Transcript Lect8Singularities

ECE 6382
Singularities
D. R. Wilton
Adapted from notes of
Prof. David R. Jackson
ECE Dept.
8/24/10
Singularity
A point zs is a singularity of the function f(z) if the function is not analytic
at zs.
(The function does not necessarily have to be infinite there.)
Recall from Liouville’s theorem that the only function that is analytic
and bounded in the entire complex plane is a constant. I.e., the only
function analytic everywhere (including the point at infinity) is a
constant. Hence, all analytic functions that are not constants have
singularities somewhere (possibly at infinity).
Taylor Series
Taylor’s theorem
If f(z) is analytic in the region
R
z  z0  R ,
z0
zs
then

f  z    an  z  z0 
n
for
n 0
an 
f
n
 z0 
n!
1

2 i
f  z
 z  z 
C
n 1
z  z0  R
dz
0
If a singularity exists at radius R, then the series diverges for
z  z0  R
Note: R is called the radius of convergence of the Taylor series.
Laurent Series
Laurent’s theorem
b
If f(z) is analytic in the region
a
a  z  z0  b ,
z0
then
f  z 

 a z  z 
n 
an 
n
n
0
f  z
1
2 j   z  z 
C
n 1
for
a  z  z0  b
dz , n  0, 1, 2
0
Note: this theorem also applies to an isolated singularity (a→ 0).
Isolated singularity: the function is singular at z0 but is analytic for 0  z  z0  
Taylor Series Example
Example:
y
1
f  z 
z 1
1
f  z 
  1  z  z 2  z 3 
1 z
zs  1

From the Taylor theorem we have:

1
z 
,

1 z
n 0
n
z 1

n
z

n 0
diverges ,
z 1
x
Taylor Series Example
Example:
y
f  z   z1/ 2

Expand about z0 = 1:
f  z    an  z  1
n
1
x
n 0
R 1
a0  1
a1 
1  1 3/ 2 
1
z



1!  2
 z 1 2
a2 
1  1  3  5/ 2 
3

z


 


2!  2  2 
8
 z 1
etc.
    
The series converges for
The series diverges for
z 1  1
z 1  1
Laurent Series Example
y
1/ 2
Example:
z
f  z 
z 1
Expand about z0 = 1:
f  z 

 an  z 1
isolated singularity
at zs = 1
1
n
n 
R 1
    
From the previous example:
a 1  1
a0 
1
2
a1  
etc.
3
8
The series converges for
The series diverges for
0  z 1  1
z 1  1
x
Singularities
Examples of singularities:
(These will be discussed in more detail later.)
T
sin  z 
removable singularity at z = 0
z
1
L
 z  z0 
L
e1/ z
N
1
1
sin  
z
N
z1 / 2
p
If expanded about
the singularity, we
can have:
T = Taylor
L = Laurent
N = Neither
pole of order p at z = z0 ( if p = 1, pole is a simple pole)
isolated essential singularity at z = z0 (pole of infinite order)
non-isolated essential singularity z = 0
branch point
(not an isolated singularity)
Isolated Singularity
Isolated singularity:
The function is singular at z0 but is analytic for
 0
z0
Examples:
sin z 1 1/ z
1
, , e ,
at z  0
z
z
sin z
0  z  z0  
Singularities
Isolated singularities
removable singularities
sin  z  1  cos  z 
,
z
z2
poles of finite order
1 1
, 2,
z z
isolated essential singularities
1
 z  z0 
m
,
1
sin   , e1/ z
z
2z  3
 z  1 ( z  2)
2
These are each discussed in more detail next.
Isolated Singularity: Removable Singularity
Removable singularity:
The limit z →z0 exists and f(z) is made analytic by defining
f ( z0 )  lim f  z 

z  z0
Example:
z0
sin  z 
z
lim
z 0
sin  z 
z
L'Hospital's
Rule

lim
z 0
cos  z 
1
1
Isolated Singularity: Pole of Finite Order
Pole of finite order (order P):
f z 

 a z  z 
n  P
n

n
s
zs
The Laurent series expanded about the singularity terminates
with a finite number of negative exponent terms.
Examples:
f  z 
1
f  z   , ( P  1)
z
3
 z  3
3

