1

Chap 8 Mapping by Elementary Functions

68. Linear Transformations

rotate by arg .A

expand or contract radius by a A

( )

, where is a nonzero constant and 0

Let ,

( )

i i

i

w AzA z

A ae z re

w ar e

1 2

1 2

Let then ( , ) ( , ) translation

w z Bw u ivz x iyB b ib

u v x b y b

位移

2

4

The general linear transformation ( 0)is a composition of and Ex: (1 ) 2

= 2 2i

w Az B AZ Az w Z B

w i z

e z

y

B

A0x

y

x

v

u0 2

3

69. The Transformation 1z

1wz

mapping between

nonzero points of z and w planes.2

2

1Since , is the composite of

1 ,

zz z wz

Z z w Zz

An inversion with respect to unit circle

a reflection in the real axis

0

1

arg arg1 1Because lim and lim 0

z z

Zz

Z z

z z

Z

Z

w1

4

To define a one to one transformation ( ) from the extended plane onto the extended plane by writing

1 (0) , ( ) 0 and ( ) for 0,

is contiuons throughout the extended

w T zz w

T T T z zz

T

2 2 2

2 2 2 2

2 2 2 2

plane.1Since is the image of under

, (3)

1 - Similarly, ,

z

w u iv z x iy wz

z x iywx yz

x yu vx y x y

u vz x yw u v u v

5

2 2

2 2

2

Let , , , be real numbers, and 4The equation

( ) 0 (5)represents an arbitrary circle or line,where 0 for a circle and 0 for a line

0

A B C D B C AD

A x y Bx Cy D

A AA

x

2

2 22 2

2

2 22 2 2

2 2

0

4 ( ) ( ) 02 2 4

4 ( ) ( ) ( )2 2 2

a circle when 4 0

B C Dy x yA A A

B C AD B Cx yA A A

B C B C ADx yA A A

B C AD

6

2 2

2 2 2 2

2 2

0 0, which means and are not both zero. 0 is a line.

substituting by , y by in (5),

we get ( ) 0 represents a

A B C B CBx Cy D

u vxu v u v

D u v Bu Cv A

circle or line

1The mapping transforms circles and lines into circles

and lines.(a) 0, 0, A circle not passing through the origin 0 is tranformed to a circle not passing through the origin 0.

(b)

wz

A D zw

T

T

T

0, 0, A circle thru 0 line not passing 0.

(c) 0, D 0, a line not passing 0 a circle through 0.

(d) 0, 0, a line thru 0 a line thru 0.

A D z w

A z w

A D z w

7

Ex1.

1 1

2 2 2

1 1

1

12 22 2

1 1

2 2 2 2

vertical line , 01

1 1( ) ( )2 2

a point ( , )

by eq.(3) ( , ) ( , )

,

x c c

wz

u vc c

z c yc yu v

c y c yx y

x y x y

v

u01 C

02 C01 C

02 C

02 C

1C02 C

1C

y

x

8

Ex2.

Ex3.

2

2 2 2

2 2

1

1 1( ) ( )2 2

y c

wz

u vc c

2 2 21

1 1

1 1( ) ( )2 2

x c u vc c

1C C12

1C

121C

9

70. Linear Fractional Transformation

(1) ( 0) , , , , , complex constantsaz bw ad bc a b c dcz d

is called a linear fractional transformation or Mobius transformation.

Eq.(1) 0 ( ( ) 0) (~ 0, - 0)

wcz dw az b bc adA zw Bz Cw D AD BC

bilinear transformation linear in zlinear in wbilinear in z and w

when 0, the condition becomes 0.

(1) a linear function

c ada bw zd d

10

when 0,

( ) (1)

a 1 =c

is a composition of1 a , , =

ca linear fractional transformation always transforms

cir

ca adcz d bc cw

cz dbc ad

c cz d

bc adZ cz d W w WZ c

cles and lines into circles and lines.

1( , )az bz

皆是

11

solving (1) for ,

( - 0)

If 0, one-to-one mapping

If 0, & has one-to-one mapping except at .

zdw bz ad bc

cw ad bc z wa a

ac z w wc

Denominator=0

12

( ) ( 0) (5)

( ) if 0

( ) if 0

( ) if 0

az bT z ad bccz d

T caT cc

dT cc

define a linear fractional tranformation

on the extended plane such that .

in the image of when 0.

aT z w cz c

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.

13

1 2 1 2

-1

-1

1

i.e., ( ) ( ) whenever and for each point in -plane, there is a point in the -plane,such that ( ) .

There is an inverse transformation

( ) iff ( )

( )

T z T z z zw w z z

T z w

T

T w z T z wdw bT w

1

1

1

( 0)

( ) if 0

( ) if 0

( )

ad bccw a

T caT cc

dTc

A linear fractional transformation

14

Ex1.

