Post on 05-Jan-2016
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.
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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)
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
Here is the other branch choice.
x
branch cut
jz r e 1/ 2 / 2jz r e
y
3
1z 1/2 1z
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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 Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
1/2 1/2 ( /2 )11/ ; 0,1j kz e k
r
Consider this function:
1/ 22( ) 1f z z
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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
Branch Cuts and Points (cont.)Branch Cuts and Points (cont.)
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.
Riemann Surface (cont.)Riemann Surface (cont.)
x
y
BD
top
bottom
BD
D
B
B
Dside view
x
y
top view
Riemann Surface (cont.)Riemann Surface (cont.)
bottom sheet
top sheet
branch cut
branch point
Riemann Surface (cont.)Riemann Surface (cont.)
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
Riemann Surface (cont.)Riemann Surface (cont.)
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
Riemann Surface (cont.)Riemann Surface (cont.)
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