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(1)

Architecture & Ocean Enginee ring

Engineering Mathematics 2

Prof. Kyu-Yeul Lee

Department of Naval Architecture and Ocean Engineering,

[2008][14-2]

December, 2008

(2)

Architecture & Ocean Enginee ring

Complex (2)

: Conformal mapping

(3)

Conformal Mapping

If a complex function w = f (z) is defined in a domain D of the z-plane, then to each point in

D there corresponds a point in the w-plane.

In this way we obtain a mapping of D onto the range of values of f (z) in the w-plane. We shall see that if f (z) is an analytic function, then the mapping given by w = f (z) is

conformal

(angle-preserving), except at points where the derivative f '(z) is zero.

Conformality is the most important geometric property of analytic functions and gives the possibility of a geometric approach to complex analysis.

we shall see that conformal mapping yields a standard method for solving boundary value

problems in (two-dimensional) potential theory by transforming a complicated region into a

simpler one.

(4)

Geometry of Analytic Functions: Conformal Mapping

(5)

Geometry of Analytic Functions: Conformal Mapping

A complex function

) , ( )

, ( )

(

) 1

( wf zu x yiv x y zxiy

of a complex variable z gives a mapping of its domain of definition D in the complex z-

plane into the complex w-plane or onto its range of values in that plane. For any point z0

in D the point w

0 = f (z0) is called the image of z0 with respect to f

.

Hence circles are mapped onto circles and rays

Using polar forms and , we have . i

re

zwR e

i

wz

2

r

2

e

2i

 2

2

, 

R r

2

r

0

rRr

02

onto rays . Figure shows this for the region

, which is mapped onto the region

0

    2 

0

2

y

v

3 , 6

2

1 3  z

w = f (z) = z 2

(6)

Geometry of Analytic Functions: Conformal Mapping

y

1

0 y

2

1

y

1 y

2

3

y

2 y

u v

5 5

In Cartesian coordinates we have and

zxiy

, )

Re( z

2

x

2

y

2

u    v  Im( z

2

)  2 xy

Hence vertical lines x = c = const are mapped onto

uc

2

y

2

, v  2 cy

) (

4

2 2

2

c c u

v  

Together,

. 4

and

2 2 2

2

2

c u v c y

y   

These parabolas open to the left.

).

( 4

2 2

2

k k u

v  

A complex function

) , ( )

, ( )

(

) 1

( wf zu x yiv x y zxiy

of a complex variable z gives a mapping of its domain of definition D in the complex z-

plane into the complex w-plane or onto its range of values in that plane. For any point z0

in D the point w

0 = f (z0) is called the image of z0 with respect to f

.

w = f (z) = z 2

(7)

Conformal Mapping

A mapping w = f (z) is called conformal if it preserves angles between oriented curves in magnitude as well as in sense.

Curves C1 and C2 and their respective images C1* and C2* under a conformal mapping w = f (z)

(8)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

has a critical point at z = 0, where and the angles are doubled (see Fig. 375), so that conformality fails.

v

z

2

wf  ( z )  2 z  0

1 2

y

0

Example of Theorem 1)

(9)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Proof of Theorem 1) The idea of proof is to consider a curve

) ( )

( )

( :

) 2

( C z tx tiy t

in the domain of f (z) and to show that w = f (z) rotates all tangents at a point z

0

(where ) through the same angle.

Now is tangent to C in (2).

) (t z z

) ( 0

1 z t t

z   )

(t0 z

0 ) (

0

z f

) ( )

( /

)

( t dz dt x t i y t z      

The image C* of C is w = f (z(t)) .

