D. R. WiltonECE Dept.

ECE 6382 ECE 6382

Functions of a Complex Variable as Mappings

8/24/10

A Function of a Complex Variable as a MappingA Function of a Complex Variable as a Mapping

w f z

z w

A function of a complex variable, , is usually viewed

as a from the complex to the complex plane.mapping

z

z

x

yw

u

vw

w f z

Simple Mappings: TranslationsSimple Mappings: Translations

a

w a z

Translation:

where is a complex constant.

z

z

x

yw

u

vww za

a

z

z

a .

The mapping translates every point in the plane

by the "vector"

Simple Mappings: RotationsSimple Mappings: Rotations

ii i iw e z e re re

Rotation:

where is a constant. real

z . The mapping rotates every point in

the plane through an angle

z

z

x

yw

u

v

w

iw e z

z

Simple Mappings: DilationsSimple Mappings: Dilations

( )i iw az a re ar e

a

Dilation (stretching) :

where is a constant. real

arg

z z

a

z A A A

A

The mapping magnifies the magnitude of a point

in the complex plane by a factor .

Multiplying by a number is equivalent to

stretching it by a factor

rotatin

complex

and arg Ag it through an angle

z

z

x

y w

u

v

ww za

z

Simple Mappings: InversionsSimple Mappings: Inversions

1 1 i

ie

wz rre

Inversion:

Inversions have a "circle preserving" property, i.e., circles

always map to circles (straight lines are "circles" of infinite radius).

Points the unit circle map to points the unit ciroutside inside cle

and vice versa.

0z x iy

Re0x

01 x

1

1 z

Im

0z x iy The straight line maps to a circle

z

Re

Im

11 z

Inversion circle -preserving property

Inversion – Line-to-Circle PropertyInversion – Line-to-Circle Property

0z x iy

Re0x

01 x

1

1 z

Im

0

002 2 2

0

02 2 2 20 0

1

1 2

1 2

z x iy

z* x iyw u iv x y

z x yz

x yu v

x y x y

y

u x

v

( is a constant, is a parameter)

To prove the straight line maps to a circle :

, ,

merely parameterizes the line, so eliminate it :

0 0 3

3 2

x v, yy u

v

Substituting from into 0x v

2

02 2

2 00

u ux vuux v

xu

2 2 2 20 0

22 2 2 2 2 20 0 0 0 0

222

0

0 0

12 01

2 0

1 12 4

1 12 2

0

u,v ,

x

x

x x

.u x v x

u x ux v x ux v x

u v

Eq. of circle of radius ,

centered at

Rearranging,

or

Re01 x

0x1 z

Im

Inversion – Line-to-Circle Property, cont’dInversion – Line-to-Circle Property, cont’d

1 1

i

ii

z e z z d it ,

w eze z

Multiplying by rotates the line by the angle

but the circle is rotated in the direct on:

iopposite

Re

d

1 z

Im z

1 d

Re1 dd

1 z

Imz d it

z d it , t ,

x d .

is the equation of vertical line

passing through on the x - axis

( ), < <

1 1< <

i

i

z

z e d it t

z

w e , tz d it

Since every straight line in the - plane may be

represented as

-

via the transformation

every straight line in the plane is mapped into

a circle

Simple Mappings: Inversions, cont’dSimple Mappings: Inversions, cont’d

1 1 i

ie

wz rre

Geometrical construction of the inversion

1w

z

w

u

v

w

z

1

z

x

y

z

w

u

v

w

A General Linear Transformation (Mapping) Is a A General Linear Transformation (Mapping) Is a Combination of Translation, Rotation, and DilationCombination of Translation, Rotation, and Dilation

Arg Arg i Bi B iw A Bz A B e re A B r e

A ,B

rotation

dilation

translation

Linear transformation:

where are complex constants .

z

z

x

y w

u

v

w

w A Bz B z

A

Arg B

A General A General BilinearBilinear Transformation (Mapping) Is a Succession Transformation (Mapping) Is a Succession of Translations, Rotations, Dilations, and Inversionsof Translations, Rotations, Dilations, and Inversions

0

A Bzw

C DzA,B,C,D

D ,

A BC D B D C DzA Bz A BC Dw B D

C Dz C Dz C Dz

f C D ,

Bilinear (Fractional or Mobius) transformation:

where are complex constants .

Note that if

If we let (translation, dilation, rota

A BC Dg

A Bzw B D g f z

C Dz

tion)

and (inversion, dilation, rotation)

then the Mobius transformation may be written as

which is just a sequence of translations, dilations, rotations, and an inversion

Sinc

z

w

e each transformation preserves circles, bilinear transformations also

have the circle -preserving property : circles in the plane are mapped into circles in

the plane, with straight lines included as circles of infinite radius.

Bilinear Transformation Example: The Smith ChartBilinear Transformation Example: The Smith Chart

0

0

Zz r jx Z R jX

Z

Z

Let where is the impedance at a

point on a transmission line of characteristic impedance and

is the generalized reflection coefficient

0

0

1 1

1 1

Z Z z z

Z Z z z

or simply

at poi nt

.