1 2 3

1 2 3

find that

map 1 0 1onte 1

az bwcz d

z z zw i w w i

1b

db d

( ( ) 0)

2 2 , c -

1 1

az bw b a ccz b

a b a bi ic b c b

ic ib a b ic ib a bic b ib

a ibi bz b iz i zwi bz b iz i z

0ba c

( 0)b

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.

15

Ex2: 1 2 3

1 2 3

1, 0, 1, , 1

0

( 0)

1

1( 1) 2 ,

2

(

z z zw i w w

daz bw bc

cza b a bi

c cic a b c a b

ii c b b c

a ic b

1 1)

2 2( 1) ( 1)

2

i ii c c

i z iwz

16

71. An Implicit Form

The equation1 2 3 1 2 3

3 2 1 3 2 1

1 2 3 1 2 3

( )( ) ( )( ) (1)( )( ) ( )( )defines (implicitly) a linear fractional transformation that maps distinct points , , onto distinct , , , respectively.Re

w w w w z z z zw w w w z z z z

z z z w w w

3 1 2 1 2 3

1 3 2 3 2 1

1

1

3

write (1) as( )( )( )( ) (2) ( )( )( )( )If , right-hand side=0 If , l

z z w w z z w wz z w w z z w w

z zw w

z z

3

eft-hand side=0 w w

17

2 1 2 3 3 2 1

2

If ( )( ) ( )( ) Expanding (2) get 0 a linear fractional transformation.

z z w w w w w w w ww w

A zw Bz Cw D

Ex1.

1 1

2 2

3 3

1 0 11

( )(1 ) ( 1)(0 1)( )(1 ) ( 1)(0 1)( )( 1)(1 ) ( )( 1)(1 )( )(1 ) ( )(1 )( 1) (

z w iz wz w iw i i zw i i zw i z i w i z iwz iz w i i wz iz w i iwz iz w i iwz z iw wz iz w i iwz z iw

1)

- 2 2 (- )2 ( ) ( - ) i zwz iw z i w z i i z wi z

18

1 1

1

1 11

2 31 2 31

0 03 2 1

3 21

equation (1) can be modified for point at infinity.suppose

1replace by , and let 0

1( )( )( 1)( )lim lim1 ( )( 1)( )( )

z z

z

z zz

z z zz z z zzz z z zz z z

z

2 3

3

1 2 3 2 3

3 2 1 3

The desired equation is( )( )( )( )

z zz z

w w w w z zw w w w z z

19

Ex2.1 1

2 2

3 3

1 2 31

3 3 2 1

1 0

1 1( )( )( )( )

( 1)(0 1)1 ( 1)(0 1)

( )( 1) ( 1)( 1)1

2 1 ( 1) ( -1)

( 1) w

z w iz wz w

z z z zw ww w z z z zw i zw z

w i z w zwz iz w i wz z wwz iz i z

i z ii z

( 1)2

iz

20

72. Mapping of the upper Half Plane

Determine all 1inear fractional transformation T thatIm 0 1

Im 0 1

T

T

z w

z w

0 1 1

Choose three points 0, 1, that will be mapped to 1

by ( - 0)

0 1, 0

only if 0 ( 0 )

zw

az bw ad bccz d

bz w b dd

az w c c wc

w

時 不在圓內

1, 0 a a cc

bza aw dc z c

21

01 0

1

0

1

1 0

1 1 0 0

1 1 0 0

1 1 0 0

Since 1 and 0

, 0 (5)

11, 11

1 1

or (1 )(1 ) (1 )(1 )

but

i

i

a b dc a c

z zw e z zz z

zz w ez

z z

z z z z

z z z z

z z z z

1 0

1 0 1 0

1 0

1 0 1 0

. . Re Re

, or if , (5) is a constant transformation

,

i e z z

z z z zz z

z z z z

22

Z

0Z

0Z

0

0

(6)i z zw ez z

0

0

0

0

0

0

0

0

0

when , 0since 0 is inside 1

is above the axis.or Im 0

if is above the -axis-

1-

if is below the -axis-

1-

z z ww w

z xz

z zw

z z

z xz z

z z

z xz z

z z

0

0

if is on the -axis

1

(6) is what we want

z xz z

z z

23

Ex1. has the above mapping propertyii z z iw ei z z i

Ex2. 1 maps 0 onto 01

0 onto 0

zw y vz

y v

(1) real real Since the image of 0 is either a line or a circle. it must be the real axis 0.

z wy

v

2

2 2

1( -1)( 1) 2(2) Im Im Im( 1)( 1) 1 1

0, 0 0, 0

z z zz z yv wz z z z

y vy v

also linear fractional transformation is onto. . . .Q E D

24

1( , 2 )c

1( ,0)c 1ce

many-to-1 mapping

73. Exponential and Logarithmic Transformations

The transformation

,

Thus , 2 , any integer

or , transformation from plane to plane

z

i x iy i

x

x

w e

e e e w e z x iy

e y n n

e y z w

1

1

1

(1) vertical line

( , ), its image , c

x c

z c y e y

25

2C

2C

1-to-1 mapping

2(2) horizontal liney c

yd

c

D C

A B

a bx

'D

'C

'B

c d 'A

, maps onto ,

z

a b

w e

a x b c y d e e c d

Ex1

26

ib

ic

ia a

c

b Ex2.