) ( ) ) (

( )

( f z z t

dt t dz dz

z df dt

w   dw    

Hence the tangent direction of C* is given by the argument (use (9) in By the chain rule,

2 1

2 1

2

1

) arg arg

( arg )

9 ( 13.2

Sec. z z      zz

(10)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Proof of Theorem 1)

2 1

2 1

2

1

) arg arg

( arg )

9 ( 13.2

Sec. z z      zz

) (

0

0

z t

z

z f

w  arg arg  arg

) 3

(   

C

1

 

C

2

C

1

*

2

* C

) ( z

0

f

 

 

 

1

1

arg arg

arg

arg arg

arg

z f

w

z f

w

 

 

  f arg Let

1

 arg z

  

plane

z w  plane

(11)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Example)

] sin

[cos )

13 ( 13.2

Sec. z

n

r

n

n   i n

The mapping is conformal, except at z = 0, where For n = 2 this is shown in Fig; we see that at 0 the angles are doubled.

, , 3 , 2 , 1

,  

z n

w

n

w   nz

n1

For general n the angles at 0 are multiplied by a factor n under the mapping. Hence the sector is mapped by z 0     / n

n

onto the upper half-plane v  0

v y

n

 /

(12)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Example)

Mapping w = z + 1/z

In terms of polar coordinates this mapping is

) sin 1 (cos

) sin

(cos    i

i r r

iv u

w      

r r a r a

r

u 1

where ,

cos 1 cos

 

 

  

   , v r 1 sin b sin , where b r 1

r   r

 

        

Hence circle are mapped onto ellipses

z   r const  1 u

22

v

22

1.

ab

the circle r = 1 is mapped onto the segment of u-axis

 2  u  2

y v

(13)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Example) Mapping w = z + 1/z

Now the derivative of w is 1

2

( 1 )(

2

1 )

1 z

z z

w z  

 

which is 0 at . These are the points at which the mapping is not conformal.

The two circles in Fig. pass through z = -1. The larger is mapped onto a Joukowski airfoil.

The dashed circle passes through both -1 and 1 and is mapped onto a curved segment.

 1

z

y

C

v

(14)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Example)

Mapping w = ez

From (10) in Sec. 13.5 we have and . Hence maps a vertical straight line x = x

0

= const onto the circle and a horizontal straight line y = y

0

= const onto the ray arg w = y

0

. The rectangle in Fig. is mapped onto a region bounded by circles and rays as shown.

x

z

e

e  Arg zy e

x

x0

e w

) sin (cos

) 1 ( 13.5

Sec. e

z

e

x

yi y

1 y

5 .

0 A B

C D

v

0 A *

* B

* C

*

D

(15)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Example)

Mapping w = ez

) sin (cos

) 1 ( 13.5

Sec. e

z

e

x

yi y

The fundamental region of in the z-plane is mapped

conformally onto the entire w-plane without the origin w = 0 (because e

z

= 0 for no z). Figure shows that the upper half of the fundamental region is mapped onto the upper half-plane , the left half being mapped inside the unit disk and the right half outside.

e

z

y 0

, arg

0  w

 1 w

z

Arg



 



 

0 when

1

0 when

1

0 when

1

x e

e

x e

e

x e

e

x z

x z

x z

(16)

Conformal Mapping

Theorem 1) Conformality of Mapping by Analytic Functions

The mapping w = f (z) by an analytic function f is conformal, except at critical points, that is, points at which the derivative is zero. f

Example)

Mapping w = Ln z

x

y

0 u

v

0

Principle. The mapping by the inverse of w = f (z) is obtained by interchanging the roles of the z-plane and the w-plane in the mapping by w = f (z).

)

1

( w f

z

Now the principal value w = f (z) = Ln z of the natural logarithm has the inverse . From Example of w =ez (with the notations z and w interchanged) we know that maps the

fundamental region of the exponential function onto the z-plane without z = 0 (because ew ≠0 for every z).

e

w

w f

z

1

( ) 

ew

w f 1( )

Hence w = f (z) = Ln z maps the z-plane without the origin and cut along the negative real axis ( ) conformally onto the horizontal strip of the w-plane , where .

