Chap. 8 - Sun Yat

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Chapter 8. Mapping by Elementary Functions
Weiqi Luo (骆伟祺)
School of Software
Sun Yat-Sen University
Email:[email protected] Office:# A313
Chapter 8: Mapping by Elementary Functions





Linear Transformations
The Transformation w=1/z
Mapping by 1/z
Linear Fractional Transformations
Mapping of the Upper Half Plane
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90. Linear Transformations
 The Mapping
w  Az
where A is a nonzero complex constant and z≠0.
We write A and z in exponential form:
A  aei , z  rei
Then
w  (ar )ei (  )
Expands or contracts the radius vector representing z by the factor a
and rotates it through the angle α about the origin.
The image of a given region is geometrically similar to that region.
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90. Linear Transformations
 The Mapping
w zB
where B is any complex constant, is a translation by means
of the vector representing B. That is, if
w  u  iv, z  x  iy, B  b1  ib2
Then the image of any point (x,y) in the z plane is the point
in the w plane
(u, v)  ( x  b1 , y  b2 )
The image of a given region is geometrically congruent to that region.
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90. Linear Transformations
 The General (non-constant) Linear Transformation
w  Az  B,( A  0)
is a composition of the transformations
Z  Az,( A  0)
and
wZ B
when z≠0, it is evidently an expansion or contraction (scaling)
and a rotation, followed by a translation.
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90. Linear Transformations
 Example
The mapping
w  (1  i) z  2
transforms the rectangular region in the z=(z, y) plane of
the figure into the rectangular region in the w=(u,v)
plane there. This is seen by expressing it as a
composition of the transformations

Z  (1  i ) z  2r exp[i (  )]
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&w  Z  2  ( X  2, Y )
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90. Linear Transformations
 Example (Cont’)
Scaling and Rotation
(x,y)-plane
(X,Y)-plane
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Translation
(u,v)-plane
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90. Homework
pp. 313
Ex. 2, Ex. 6
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91. The Transformation w=1/z
 The Equation
1
w
z
establishes a one to one correspondence between the nonzero
points of the z and the w planes.
2
Since zz | z |, the mapping can be described by means of the
successive transformations
z
Z  2 ,w  Z
|z|
To make the transformation continuous
on the extended plane, we let
w(0)  , w()  0
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92. Mapping by w=1/z
 The Mapping
1 z
z
w 
 2
z zz | z |
reveals that
u
x
y
,
v

x2  y 2
x2  y 2
Similarly, we have that
1
w
w
z 

w ww | w |2
u
v
x 2 2,y 2 2
u v
u v
Based on these relations between coordinates, the mapping w=1/z transforms
circles and lines into circles and lines
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92. Mapping by w=1/z
 Consider the Equation
A( x  y )  Bx  Cy  D  0
2
2
represents an arbitrary circle or line (B2+C2>4AD)
Circle:
Line:
B 2
C 2
B 2  C 2  4 AD 2
(x 
)  (y 
) (
) ,( A  0)
2A
2A
2A
Bx  Cy  D  0,( A  0)
Note: Line can be regarded as a special circle with a infinite radius.
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92. Mapping by w=1/z
 The Mapping by w=1/z
If x and y satisfy
A( x2  y2 )  Bx  Cy  D  0
(a circle or line in (x,y)-plane )
u
v
then after the mapping by w=1/z, i.e.x  2 2 , y  2 2
u v
u v
we get that
D(u  v )  Bu  Cv  A  0
2
2
(also a circle or line in (u,v)-plane )
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92. Mapping by w=1/z
 Four Cases
Case #1: A circle (A ≠ 0) not passing through the origin (D ≠ 0)
in the z plane is transformed into a circle not passing through
the origin in the w plane;
Case #2: A circle (A ≠ 0) through the origin (D = 0) in the z
plane is transformed into a line that does not pass through the
origin in the w plane;
Case #3: A line (A = 0) not passing through the origin (D ≠ 0) in
the z plane is transformed into a circle through the origin in
the w plane;
Case #4: A line (A = 0) through the origin (D = 0) in the z plane
is transformed into a line through the origin in the w plane.
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92. Mapping by w=1/z
 Example 1
A vertical line x=c1 (c1≠0) is transformed by w=1/z into
the circle –c1(u2+v2)+u=0, or
(u 
1 2 2
1
)  v  ( )2
2c1
2c1
 Example 2
A horizontal line y=c2 (c2≠0) is transformed by w=1/z
into the circle
1 2
1 2
u  (v 
) ( )
2c2
2c2
2
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92. Mapping by w=1/z
 Illustrations
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92. Mapping by w=1/z
 Example 3
When w=1/z, the half plane x≥c1 (c1>0) is mapped onto
the disk
1 2 2
1 2
(u 
2c1
) v (
2c1
)
For any line x=c (c ≥c1) is transformed into the circle
1 2 2
1 2
(u  )  v  ( )
2c
2c
Furthermore, as c increases through all values greater than c1, the
lines x = c move to the right and the image circles shrink in size.
Since the lines x = c pass through all points in the half plane x ≥
c1 and the circles pass through all points in the disk.
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92. Mapping by w=1/z
 Illustrations
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92. Homework
pp. 318
Ex. 5, Ex. 8, Ex. 12
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93. Linear Fractional Transformations
 The Transformation
az  b
w
, (ad  bc  0)
cz  d
where a, b, c, and d are complex constants, is called a linear
fractional (Möbius) transformation.
We write the transformation in the following form
Azw  Bz  Cw  D  0,( AD-BC  0)
this form is linear in z and linear w, another name for a linear
fractional transformation is bilinear transformation.
Note: If ad-bc=0, the bilinear transform becomes a constant function.
dw ad  bc

