Notes 9 Singularitiescourses.egr.uh.edu/ECE/ECE6382/Class Notes/Notes 9... · Singularity. A point...
Transcript of Notes 9 Singularitiescourses.egr.uh.edu/ECE/ECE6382/Class Notes/Notes 9... · Singularity. A point...
Singularities
ECE 6382
Notes are from D. R. Wilton, Dept. of ECE
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David R. Jackson
Fall 2019
Notes 9
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.
Hence, all non-constant analytic functions have singularities somewhere (possibly at infinity).
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Taylor Series
Since f (z) is analytic in the region
0 cz z R− <
then
( ) ( )00
nn
nf z a z z
∞
=
= −∑ 0 cz z R− <for
The series converges for |z-z0| < Rc.
0zcR
sz
( ) ( ) ( )( )
01
0
1! 2
n
n nC
f z f za dz
n i z zπ += =−∫
The radius of convergence Rc is the distance to the closest singularity.
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The series diverges for |z-z0| > Rc (proof omitted).
Laurent Series
If f(z) is analytic in the region 0a z z b< − <
then
( ) ( )0n
nn
f z a z z∞
=−∞
= −∑ 0a z z b< − <for
( )( ) 1
0
1 , 0, 1, 22n n
C
f za dz n
i z zπ += = ± ±−∫
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0z
ba
szsz
The series diverges outside the annulus (proof omitted).
The series converges inside the annulus.
Taylor Series Example
Example: ( ) 11
f zz
=−
( ) 2 31 11
f z z z zz
− = = − + + + + −
0
0
1 , 11
, 1diverges
n
n
n
n
z zz
z z
∞
=
∞
=
= <−
>
∑
∑
From the property of Taylor series we have:
The point z = 1 is a singularity (a first-order pole).
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1sz =
x
y
×0z
Example: ( )f z z=
( ) ( )0
1 nn
nf z a z
∞
=
= −∑
1cR =
0
3/21
1
5/22
1
1
1 1 11! 2 2
1 1 3 32! 2 2 8
z
z
a
a z
a z
−
=
−
=
=
= =
= − = −
etc.
Expand about z0 = 1:
The series converges for 1 1z − <
The series diverges for 1 1z − >
Taylor Series Example
π θ π− < <
1 x
y
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( ) ( )21 31 1 12 8
z z z= + − − − +
Example: ( )1
zf zz
=−
( ) ( )1 nn
nf z a z
∞
=−∞
= −∑
1
0
1
112
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a
a
a
− =
=
= −
etc.
Expand about z0 = 1:
The series converges for 0 1 1z< − <
The series diverges for 1 1z − >
Laurent Series Example
Using the previous example, we have:
π θ π− < <
(The coefficients are shifted by 1 from the previous example.)
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( )1 1 3 11 1 2 8
z zz z
= + − − +− −
1x
y
×
Isolated SingularityIsolated singularity:
The function is singular at zs but is analytic for 0 sz z δ< − <
Examples: 1/sin 1 1, , , 0sin
zz e zz z z
=at
sz
0δ >
A Laurent series expansion about zs is always possible!
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0,a a b δ> < <This is a special case of a Laurent series with .
(for some δ)
Non-Isolated Singularity
Non-Isolated Singularity:By definition, this is a singularity that is not isolated.
Example:
( ) 11sin
f z
z
=
Simple poles at:
1zmπ
=
Note: A Laurent series expansion in a neighborhood of zs = 0 is not possible!
(Distance between successive poles decreases with m !)
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Note: The function is not analytic in any region 0 < |z| < δ.
( )0sz = non - isolated
XX XX XXππ−
x
y
XX X XX X XX
Branch Point:
This is a type of non-isolated singularity.
Example:
( ) 1/ 2f z z=
0sz =Not analytic at the branch point.
x
y
Non-Isolated Singularity (cont.)
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Note: The function is not analytic in any region 0 < |z| < δ.
Note: A Laurent series expansion in a neighborhood of zs = 0 is not possible!
Examples of SingularitiesExamples: (These will be discussed in more detail later.)
( )1
psz z−
pole of order p at z = zs ( if p = 1, pole is a simple pole)
1/ ze essential singularity at z = z0 (pole of infinite order)
1/ 2z branch point (not an isolated singularity)
( )sin zz
removable singularity at z = 0
11sinz
non-isolated singularity z = 0
T
L
L
N
N
If expanded about the singularity, we can have: T = TaylorL = LaurentN = Neither
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Classification of Isolated Singularities
Isolated singularities
Removable singularities Poles of finite order
( ) ( )2
sin 1 cos,
z zz z
−
( )
( )
2
2
1 1 1, , ,1
2 31 ( 2)
mz z zz
z z
−
+
− +
Essential singularities(poles of infinite order)
1/1sin , zez
These are each discussed in more detail next.
