z − z
( z -1)( z + 1)
2y
= Im
=
2
2
( z + 1)( z + 1)
z +1
z +1
2
(2) v = Im w = Im
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
→ ρ eiφ = e x eiy
Thus ρ = e x ,
or ρ = e ,
x
(1)
w = ρ eiφ , z = x + iy
φ = y + 2nπ ,
n any integer
φ = y transformation from z plane to w plane
x = c1 vertical line
z = (c1 , y ), its image ρ = ec1 , φ = y
(c1 , 2π )
ec1
(c1 , 0)
many-to-1 mapping
24
12
(2)
y = c2 horizontal line
C2
C2
1-to-1 mapping
Ex1
w = ez
a ≤ x ≤ b, c ≤ y ≤ d maps onto e a ≤ ρ ≤ eb , c ≤ φ ≤ d
C'
y
D
d
C
D'
B'
c
A
B
a
b
φ=d
x
A'
φ =c
25
Ex2.
φ =b
ib
ic
φ=c
φ =a
ia
w = log z = ln r + iθ
(r > 0, α < θ < α + 2π )
any branch of log z , maps onto a strip
v
y
i (α + 2π )
θ0
iθ 0
α
iα
x
0
u
26
13
Ex3.
w = log
z −1
z +1
↑
principal branch
z −1
and w = log Z
z +1
↑
is a composition of Z =
maps upper half plane y > 0 onto
upper half plane v > 0
(0 < θ < π )
maps upper half plane
onto the strip 0 < v < π
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
−
π
2
maps
≤x≤
π
2
(1-to-1)
, y≥0
v≥0
onto
y
E
M'
A
M
L'
L
1
D
−
B
π
2
c
π
E'
D'
B'
A'
x
2
28
14
A. boundary of the strip → real axis
(1) BA segment
x=
π
2
e y + e− y
2
e y − e− y
sinh y =
2
, y≥0
cosh y =
u = cosh y, v = 0
(2) DB segment
y=0
u = sin x
v=0
(3) DE segment
π
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 <
π
2
u = sin c1 cosh y, v = cos c1 sinh y
2
(-∞ < y < ∞)
2
u
v
−
= 1 hyperbola
sin 2 c1 cos 2 c1
with foci at the points w = ± sin 2 c1 + cos 2 c1 = ±1
30
15
Consider a horizontal line segment y = c2 , − π ≤ x ≤ π , c2 > 0
its image is u = sin x cosh c2 ,
2
v = cos x sinh c2
2
u
v
+
=1
2
cosh c1 sinh 2 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'
−π
−
0
π
2
π
−1
u
1
x
π
A' E '
2
31
Ex2.
bi
C
C'
B
D
L'
L
F
E
−
π
π
2
2
c2 = 0,
Ex3.
A
u = sin x,
v=0
(−
π
π
2
E'
A'
−1
1
≤x≤
π
2
B'
)
cos z = sin( z + )
2
⇒Z = z+
Ex4.
D'
π
2
, w = sin Z
w = sinh z
w = −i sin(iz )
Z = iz , W = sin Z
w = −iW
w = cosh z
= cos(iz )
32
16
z
75. Mapping by Branches of
z
1
then z
1
2
2
(r > 0, −π < θ ≤ π )
i (θ + 2kπ )
2
iθ
r exp
2
= r exp
principal root
1
2
are the two square roots of z when z ≠ 0
2
if z = r exp(iθ )
z
1
(k = 0,1)
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
iθ
or F0 ( z ) = r exp
2
( z > 0, −π < Argz < π )
(r > 0, −π < θ < π )
33
Ex1
v
C
C'
B
R2
w=z
R1
1
R2 '
2
R1'
2
D
D'
A
0 ≤ r ≤ 2,0 ≤ θ ≤
B'
π
2
u
A'
0 ≤ ρ ≤ 2, 0 ≤ φ ≤
2
π
4
w = F0 (sin z )
Ex2
( z > 0, −π < Arg z < π )
⇒ Z = sin z , w = F0 ( Z )
y
y
v
D' '
D
D'
A
sin Z
π
2
C
B
x
F0 ( z )
C'
x
B'
A'
C' '
u
B' '
A' '
34
17
when − π < θ < π and the branch
log z = ln r + i (θ + 2π ) is used,
z
1
2
= F1 ( z ) = r exp
= − r exp
iθ
2
π < θ + 2π < 3π
i (θ + 2π )
2
= − F0 ( z )
other branches of z
f a ( z ) = r exp
z
1
n
1
2
iθ
2
(r > 0, α < θ < α + 2π )
1
i (θ + 2kπ )
= 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 )
When Z = R eiθ , branches of Z
Z
1
2
= R exp
iθ
2
1
2
1
2
are
( R > 0, α < θ < α + 2π )
If we write
R = z − z0 , Θ = Arg ( z - z0 ) and θ = arg( z − z0 )
two branches of ( z − z0 )
iΘ
2
iθ
and g 0 ( z ) = R exp
2
G0 ( z ) = R exp
1
2
are
( R > 0, −π < Θ < π )
( R > 0,0 < θ < 2π )
36
18
G0 ( z ) is defined at all points in the z plance except z = 0
and the ray Argz = π .
The transformation w = G0 ( z ) is a one-to-one
mapping of the domain
− π < Arg ( z − z0 ) < π
z - z0 > 0,
onto the right half Re w > 0 of the w-plane
y
y
v
z
z0
R
Z
θ
w
R
R
θ
x
x
θ
2
u
The transformation w = g 0 ( z ) maps the domain
0 < arg( z − z0 ) < 2π
z − z0 > 0,
in a ont-to-one manner onto the upper half plane Im w > 0
Ex.2
( z 2 − 1)
1
2
1
= ( z − 1) 2 ( z + 1)
1
2
( z ≠ ±1)
37
19