0
2
dz (cz  d )
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93. Linear Fractional Transformations
az  b
w
, (ad  bc  0)
cz  d
When c=0
When c≠0
w
az  b a
b
 z  , (ad  0)
d
d
d
a bc  ad
1
w 

, (ad  bc  0)
c
c
cz  d
which includes three basic mappings
1
a bc  ad
Z  cz  d ,W  , w  
W
Z
c
c
It thus follows that, regardless of whether c is zero or not, any linear fractional
transformation transforms circles and lines into circles and lines.
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93. Linear Fractional Transformations
az  b
T ( z) 
, (ad  bc  0)
cz  d
To make T continuous on the extended z plane, we let
T ()  ,(c  0)
a
d
T ()  & T ( )  , (c  0)
c
c
There is an inverse transformation (one to one mapping) T-1
dw  b
T ( w) 
, (ad  bc  0)
cw  a
d
1 a
1
1
T
(
)


&
T
(

)


, (c  0)
T ()  ,(c  0)
c
c
1
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93. Linear Fractional Transformations
 Example 1
Let us find the special case of linear fractional
transformation that maps the points
z1 = −1, z2 = 0, and z3 = 1
onto the points w1 = −i, w2 = 1, and w3 = i.
T (0)  1
T (1)  i, T (1)  i
T ( z) 
ad  bc  0  b(a  c)  0  b  0
a  ib
ic  ib  a  b ic  ib  a  b
c  ib
bd
ibz  b
b(iz  1)

, (b  0)
ibz  b b(iz  1)
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T ( z) 
iz  1 i  z

iz  1 i  z
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93. Linear Fractional Transformations
 Example 2
Suppose that the points z1 = 1, z2 = 0, and z3 = −1
are to be mapped onto w1 = i, w2 =∞, and w3 = 1.
T (0)  
T (1)  i
T (1)  1
c  0, d  0
az  b
T ( z) 
, (bc  0)
cz
ic  a+b, c  a  b
T ( z) 
2a  (1  i)c, 2b  (i  1)c
(i  1) z  (i  1)
2z
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94. An Implicit Form
 The Equation
( w  w1 )( w2  w3 ) ( z  z1 )( z2  z3 )

( w  w3 )( w2  w1 ) ( z  z3 )( z2  z1 )
defines (implicitly) a linear fractional transformation that maps
distinct points z1, z2, and z3 in the finite z plane onto distinct
points w1, w2, and w3, respectively, in the finite w plane.
Verify this Equation
Why three rather than four distinct points?
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94. An Implicit Form
 Example 1
The transformation found in Example 1, Sec. 93, required
that z1 = −1, z2 = 0, z3 =1 and w1 = −i, w2 = 1, w3 = i.
Using the implicit form to write
( w  i)(1  i) ( z  1)(0  1)