0

x

y v

)2( i

0i

i0

u

log ln ( 0, 2 )w z r i r

any branch of log z , maps onto a strip

27

Ex3.1log1

principal branch

1is a composition of and log1

maps upper half plane 0 onto

zwz

zZ w Zz

y

upper half plane 0v

maps upper half planeonto the strip 0 v

(0 )

28

74. The transformation sinw z

Since sin sin cosh cos sinh sin

sin cosh , cos sinh

z x y i x yw zu x y v x y

Ex1.

sin maps

, 0 onto 02 2

w z

x y v

(1-to-1)

E

D

2

2

B

AM L

c

y

x

'M 'L

'E 'D 'B 'A

1

29

. boundary of the strip real axis (1) BA segment

, 0 cosh2 2

cosh , 0 sinh2

(2) DB segment 0

y y

y y

A

e ex y y

e eu y v y

y

sin 0

(3) DE segment

- , 02

cosh , 0

u x v

x y

u y v

30

1 1

1 1

2 2

2 21 1

B. Interior of strip maps onto upper half 0 of plane

line 02

sin cosh , cos sinh (- )

1 hyperbolasin cos

with foci at the points

v w

x c c

u c y v c y y

u vc c

w

2 21 1sin cos 1c c

31

2 2

2 2

2 2

2 21 2

2 22 2

Consider a horizontal line segment , , 0its image is sin cosh , cos sinh

1 an ellipsecosh sinh

with foci at cosh sinh 1

y c x cu x c v x c

u vc c

w c c

A B C D E

2

0

2

x

y

02 Cy

v

u'B

'A 'E

'D

'C

1 1

32

D

E

bi C B

LAF

2

2

'D 'E

'C

'L

'A 'B1 1

Ex2.

2 0, sin , 0 ( )2 2

c u x v x

Ex3. cos sin( )2

, sin2

z z

Z z w Z

Ex4. sinhsin( ), sin

cosh cos( )

w zw i izZ iz W Z w iW

w ziz

33

75. Mapping by Branches of1

2z1

2

12

12

12

are the two square roots of when 0if exp( ) ( 0, )

( 2 )then exp ( 0,1)2

principal root exp2

can also be written1 exp( log ) 02

The principal bra

z z zz r i r

i kz r k

ir

z

z z z

12

0

0

0

nch ( ) of is obtained bytaking the principal branch of log

1 ( ) exp( log ) ( 0, )2

or ( ) exp ( 0, )2

F z zz

F z z z Argz

iF z r r

34

C

D

2R

1R

B

A2

12w z

v

'D

'C

'B

2'A

'2R

'1Ru

y

x

D

C

A

B2 Zsin

y

x

'D

'C 'B 'A

)(0 zF

v

u''C ''B ''A

''D

Ex1

Ex2

0 2,02

r 0 2, 04

0

0

(sin )sin , ( ) ( 0, )

w F zZ z w F Z z Arg z

35

12

1

0

12

when and the branch log ln ( 2 ) is used,

( 2 ) ( ) exp2

exp2

= ( )

other branches of

( ) exp 2a

z r iiz F z r

ir

F z

zif z r

1

( 0, 2 )

1 ( 2 )exp( log ) exp , 0,1,2,... 1nn

r

i kz z r k nn n

2 3

36

76. Square roots of polynomials

Ex1. 12

0

12

0

1 12 2

0

12

12

Branches of ( ) is a composition of

with

Each branch of yields a branch of ( )

When e , branches of are

exp ( 0, 2 )2

If we write

i

z z

Z z z w Z

Z z z

Z R ZiZ R R

0 0 0

12

0

0

0

, ( - ) and arg( )

two branches of ( ) are

( ) exp ( 0, )2

and ( ) exp ( 0,0 2 )2

R z z Arg z z z z

z ziG z R R

ig z R R

37

0 ( ) is defined at all points in the plance except 0and the ray .G z z z

Argz

0

0 0

The transformation ( ) is a one-to-onemapping of the domain - 0, ( )

onto the right half Re 0 of the -plane

w G z

z z Arg z z

w w

y

x

v

u

0z

z

R

x

y

R

Z

R

2

w

0

0 0

The transformation ( ) maps the domain 0, 0 arg( ) 2

in a ont-to-one manner onto the upper half plane Im 0

w g z

z z z z

w

1 1 12 2 2 2( 1) ( 1) ( 1) ( 1)z z z z Ex.2