Re

Im1

1

z

z

For an interpretation of mobius transformations as projections on a sphere, seehttp://www.youtube.com/watch?v=JX3VmDgiFnY

zr

jx

The Squaring TransformationThe Squaring Transformation

2 2 2iw f z z r e

x

yz

z

z

w

u

v

w

z w

z w

The transformation maps the -plane into the -plane

The entire -plane covers the -plane twice

The transformation is said to be

half entire

many - to - one

Another Mapping of the Squaring Another Mapping of the Squaring TransformationTransformation

2 2 2iw f z z r e

Re

Im

1 2 3

3

2

1

9

4

1

90o

180o

270o

360o

-180o-270o

-360o 0o

-90o

Re

Im

2z

The Square Root TransformationThe Square Root Transformation

21 1

2 2 2 0 1ki

w f z z r e , k ,

w

u

v

w

z w

The transformation maps the entire -plane into one half of the -plane

The transformation is said to be The inverse of the square

function (i.e., the square root function) is

one - to - many.

no . t unique

wx

yz

z

top sheet

bottom sheet

3

Connecting the Two Sheets at a Branch Cut to Form a Connecting the Two Sheets at a Branch Cut to Form a Riemann SurfaceRiemann Surface

21 1

2 2 2 0 1ki

w f z z r e , k ,

bottom sheet

branch point

branch cut

arg

arg

z

If we agree to limit to one sheet, never crossing the branch cut,

the inverse is unique. Physics usually dictates the branch choice and cut.

The two sheets shown may be defined as

1)

arg 3

z

z

(principal branch)

2)

x

yz

z

3

Only at a branch point are multiple cycles encircling the point required to return to the starting value

top sheet

The Branch Cut Allows Us to Map the Square Root FunctionThe Branch Cut Allows Us to Map the Square Root Function

1 122 2 10

i kw f z z r e , k , ,

Re

Im

1 2 3

3

2

11

22.5o

45o

67.5o

90o

-45o-67.5o

-90o 0o

2

3

-22.5o

top sheet, k = 0

Re

Im

1 2 3

3

2

11

202.5o

225o

247.5o

270o

135o

90o 180o

2

3

157.5o

bottom sheet , k = 1

112.5o

Re

Im1

2z

Constant Constant uu and and vv Contours are Orthogonal Contours are Orthogonal z x, y u x, y v x, y

u uˆ ˆux y

v vˆvx y

x y

x

Consider contours in the plane on which the real quantities and

are constant.

The directions normal to these contours are along the gradient direction :

ˆ

u u u uˆ ˆ ˆ

vˆ ˆ ˆv u

y xˆ ˆu v

x y xx yy

u

y

x y x y x y x y

The gradients, and therefore the contours, are orthogonal (perpendicular) by the C. R. conditions : C.R. cond's

0u u u u

y x y x

x

yz

u

constantu

constantv

v

u

vw

Mappings of Analytic Functions are Mappings of Analytic Functions are ConformalConformal (Angle-Preserving) (Angle-Preserving)

1 2

0 0 00

C C z x iy

w u iv

f z w f z

Consider a pair of intersecting paths in the plane mapped

onto the plane.

If at the intersection of their mapping, then the angles

between the intersecting c

,

,

onto

1 0 1 1 0 1 1 1

2 0 2 2 0 2 2 2

2 1 2 1

2 1

Arg Arg Arg

Arg Arg Arg

Arg Arg Arg Arg

0

w f z z w f z z , z C

w f z z w f z z , z C

w w z z

z , z ,

urs are preserved:

along

along

The approximation improves as with

2 1

2 1

Arg Arg

Arg Arg

z z

w w

and the above approximation yielding

u

v

w 1C

2C0w

x

y

z1C

2C

0z

0

2

0

Arg

0

0

Arg0

0

2 0 0

Arg Arg Arg 2Argz

z

z

f z

w f z z z

w f z z z z; f

w z z z

The condition below is

needed since is !

E.g., consider at :

since ,

indeterminant

Constant |Constant |w|w| and Arg and Arg ww Contours are also Contours are also Orthogonal Orthogonal

1

iw Re R

z f w z

If the constant and contours are orthogonal.

If is a mapping back to (a branch of) the plane,

the mapping preserves the orthogonality.

-45o

Re

Im

1 2 3

3

2

11

22.5o

45o

67.5o

90o

-67.5o

-90o 0o

2

3

-22.5o

z

12f z z Magnitude, phase plot of :

w

u

v

The Logarithm FunctionThe Logarithm Function2 ln

2 1 2

z z i k

z

w e e z w

w z e w

z z i k k , ,

Since , the inverse function cannot be unique.

I.e., given , is the complex number such that . But this

number cannot be unique, since , ,

has the propalso

2

( 2 )2 2

ln

ln 2 0 1 2

ln 2 0 1 2

z z i k z

i v ki w w i k u iv i k u

u

e e e w

w z w

z z e e e e e e

z e u z , v k , k , , ,

w u iv z i k , k , , ,

erty that .

Let . To evaluate , note that we have

so that

and hence

which has 0z a branch point with an infinite number of branches.

x

yz

x

y

Infinite # of sheetsBranch cut

Branch point

Riemann surface for the log function

Arbitrary Powers of Complex NumbersArbitrary Powers of Complex Numbers

ln

ln 2ln

ln

0

Since ,

also has an infinite number of values (branches), unless integer for some ,

i.e., unless is real ( ) and rational. r i i

z z

a z ai ka a z

z e e

z e e

ak k

a a i a a

x

y

x

yz

Infinite # of sheetsBranch cut

Branch point

Riemann surfaces for za