Courtesy of Prof. David R. Jackson ECE Dept. Branch Points and Branch Cuts ECE 6382 8/24/10.

Courtesy of Prof. David R. JacksonECE Dept.

Branch Points and Branch Cuts

ECE 6382 ECE 6382

8/24/10

PreliminaryPreliminary

Consider 1/ 2f z z jz r e

1/21/2 /2j jz r e r e

1z r 0 : 1/2 1z

2 : 1/2 1z

4 : 1/2 1z

There are two possible values.

Choose

Branch Cuts and PointsBranch Cuts and Points

The concept is illustrated for 1/ 2f z z jz r e

1/ 2 / 2jz r e

x

y

AB C

r = 1

Consider what happens if we encircle the origin:

Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)1/ 2 / 2jz r e

1/2

A 0 1

B +

C 2 -1

z

j

point

We don’t get back the same result!

x

y

AB C

r = 1

r

Now consider encircling the origin twice:

1/ 2 / 2jz r e

1/2

A 0 1

B +

C 2 -1

D 3 -

E 4 1

z

j

j

point

We now get back the same result!

Hence the square-root function is a double-valued function.

Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)

x

y

AB CD

E

r = 1

r

Now consider encircling a point z0 not at the origin

1/ 2 / 2jz r e

Unlike encircling the origin, now we return to the same result! Only the point at the origin must be encircled twice to return to the starting value.


x

y

A

B increasing

decreasing

0z

In order to make the square-root function single-valued, we insert a “barrier” or “branch cut”.

The origin is called a branch point: we are not allowed to encircle it if we wish to make the square-root function single-valued.

x

branch cut

y

Here the branch cut is chosen to lie on the negative real axis (an arbitrary choice)


We must now choose what “branch” of the function we want.

x

branch cut

jz r e 1/ 2 / 2jz r e

y

1z 1/2 1z


Here is the other branch choice.

x

branch cut

jz r e 1/ 2 / 2jz r e

y

3

1z 1/2 1z


Note that the function is discontinuous across the branch cut.

x

branch cut

jz r e 1/ 2 / 2jz r e

y

1z 1/2 1z

1,z

1,z

1/2z j

1/2z j


The shape of the branch cut is arbitrary.

x

branch cut

jz r e

1/ 2 / 2jz r e y

/ 2 3 / 2

1z 1/2 1z


Second branch is

3 / 2 7 / 2

The branch cut does not even have to be a straight line

jz r e 1/ 2 / 2jz r e

x

branch cut

y

1z 1/2 1 ( 0)z

In this case the branch is determined by requiring that the square-root function change continuously as we start from

a specified value (e.g., z = 1).

1z 1/ 2z j

z j 1/ 2 / 4 1 / 2jz e j

z j

1/ 2 / 4 1 / 2jz e j


Branch points appear in pairs; here one is at z=0 and the other at z= ∞ as determined by examining ζ = 1/ z at ζ=0

jz r e 1/2 ( /2 ) ,

0,1

j kz r e

k

Hence the branch cut for the square-root function connects the origin and the point at infinity


1/2 1/2 ( /2 )11/ ; 0,1j kz e k

r

Consider this function:

1/ 22( ) 1f z z


What do the branch points and branch cuts look like for this function?

1/ 2 1/ 21/ 2 1/ 2 1/ 22( ) 1 1 1 1 1f z z z z z z


x

y

1

1

There are two branch cuts: we are not allowed to encircle either branch point.

1/ 21/ 2 1/ 2 1/ 21 2( ) 1 1f z z z w w


Geometric interpretation

x

y

1

11w

2w

12

1 2/ 2 / 21 2( ) j jf z r e r e

1

2

1 1

2 2

1

( 1)

j

j

w z r e

w z r e

1/ 2 1/ 21/ 22( ) 1 1 1f z z z z


x

y

1

1

We can rotate both branch cuts to the real axis.

1/ 2 1/ 21/ 22( ) 1 1 1f z z z z


x

y

1

1

Note that the function is the same at the two points shown.

The two branch cuts “cancel”

1x

1/ 2 1/ 21/ 22( ) 1 1 1f z z z z


An alternative branch cut

x

y

1

1

Note: we are allowed to encircle both branch points, but not only one of them!

1/ 2 1/ 21/ 22( ) 1 1 1f z z z z


x

y

1

1 1234

5 6 7

Suppose we agree that at point 1, 1 = 2 = 0. This should uniquely determine the branch of the function everywhere in the complex plane.

Find the angles 1 and 2 at the other points labeled.

1/ 2 1/ 21/ 22( ) 1 1 1f z z z z


2

1 0 0

2 0

3 0

4

5 2

6 2

7 2 2

1point

x

y

1

1 1234

5 6 7

Riemann SurfaceRiemann Surface

A Riemann surface is a surface that combines the different sheets of a multi-valued function.

It is useful since it displays all possible values of the function at one time.

Riemann Surface (cont.)Riemann Surface (cont.)

The concept of the Riemann surface is illustrated for

1/ 2f z z

The function z½ is analytic everywhere on this surface (there are no branch cuts). It also assumes all possible values on the surface.

1/ 2

1/ 2

(1 1)

3 (1 1)

Top sheet:

Bottom sheet:

Consider this choice:

jz r e

The Riemann surface is really two complex planes connected together.


x

y

BD

top

bottom

BD

D

B

B

Dside view

x

y

top view


bottom sheet

top sheet

branch cut

branch point


connection betweensheets

1/2

A 0 1

B +

C 2 -1

D 3 -

E 4 1

z

j

j

point

1/ 2 / 2 / 2( 1) j jf z z e e

jz re

x

y

AB CD

EBD

r = 1

r


1/22 1f z z

There are two “escalators” that now connect the top and bottom sheets of the surface.

y

x1

1

escalator

escalator


1/22 1f z z

bottom sheet

y

x1

1

3f z

3f z

2z

top sheet:

Define by 1 = 2 = 0 on real axis for x > 1

1/21/2( ) 1 1f z z z

The angle 1 has changed by 2 as we go back to the point z = 2.

Other Multiple-Branch FunctionsOther Multiple-Branch Functions

1/3 1/3 /3jf z z r e three sheets

ln( ) ln lnjf z z r e r j infinite number of sheets

lnln lnr j jz rf z z e e e e

infinite number of sheets

The power is an irrational number.

4/5 4/5 4 /5jf z z r e five sheets

Courtesy of Prof. David R. Jackson ECE Dept. Branch Points and Branch Cuts ECE 6382 8/24/10.

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Transcript of Courtesy of Prof. David R. Jackson ECE Dept. Branch Points and Branch Cuts ECE 6382 8/24/10.