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Isolated Singularity: Removable Singularity
The limit z → z0 exists and f(z) is made analytic by defining
Example: ( )sin zz
( )0
0( ) limz z
f z f z→
≡
sz
δ
( ) ( )0 0
sin coslim lim 1
1z z
z zz→ →
= =
L'Hospital's Rule
Removable singularity:
Laurent series → Taylor series
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Pole of finite order (order P):
( ) ( )P
nn s
nf z a z z
∞
=−
= −∑sz
δ
The Laurent series expanded about the singularity terminateswith a finite number of negative exponent terms.
Examples:( ) 1 , ( 1)f z
zP= =
( )( ) ( ) ( ) ( )3 2
3 2 1 1 3 , ( 3)33 3
f z zzz
Pz
= + + + + − + =−− −
Isolated Singularity: Pole of Finite Order
simple pole at z = 0
pole of order 3 at z = 3
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Isolated Singularity: Essential Singularity
Essential Singularity(pole of infinite order):
( ) ( )nn s
nf z a z z
−∞
∞
=
= −∑
The Laurent series expanded about the singularity has an infinite number of negative exponent terms.
Examples:
( ) 1/z2 3
0
1 1 1 1 11! 2 6
n
nf z e
n z z z z
∞
=
= = = + + + +
∑
( ) ( ) 13 5
1
1 1 1 1 1 1sin 1! 6 120
nn
nodd
f zz n z z z z
∞+
=
= = − = − + +
∑
sz
δ
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Graphical Classification of an Isolated Singularity at zs
( ) ( ) ( ) ( ) ( ) ( )1 21 0 1 2
n pn s p s s s s
nf z a z z a z z a z z a a z z a z z
∞− −
− −=−∞
= − = + − + + − + + − + −∑
Analytic or removable
Simple pole
Pole of order p
Essential singularity
Isolated
singularities
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Laurent series:
Picard’s TheoremThe behavior near an essential singularity is pretty wild.
Picard’s theorem:In any neighborhood of an essential singularity, the function will assume every complex number (with
possibly a single exception) an infinite number of times.sz
δ
( ) 1/zf z e=
For example:
No matter how small δ is, this function will assume all possible complex values (except possibly one).
(Please see the next slide.)
17Picard
Picard’s Theorem (cont.)
( )
( )( )
0
01/
22 2 2
0
0 0
20 0
0
0 02
cos sin
1/
cos ln , sin 2
cos s n
(
in 1 l 2
iiz i ni
i
er r r
ze
r
w R e
w R ez re e e e er n
r
R
nR
θπθ θ θ
θ θ π
θ θ π
−Θ−
Θ
+
=
= = = =
= = − +
+ = = + +
=
⇒ =
⇒ Θ
⇒ Θ
a given arbitrary complex number)
Example: ( ) 1/zf z e=
Set
( )( )1
22
0
00 0
21 0, tan / 2lnln 2
n nnr
RR n
πθ π
π
→∞ →∞− Θ− +
= → = → −+ Θ +
Hence
The “exception” here is w0 = 0 (R0 = 0). 18
0sz =
δ
Take the ln of both sides.
Picard’s Theorem (cont.)
0, / 2n n
r θ π→∞ →∞
= → = → −
Example (cont.)
This sketch shows that as n increases, the points where the function exp(1/z)equals the given value w0 “spirals in” to the (essential) singularity.
You can always find a solution for z now matter how small δ (the “neighborhood”) is!
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( ) 1/zf z e=
0sz = x
y
δ
Picard’s Theorem (cont.)
Example (cont.)
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( ) 1/zf z e=
Plot of the function exp(1/z), centered on the essential singularity at z = 0. The color represents the phase, the brightness represents the magnitude. This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).
https://en.wikipedia.org/wiki/Essential_singularity
cos sin1/ ir rze e eθ θ−=
iz re θ=
Singularity at Infinity
Example:
( ) 3f z z=
( ) ( ) 3
1f z g ww
= = 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.
We classify the types of singularities at infinity by letting w = 1/z
and analyzing the resulting function at w = 0.
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Other Definitions
Meromorphic: The function is analytic everywhere in the finite plane except for isolated poles of finite order.
( )( )( )3
sin 1, ( )sin1 1
zf z g zzz z
= =− +
Examples:
Entire: The function is analytic everywhere in the finite plane.
Examples:( ) 2, sin , 2 3 1zf z e z z z= + +
Meromorphic functions can always be expressed as the ratio of two entire functions, with the zeros of the denominator function as the poles (proof omitted).
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