2
 z  3
2

simple pole at z = 0
1
 1   z  3 
 z  3
, ( P  3)
pole of order 3 at z = 3
Isolated Singularity: Isolated Essential Singularity

Isolated Essential Singularity:
f z 


n  
an  z  z s 
n
zs
The Laurent series expanded about the singularity has an
infinite number of negative exponent terms
Examples:
n
1 1
1
n 1  1 
1  1
f  z   sin      1     3 

5
z 6 z 120 z
 z  n 1 n !
z
odd
f  z  e
1/  z  zs 
n
1 1 
1
1
1
 

1





2
3
n
!
z

z
z

z
 s  2  z  zs  6  z  zs 
n 0
s 


Isolated Essential Singularity: Picard’s Theorem
The behavior near an isolated essential singularity is pretty wild:
Picard’s theorem:
In any neighborhood of an isolated essential singularity, the
function will come arbitrarily close to every complex number.

zs
Example :
1
z
e e
e i
r
e
cos
r
e
 i sinr 
 z0  r0ei0 , a given arbitrary complex number
 cos   r ln r0 , sin    r 0  2n 
2
cos2  sin 2  1  r 2 ln 2 r0  0  2n  


n 
n 
1
1   0  2n 
r
 0,   tan
  / 2
1
2
2
ln r0
ln 2 r0  0  2n  


Picard’s Theorem (cont.)
Example (cont.)
zs

r
n 
1
ln 2 r0   0  2n  


2
1
 0,
2
  0  2n  n
  tan
  / 2
ln r0
1
( z0  r0ei0 , a given arbitrary complex number)
This sketch shows that as n increases, the points where the function exp(1/z)
equals the given value z0 "spiral in" to the (essential) singularity.
You can always find a solution now matter how small  (the “neighborhood”) is!
Essential Singularities
Essential Singularities
A singularity that is not a removable singularity, a pole of finite order, or a
branch point singularity is called an essential singularity.
They are the singularities where the behavior is the “wildest”.
Two types: isolated and non-isolated.
Laurent series about the
singularity has an infinite
number of negative exponent
terms.
A Laurent series about the
singularity is not possible,
in general.
Classification of an
Isolated Singularity at zs
f z 

 a z  z 
n  
n
s
n

 a p  z  z s 
p

 a1  z  z s   a0  a1  z  z s   a2  z  z s 
1
Analytic
Simple Pole
Pole of Order p
Isolated Essential Singularity
Remember that in classifying a singularity, the series is expanded
about the singular point, z0  zs !
2
Non-Isolated Essential Singularity
Non-Isolated Essential Singularity:
By definition, this is an essential singularity that is not isolated.
y
Example:
f z 
1
1
sin  
z
X
X
X

X

X
X
x
zs  0
simple poles at:
1
z
m
X XXX XXX X
(Distance between successive
poles decreases with m !)
Note: A Laurent series expansion in a neighborhood of zs = 0 is not possible!
Branch Point
Branch Point:
This is another type of singularity for which a Laurent expansion
about the point is not possible (not an isolated singularity).
y
Example:
f  z   z1/ 2
x
zs  0
not analytic on
the branch cut
Singularity at Infinity
We classify the types of singularities at infinity by letting
w
1
z
Example:
f  z   z3
f  z   g  w 
1
w3
pole of order 3 at w = 0
The function f(z) has a pole of order 3 at infinity.
Note: when we say “finite plane” we mean everywhere except at infinity.
The function f(z) in the example above is analytic in the finite plane.
Other Definitions
Entire:
The function is analytic everywhere in the finite plane.
Examples:
f  z   ez , sin z, 2z 2  3z  1
Meromorphic:
The function is analytic everywhere in the finite plane
except for a finite number of poles of finite order.
Example:
f  z 
sin z
 z  1 z  1
3