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Chapter 8. Mapping by Elementary Functions. Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University Email : weiqi.luo@yahoo.com Office : # A313. Chapter 8: Mapping by Elementary Functions. Linear Transformations The Transformation w=1/z Mapping by 1/z - PowerPoint PPT Presentation
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Chapter 8. Mapping by Elementary Functions
Weiqi Luo (骆伟祺 )School of Software
Sun Yat-Sen UniversityEmail : weiqi.luo@yahoo.com Office : # A313
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Linear Transformations The Transformation w=1/z Mapping by 1/z Linear Fractional Transformations Mapping of the Upper Half Plane
2
Chapter 8: Mapping by Elementary Functions
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The Mapping
where A is a nonzero complex constant and z≠0.
We write A and z in exponential form:
Then
Expands or contracts the radius vector representing z by the factor a and rotates it through the angle α about the origin.
90. Linear Transformations
3
w Az
( )( ) iw ar e
,i iA ae z re
The image of a given region is geometrically similar to that region.
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The Mapping
where B is any complex constant, is a translation by means of the vector representing B. That is, if
Then the image of any point (x,y) in the z plane is the point
in the w plane
90. Linear Transformations
4
w z B
1 2, ,w u iv z x iy B b ib
1 2( , ) ( , )u v x b y b
The image of a given region is geometrically congruent to that region.
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The General (non-constant) Linear Transformation
is a composition of the transformations
90. Linear Transformations
5
, ( 0)w Az B A
, ( 0)Z Az A w Z B
when z≠0, it is evidently an expansion or contraction (scaling) and a rotation, followed by a translation.
and
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Example The mapping
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
90. Linear Transformations
6
(1 ) 2w i z
(1 ) 2 exp[ ( )]4
Z i z r i
& 2 ( 2, )w Z X Y
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Example (Cont’)
90. Linear Transformations
7
(x,y)-plane (X,Y)-plane (u,v)-plane
Scaling and Rotation Translation
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pp. 313
Ex. 2, Ex. 6
90. Homework
8
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The Equation
establishes a one to one correspondence between the nonzero points of the z and the w planes.
Since , the mapping can be described by means of the successive transformations
91. The Transformation w=1/z
9
1w
z
2| |zz z
2
z,
| |Z w Z
z
(0) , ( ) 0w w
To make the transformation continuous on the extended plane, we let
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The Mapping
reveals that
92. Mapping by w=1/z
10
2
1 z z
| |w
z zzz
2 2 2 2,
x yu v
x y x y
2
1 w w
| |z
w www
Similarly, we have that
2 2 2 2,
u vx y
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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Consider the Equation
represents an arbitrary circle or line ( B2+C2>4AD)
92. Mapping by w=1/z
11
2 2( ) 0A x y Bx Cy D
2 22 2 24
( ) ( ) ( ) , ( 0)2 2 2
B C B C ADx y A
A A A
Circle:
0, ( 0)Bx Cy D A Line:
Note: Line can be regarded as a special circle with a infinite radius.
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The Mapping by w=1/z If x and y satisfy
then after the mapping by w=1/z,
we get that
92. Mapping by w=1/z
12
2 2( ) 0A x y Bx Cy D
2 2( ) 0D u v Bu Cv A
(a circle or line in (x,y)-plane )
(also a circle or line in (u,v)-plane )
2 2 2 2. . ,
u vi e x y
u v u v
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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.
92. Mapping by w=1/z
13
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Example 1
A vertical line x=c1 (c1≠0) is transformed by w=1/z into the circle –c1(u2+v2)+u=0, or
Example 2 A horizontal line y=c2 (c2≠0) is transformed by w=1/z
into the circle
92. Mapping by w=1/z
14
2 2 2
1 1
1 1( ) ( )
2 2u v
c c
2 2 2
2 2
1 1( ) ( )
2 2 u v
c c
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Illustrations
92. Mapping by w=1/z
15
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Example 3 When w=1/z, the half plane x≥c1 (c1>0) is mapped onto
the disk
For any line x=c (c ≥c1) is transformed into the circle
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.
92. Mapping by w=1/z
16
2 2 2
1 1
1 1( ) ( )
2 2u v
c c
2 2 21 1( ) ( )
2 2u v
c c
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Illustrations
92. Mapping by w=1/z
17
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pp. 318
Ex. 5, Ex. 8, Ex. 12
92. Homework
18
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The Transformation
where a, b, c, and d are complex constants, is called a linear fractional (Möbius) transformation.
We write the transformation in the following form
this form is linear in z and linear w, another name for a linear fractional transformation is bilinear transformation.
93. Linear Fractional Transformations
19
, ( 0)az b
w ad bccz d
0, ( D-BC 0)Azw Bz Cw D A
Note: If ad-bc=0, the bilinear transform becomes a constant function.
