Chapter 8. Mapping by Elementary Functions

Chapter 8. Mapping by Elementary Functions

Weiqi Luo (骆伟祺 )School of Software

Sun Yat-Sen UniversityEmail ： [email protected] Office ： # A313

mailto:[email protected]

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Linear Transformations The Transformation w=1/z Mapping by 1/z Linear Fractional Transformations Mapping of the Upper Half Plane

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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

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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


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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


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, ( 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


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(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’)


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(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

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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

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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

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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）


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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


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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.


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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


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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


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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.


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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


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pp. 318

Ex. 5, Ex. 8, Ex. 12

92. Homework

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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

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, ( 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.

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( )

dw ad bc

dz cz d

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, ( 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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( ) , ( 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.


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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.


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(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

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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


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( )(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 ∞


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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.


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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

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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

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, ( 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


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, ( 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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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


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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


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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.


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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.


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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;


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1

1

zw Log

z

1&

1

zZ w LogZ

z

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Example 3 (Cont’)


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Re ,( 0,0 )iZ R

(Re ) lniw Log R i

0,0R

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pp. 329

Ex. 1, Ex. 2

95. Homework

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Chapter 8. Mapping by Elementary Functions