 ImLn jumps by 2

where  z

v

iv u w 

3 

w =Ln z w =Ln z +2π

(17)

Linear Fractional Transformation

(18)

Linear Fractional Transformation

Linear fractional transformations d cz

b w az

  )

1

( ( adbc  0 )

where a, b, c, d are complex or real numbers. Differentiation gives

2

2

( )

) (

) (

) ) (

2

( cz d

bc ad

d cz

b az c d cz w a

 

 

circle) unit

in the (Inversion

/ 1

ons) ansformati

(Linear tr ) (Rotations 1

with

ons) (Translati

) 3 (

z w

b az w

a az

w

b z w

, 0

if adbc   w

: Not conformal

(19)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

real) , , , ( 0

)

( x

2

y

2

Bx Cy D A B C D

A     

2 0

2   

 

D

i z C z

z B z

z Az



 

iy x z

iy x z

A = 0 gives a straight line and A ≠ 0 a circle. In terms of and this equation

becomes

z z

Now w = 1/z. Substitution of z = 1/w and multiplication by gives the equation

w w

This represents a circle (if ) or a straight line (if ) in the w-plane.D0 D0

2 0

2     

 

w w D w w

i z C z

w z w B z

w w z Az

d cz

b w az

  )

1 (

)

0

( adbc

(20)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

 1

r

Hence the unit circle is mapped onto the unit circle

z wR  1 ; we

i

e

i

.

For a general z the image w = 1/z can be found geometrically by marking

R r w   1

re

i

z  1 wR e

i

wz  1 ,    

R r

d cz

b w az

  )

1 (

) 0 ( adbc

x y

u v

1

rR 1

1 r

1 R 1

r

1 R

1 1

(21)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

y

In case x=0

v

2 x 2 , 2 y 2

u v

x y x y

  

 

0, 1

u v

   y

1 2

(22)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

x y

In case x=0

u v

2 x 2 , 2 y 2

u v

x y x y

  

 

0, 1

u v

   y

3 3

4 4

In case y=0 1, 0

u v

x

1 2

(23)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

y

In case x=1/2

v

2 x 2 , 2 y 2

u v

x y x y

  

 

1 2

2 2

: ( ) 0 ( , , , real)

z A xyBx Cy  D A B C D

2 2

: ( ) 0

w A Bu   CvD uv

, for instance : A=0, B=2, C=0, D=-1

2 2

2 u  ( uv )  0

u 1

2

v

2

1

2

2

 1

x

(24)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

x y

In case x=1/2

u v

2 x 2 , 2 y 2

u v

x y x y

  

 

3 3

4 4

1 2

2 2

: ( ) 0 ( , , , real)

z A xyBx Cy  D A B C D

2 2

: ( ) 0

w A Bu   CvD uv

, for instance : A=0, B=2, C=0, D=-1

2 2

2 u  ( uv )  0

u 1

2

v

2

1

2

1 2

2

 1 x

1

and in case y=0 2, 0 u v

(25)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

y

In case x=1/2

v

2 x 2 , 2 y 2

u v

x y x y

  

 

1 2

2 2

: ( ) 0 ( , , , real)

z A xyBx Cy  D A B C D

2 2

: ( ) 0

w A Bu   CvD uv

, for instance : A=0, B=2, C=0, D=-1

2 2

2 u  ( uv )  0

u 1

2

v

2

1

2

2

 1 x

and in case y=1/2 1,

u

1 2

(26)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

x y

In case x=1/2

u v

2 x 2 , 2 y 2

u v

x y x y

  

 

3 3

4 4

1 2

2 2

: ( ) 0 ( , , , real)

z A xyBx Cy  D A B C D

2 2

: ( ) 0

w A Bu   CvD uv

, for instance : A=0, B=2, C=0, D=-1

2 2

2 u  ( uv )  0

u 1

2

v

2

1

2

1 2

2

 1 x

1

and in case y

0, 0 u v

1 2

(27)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

y

In case x=1/2

v

2 x 2 , 2 y 2

u v

x y x y

  

 