20
( )
dw ad bc
dz cz d
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93. Linear Fractional Transformations
20
, ( 0)az b
w ad bccz d
When c=0
, ( 0)az b a b
w z add d d
When c≠01
, ( 0)
a bc adw ad bc
c c cz d
1, ,
a bc adZ cz d W w W
Z c c
which includes three basic mappings
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
21
( ) , ( 0)az b
T z ad bccz d
( ) , ( 0)T c ( ) & ( ) , ( 0)a d
T T cc c
To make T continuous on the extended z plane, we let
1( ) , ( 0)dw b
T w ad bccw a
There is an inverse transformation (one to one mapping) T-1
1( ) , ( 0)T c 1 1( ) & ( ) , ( 0)
a dT T c
c c
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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.
93. Linear Fractional Transformations
22
b d
ic ib a b ic ib a b
0 ( ) 0 0ad bc b a c b
( 1)( ) , ( 0)
( 1)
ibz b b izT z b
ibz b b iz
c iba ib
1( )
1
iz i zT z
iz i z
(0) 1T
( 1) , (1)T i T i
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Example 2
Suppose that the points z1 = 1, z2 = 0, and z3 = −1
are to be mapped onto w1 = i, w2 =∞, and w3 = 1.
93. Linear Fractional Transformations
23
(0)T 0, 0c d
( ) , ( 0)az b
T z bccz
(1)T i
( 1) 1T +b,ic a c a b 2 (1 ) , 2 ( 1)a i c b i c
( 1) ( 1)( )
2
i z iT z
z
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The Equation
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.
94. An Implicit Form
24
1 2 3 1 2 3
3 2 1 3 2 1
( )( ) ( )( )
( )( ) ( )( )
w w w w z z z z
w w w w z z z z
Verify this Equation
Why three rather than four distinct points?
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Example 1The 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
Then solving for w in terms of z, we have
94. An Implicit Form
25
( )(1 ) ( 1)(0 1)
( )(1 ) ( 1)(0 1)
w i i z
w i i z
i zw
i z
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For the point at infinity For instance, z1=∞,
Then the desired modification of the implicit form becomes
The same formal approach applies when any of the other prescribed points is ∞
94. An Implicit Form
26
1 1
2 31 2 3 1 2 3 2 31 1
0 03 2 1 1 3 1 2 3
3 21
1( )( )
( )( ) ( 1)( )lim lim
1( )( ) ( )( 1)( )( )z z
z z zz z z z z z z z z zz z
z z z z z z z z z z zz z zz
1 2 3 2 3
3 2 1 3
( )( )
( )( )
w w w w z z
w w w w z z
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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
of the implicit form, which tells us that
Solving here for w, we have the transformation obtained earlier.
94. An Implicit Form
27
1 2 31
3 3 2 1
( )( )
( )( )
z z z zw w
w w z z z z
( 1)(0 1)
1 ( 1)(0 1)
w i z
w z
( 1) ( 1)
2
i z iw
z
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pp. 324
Ex. 1, Ex. 4, Ex. 6
94. Homework
28
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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
95. Mappings of The Upper Half Plane
29
, ( 0)az b
w ad bccz d
x
y
u
v
1
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Imz=0 are transformed into circle |w|=1
when points z=0, z=∞ we get that
95. Mappings of The Upper Half Plane
30
, ( 0)az b
w ad bccz d
| | | | 0b d
where α is a real constant, and z0 and z1 are nonzero complex constants.
| | | | 0a c
00 1 1 0
1
( ), ( , , ,| | | | 0)
i iz z a b d
w e e z z z zz z c a c
( / )( )
( / )
az b a z b aw
cz d c z d c
Rewrite | | | | 0 b d
a c
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95. Mappings of The Upper Half Plane
31
01 0
1
( ), (| | | | 0)i z zw e z z
z z
when points z=1, we get that 0 1|1 | |1 |z z
0 0 1 1(1 )(1 ) (1 )(1 )z z z z 1 1 0 0 1 0, (| | | |)z z z z z z
1 0 1 0z z orz z
If z1=z0, then is a constant function iw e
Therefore, 1 0z z
Finally, we obtain the mapping 00
0
( ), (Im 0)i z zw e z
z z
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Mappings of The Upper Half Plane
95. Mappings of The Upper Half Plane
32
00
0
( ), (Im 0)i z zw e z
z z
w
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Example 1 The transform
in Examples 1 in Sections. 93 and 94 can be written
95. Mappings of The Upper Half Plane
33
i zw
i z
( )i z iw e
z i
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Example 2By writing z = x + iy and w = u + iv, we can readily show
that the transformation
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.
95. Mappings of The Upper Half Plane
34
1
1
zw
z
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Example 2 (Cont’) Furthermore, for any point w in the finite w plane,
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.
95. Mappings of The Upper Half Plane
35
2
( 1)( 1) 2Im Im ,( 1)
| 1|( 1)( 1)
z z yv w z
zz z
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Example 3 The transformation
where the principal branch of the logarithmic function is used, is a composition of the function
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;
95. Mappings of The Upper Half Plane
36
1
1
zw Log
z
1&
1
zZ w LogZ
z
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Example 3 (Cont’)
95. Mappings of The Upper Half Plane
37
Re ,( 0,0 )iZ R
(Re ) lniw Log R i
0,0R
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pp. 329
Ex. 1, Ex. 2
95. Homework
38
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