( w  i)(1  i) ( z  1)(0  1)
Then solving for w in terms of z, we have
iz
w
iz
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94. An Implicit Form
 For the point at infinity
For instance, z1=∞,
1
)( z2  z3 )
( z  z1 )( z2  z3 )
( z1 z  1)( z2  z3 ) z2  z3
z1
z1
 lim
  lim

z

0
z

0
1
( z  z3 )( z2  z1 ) 1 ( z  z )( z  ) z1 1 ( z  z3 )( z1 z2  1) z  z3
3
2
z1
(z 
Then the desired modification of the implicit form becomes
( w  w1 )( w2  w3 ) z2  z3

( w  w3 )( w2  w1 ) z  z3
The same formal approach applies when any of the other
prescribed points is ∞
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94. An Implicit Form
 Example 2
In Example 2, Sec. 93, the prescribed points were
z1 = 1, z2 = 0, z3 = −1 and w1 = i, w2 =∞, w3 = 1.
In this case, we use the modification
w  w1 ( z  z1 )( z2  z3 )

w  w3 ( z  z3 )( z2  z1 )
of the implicit form, which tells us that w  i  ( z  1)(0  1)
w 1
( z  1)(0  1)
Solving here for w, we have the transformation obtained
earlier.
(i  1) z  (i  1)
w
2z
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94. Homework
pp. 324
Ex. 1, Ex. 4, Ex. 6
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95. Mappings of The Upper Half Plane
 Mappings of the Upper Half Plane
We try to determine all linear fractional transformations
that map the upper plane (Imz>0) onto the open disk
|w|<1 and the boundary Imz=0 of the half plane onto the
boundary |w|=1 of the disk
y
az  b
w
, (ad  bc  0)
cz  d
v
1
x
u
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95. Mappings of The Upper Half Plane
 Imz=0 are transformed into circle |w|=1
az  b
w
, (ad  bc  0)
cz  d
when points z=0, z=∞ we get that
| b || d | 0
Rewrite
| a || c | 0
az  b
a z  (b / a)
w
 ( )
cz  d
c z  (d / c)
b d
| || | 0
a
c
z  z0
a
b
d
i
we (
), (e  , z0   , z1   ,| z1 || z0 | 0)
z  z1
c
a
c
i
where α is a real constant, and z0 and z1 are nonzero complex constants.
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95. Mappings of The Upper Half Plane
z  z0
we (
), (| z1 || z0 | 0)
z  z1
i
when points z=1, we get that |1  z0 ||1  z1 |
(1  z0 )(1  z0 )  (1  z1 )(1  z1 )
z1  z1  z0  z0 ,(| z1 || z0 |)
z1  z0orz1  z0
If z1=z0, then w  ei is a constant function
Therefore, z1  z0
z  z0
we (
), (Im z0  0)
z  z0
i
Finally, we obtain the mapping
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95. Mappings of The Upper Half Plane
 Mappings of The Upper Half Plane
z  z0
we (
), (Im z0  0)
z  z0
i
w
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95. Mappings of The Upper Half Plane
 Example 1
The transform
iz
w
iz
in Examples 1 in Sections. 93 and 94 can be written
z i
we (
)
z i
i
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95. Mappings of The Upper Half Plane
 Example 2
By writing z = x + iy and w = u + iv, we can readily show
that the transformation
z 1
w
z 1
maps the half plane y > 0 onto the half plane v > 0 and the
x axis onto the u axis.
Firstly, when the number z is real, so is the number w.
Since the image of the real axis y=0 is either a circle or a
line, it must be the real axis v=0.
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95. Mappings of The Upper Half Plane
 Example 2 (Cont’)
Furthermore, for any point w in the finite w plane,
( z  1)( z  1)
2y
v  Im w  Im

,( z  1)
2
( z  1)( z  1) | z  1|
which means that y and v have the same sign, and points
above the x axis correspond to points above the u axis.
Finally, since point on x axis correspond to points on the u axis and
since a linear fractional transformation is a one to one mapping
of the extended plane onto the extended plane, the stated
mapping property of the given transformation is established.
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95. Mappings of The Upper Half Plane
 Example 3
The transformation
z 1
w  Log
z 1
where the principal branch of the logarithmic function is
used, is a composition of the function
z 1
Z
& w  LogZ
z 1
According to Example 2, Z=(z-1)/(z+1) maps the upper half plane
y>0 onto the upper half plane Y>0, where z=x+iy, Z=X+iY;
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95. Mappings of The Upper Half Plane
 Example 3 (Cont’)
Z  Rei ,( R  0,0     )
w  Log (Rei )  ln R  i
R  0, 0    
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95. Homework
pp. 329
Ex. 1, Ex. 2
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