1 2

2 2

: ( ) 0 ( , , , real)

z A xyBx Cy  D A B C D

2 2

: ( ) 0

w A Bu   CvD uv

, for instance : A=0, B=2, C=0, D=-1

2 2

2 u  ( uv )  0

u 1

2

v

2

1

2

2

 1 x

In conclusion, 1

2

5

6

(28)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

x y

similarly

u v

2 x 2 , 2 y 2

u v

x y x y

  

 

3 3

4 4

1 2

2 2

: ( ) 0 ( , , , real)

z A xyBx Cy  D A B C D

1 2

2

 1 x

1

1 2

5

6

2 2

: ( ) 0

w A Bu   CvD uv

(29)

Linear Fractional Transformation

Theorem 1) Circles and Straight Lines

Every linear fractional transformation (1) maps the totality of circles and straight lines in the z-plane onto the totality of circles and straight lines in the w-plane.

Figure shows that w = 1/z maps horizontal and vertical straight lines onto circles or straight lines.

u v

1 2

 1

 2

1 2

y

1 y  0

2

 1

x 2

 1 x 2

 1

y

2 2 2 2

1 1 ( )

( )( )

x iy x y

w i u iv

z x iy x iy x iy x y x y

       

    

2 2 , 2 2

x y

u v

x y x y

  

 

(30)

Special Linear Fractional Transformation

(31)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given*

Three given distinct points z

1, z2, z3

can always be mapped onto three prescribed distinct points

w1, w2, w3

by on, and only one, linear fractional transformation w = f (z) . This mapping is given implicitly by the equation

(If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2

3 1 1

2

3 2

3

1

) 2

( z z

z z

z z

z z w

w

w w

w w

w w

 

 

 

d cz

b w az

 

)

1

(

(32)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mapping of Standard Domains Half-Plane onto a Disk

Find the linear fractional transformation (1) that maps

1 ,

0 ,

1

2 3

1

  zz

z w

1

  1 , w

2

  i , w

3

 1

Solution) From (2) we obtain

d cz

b w az

 

)

1

(

(33)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mapping of Standard Domains Half-Plane onto a Disk

Find the linear fractional transformation (1) that maps

1 ,

0 ,

1

2 3

1

  zz

z w

1

  1 , w

2

  i , w

3

 1

Solution) From (2) we obtain

) 1 ( 0

1 0 1

) 1 ( )

1 (

1 1

) 1 (

 

 

 

z z i

i w

w

d cz

b w az

 

)

1

(

(34)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mapping of Standard Domains Half-Plane onto a Disk

Find the linear fractional transformation (1) that maps

1 ,

0 ,

1

2 3

1

  zz

z w

1

  1 , w

2

  i , w

3

 1

Solution) From (2) we obtain

) 1 ( 0

1 0 1

) 1 ( )

1 (

1 1

) 1 (

 

 

 

z z i

i w

w

) 1 1 ( 1 1

1 1

1  

 

 

z z i

i w

w

d cz

b w az

 

)

1

(

(35)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mapping of Standard Domains Half-Plane onto a Disk

Find the linear fractional transformation (1) that maps

1 ,

0 ,

1

2 3

1

  zz

z w

1

  1 , w

2

  i , w

3

 1

Solution) From (2) we obtain

) 1 ( 0

1 0 1

) 1 ( )

1 (

1 1

) 1 (

 

 

 

z z i

i w

w

) 1 1 ( 1

1  

 

 

i z

w

d cz

b w az

 

)

1

(

(36)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mapping of Standard Domains Half-Plane onto a Disk

Find the linear fractional transformation (1) that maps

1 ,

0 ,

1

2 3

1

  zz

z w

1

  1 , w

2

  i , w

3

 1

Solution) From (2) we obtain

) 1 ( 0

1 0 1

) 1 ( )

1 (

1 1

) 1 (

 

 

 

z z i

i w

w

) 1 1 ( 1 1

1 1

1  

 

 

z z i

i w

w

d cz

b w az

 

)

1

(

(37)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mappings of disks onto disks

d cz

b w az

  ) 1 (

) 1 3

(

0

  cz

z

w z cz

0

, z  1 .

