Singularities

ECE 6382

Notes are from D. R. Wilton, Dept. of ECE

1

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 += = ± ±−∫

4

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

6

( ) ( )21 31 1 12 8

z z z= + − − − +

Example: ( )1

zf zz

=−

( ) ( )1 nn

nf z a z

∞

=−∞

= −∑

1

0

1

112

38

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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