 1

z cz

0

0 0 0

zz   z z   z z   z c  1  zcz zc

take (unit circle), obtaining, with as in (3),

the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following

function, which maps z

0

onto the center w = 0.

(38)

Special Linear Fractional Transformation

Theorem 1) Three Points and Their Images Given

Three given z1, z2, z3 can always be mapped onto three prescribed distinct points w1, w2, w3 by on, and only one, linear fractional transformation w = f (z) . (If one of these points is the point ∞, the quotient of the two differences containing this point must be replaced by 1.)

1 2

3 2 3 1 1

2 3 2 3

1

) 2

( z z

z z z z

z z w w

w w w w

w w

 

 

 

Mappings of disks onto disks

d cz

b w az

  ) 1 (

) 1 3

(

0

  cz

z w z

0

,

0

1

cz z

the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following

function, which maps z

0

onto the center w = 0.

(39)

Special Linear Fractional Transformation

Mappings of disks onto disks

d cz

b w az

  ) 1 (

) 1 3

(

0

  cz

z

w z cz

0

, z  1 .

the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following function, which maps z

0

onto the center w = 0.

Taking z0 = ½ in (3), we obtain 1

2 1 2

2 1

1 2

z z

w z z

 

 

 

iy x

y i w x

 

2 2 1 2

) 2

)(

2 (

) 2

)(

2 1 2 (

iy x

iy x

iy x

y i x

 

2 2

2 2

2 2

) 2 (

3 )

2 (

2 5

2 2

y x

i y y

x

x y

x

 

 

iv u

y

B A

1 x

*

A B *

v

(40)

Special Linear Fractional Transformation

Mappings of disks onto disks

d cz

b w az

  ) 1 (

) 1 3

(

0

  cz

z

w z cz

0

, z  1 .

the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following function, which maps z

0

onto the center w = 0.

Taking z0 = ½ in (3), we obtain 1

2 1 2

2 1

1 2

z z

w z z

 

 

 

iy x

y i w x

 

2 2 1 2

) 2

)(

2 (

) 2

)(

2 1 2 (

iy x

iy x

iy x

y i x

 

2 2

2 2

2 2

) 2 (

3 )

2 (

2 5

2 2

y x

i y y

x

x y

x

 

 

iv u

2 2

2 2 5 2 3

x y x , y

u     v  

   

y

B A

1 x

*

A B *

1 2 u

v

(41)

Special Linear Fractional Transformation

Mappings of disks onto disks

d cz

b w az

  ) 1 (

) 1 3

(

0

  cz

z

w z cz

0

, z  1 .

the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following function, which maps z

0

onto the center w = 0.

Taking z0 = ½ in (3), we obtain 1

2 1 2

2 1

1 2

z z

w z z

 

 

 

iy x

y i w x

 

2 2 1 2

) 2

)(

2 (

) 2

)(

2 1 2 (

iy x

iy x

iy x

y i x

 

2 2

2 2

2 2

) 2 (

3 )

2 (

2 5

2 2

y x

i y y

x

x y

x

 

 

iv u

y

B A

1 x

*

A B *

v

 0

x

(42)

Special Linear Fractional Transformation

Mappings of disks onto disks

d cz

b w az

  ) 1 (

) 1 3

(

0

  cz

z

w z cz

0

, z  1 .

the unit disk in the z-plane is mapped onto the unit disk in the w-plane by the following function, which maps z

0

onto the center w = 0.

Taking z0 = ½ in (3), we obtain 1

2 1 2

2 1

1 2

z z

w z z

 

 

 

iy x

y i w x

 

2 2 1 2

) 2

)(

2 (

) 2

)(

2 1 2 (

iy x

iy x

iy x

y i x

 

2 2

2 2

2 2

) 2 (

3 )

2 (

2 5

2 2

y x

i y y

x

x y

x

 

 

iv u

2 2

2 2 5 2 3

x y x , y

u     v  

   

2

 1

y

 0 y

2

 1 x

2

 1 x

y

B A

1 x

*

A B *

1 2 u

v

 0

x

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