D. R. Wilton ECE Dept. ECE 6382 Pole and Product Expansions, Series Summation 8/24/10.

D. R. WiltonECE Dept.

ECE 6382 ECE 6382

Pole and Product Expansions, Series Summation

8/24/10

Pole Expansion of Pole Expansion of Meromorphic FunctionsMeromorphic Functions

1 2 3

1

( ) , 1, , 0

( ) , 1,

( )

n

n n

N N

f z z a n a a a

f z b z a n

f z M N C R

N

Mittag-Leffler Theorem:

has simple poles at where

has residue at

, independent of , on circles of of radius that enclose

p n

oles,

1

( ) (0)

N N

n n

n n n

R

b bf z f

z a a

ot passing through any of them and such that

Then

Note that a pole at the origin is not allowed!

1Historical note: It is often claimed that friction between Mittag-Leffler and Alfred Nobel resulted in there being no Nobel Prize in mathematics. However, it seems this is not likely the case; see, for example, www.snopes.com/science/nobel.asp

http://www.snopes.com/science/nobel.asp

Proof of Mittag-Leffler TheoremProof of Mittag-Leffler Theorem

1

2

0

1 ( ) (0) ( )

2 ( ) ( )

( )1 ( ) 1

2 ( ) 2 ( )

N

N

N n

Nn

Nn n nC

iN N

iN N

NN NC

z a z a

bf w dw f f zI

i w w z z z a a z

w R e C

f R e R df w dwI

i w w z R R z

For , , consider the sequence of contour integrals

But we also have for on

1

2

2 NM R

NR

1 1

0( )

( ) (0) (0)( ) ( )

NN

n n n

n nn n n n

R z

zb b bf z f f

a a z z a a

Hence, taking the limit in the first equation above, NC

4C 2C

2Na 1Na

2a 1a

0w

w z

Rew

Imw

Extended Form of the Extended Form of the Mittag-Leffler TheoremMittag-Leffler Theorem

1 2 3( ) , 1, , 0

( ) , 1,

( )

n

n n

p

N N

f z z a n a a a

f z b z a n

f z M z N C R

Extended Mittag-Leffler Theorem:

has simple poles at where

has residue at

, independent of , on circles of of radius that encloses

1

1 1( )

1 ( )

2 ( )

/(0)( ) (0) (0)

!

N

N N

N pC

p pp pn n

n

N R

f w dwI

i w w z

b z az ff z f zf

p z a

poles, not passing through any of them and such that

Then considerations of the integral

lead to the expansion

1n

Example: Pole Expansion of cot Example: Pole Expansion of cot zz

coscot , 0, 1, 2,

sin( ) cos ( ) cos

lim lim 1sin sin 1

0

1cot

nz n z n

zz z n n

zz n z z n z

z z n

z

zz

has poles with residues

Then since a pole at is not allowed, consider

for which the singula

rity

at th

0 0

0

1 cos sinlim cot lim

sin

1

lim

z z

z

z z zzz z z

z

e origin has been removed and which has

the (finite) t

limi

2

2!z

z

33 5

2 43 0

1 1( )

3! 2 6lim 0

( )

3!

z

zz z

z zzz z

O

O

Example: Pole Expansion of cot Example: Pole Expansion of cot z (cont.)z (cont.)

2

2

coscot ,

sin(2 1)

2

cos cos cos sincot

sin cos sisin

iN N

N N

iN N

iNN

zz C z R e

zN

R C z n

R e R iz

R iR e

To check that is bounded on note that where

so that threads between the poles at , and that

2

2

2 2 2 2

2 2 2 2

2

n

cos cos cosh sin sin cos sinh sin

sin cos cosh sin cos cos sinh sin

cos cos cosh sin sin cos sinh sin

sin cos cosh sin cos cos sinh sin

cosh sin sin

N N N N

N N N N

N N N N

N N N N

N

R R i R R

R R i R R

R R R R

R R R R

R

2

2 2

2 2

cos

cosh sin cos cos

sinh cosh 1

N

N N

R

R R

x x

where we've used in both numerator and denom

inator.


2 22

2 2

2

cosh sin sin coscot

cosh sin cos cos

0 / 2 cot 1.88823

cot 1.88823 .

N N

N N

R Rz

R R

N z

z M N

Now if we plot the expression

for , for various values of , it appears that

so that , independent

of

|cot z|^2 on circular arcs R_n =(N-1/2)pi/2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2

Theta (radians)

|co

t z|

^2

N=1

N=2

N=3

N=4

N=8

• Actually, it isn’t necessary that the paths CN be circular; indeed it is simpler in this case to estimate the maximum value on a sequence of square paths of increasing size that pass between the poles


2 2 2 2 2 22

2 2 2 2

2

cos( ) cos cosh sin sinh cos cosh sin sinhcot

sin( ) sin cosh cos sinh sin cosh cos sinh

tanh

x iy x y i x y x y x yz

x iy x y i x y x y x y

y

Alternatively, we could consider the expression on the contours shown below :

2 2 2

2 22

2 222

1, 2 1 2

cos cosh sincoth coth 2, 2 1 2

sin cos sinh

1

cosh

si

.( )

02

h

( )

n

N

N NNC

x N

x y xy y N

x x y

f w dwI C

i w w z

y

y

It is easy to show that on these also

1C

3C

2C

cot coth2

z

3

2

2

5

2

cot coth2

z

cot coth2

z

coth (x) ―


12

1( ) cot

1 1 1 2cot cot

( ) , 1, 2, 1

(0) 0

( ) cot

N

n n

f z zz

C z z M Mz z N

f z a n n b

f

f z

Collecting our results, we now have (1) , where

on , , and

where (2) has poles at with (3) residues

and (4) . Hence

1 1

1 1

2 2 21

1(0)

( ) ( )

1 1 1 10

( ) ( )

1 2cot

n n n n

n nn n n n

n n

n

b b b bz fz z a a z a a

z n n z n n

zz

z z n

or rearranging and combining the two series term - by - term,

Ques. ( ) cotf z z z: Can we alternatively consider expanding ?

Other Pole ExpansionsOther Pole Expansions

2 2 2 2 2 21 1

2 20 02 2

2 20 02 2

2 2 2

1 1 2 1 1 2

sin sinh

2 1 2 11 1

cos cosh2 1 2 1

2 2

2 2tan tanh

2 1 2 1

2 2

1 2cot coth

n n

n n

n n

z z

z z z n z z z n

n n

z zn nz z

z zz z

n nz z

zz z

z z n

2 2 21 1

1 2

n n

z

z z n

• The Mittag-Leffler theorem generalizes the partial fraction representation of a rational function to meromorphic functions

Infinite Product Expansion of Infinite Product Expansion of Entire FunctionsEntire Functions

1 2 3

( )

( ) , 1, , 0

( )

( )

n

N N

f z

f z z a n a a a

f zM N C R

f z

Weierstrass's Factor Theorem:

is an entire function

has simple zeros at where

, independent of , on circles of radius that do not

pass

through

(0)

(0)

1

( )

( ) (0) 1 n

N N

zz faf

n n

f z R

zf z f e e

a

zeros of and such that

Then

There exists a generalization to multiple order zeros (Schaum's, p. 267)

Product Expansion Formula Product Expansion Formula

( ) ( ) ( )

( )

( ) 1 (ln ( ) ln ln ( )

( )

n

n

nn

f z f z z a g z

g z z a

f z d d d g zf z z a g z

f z dz dz dz z a

Proof :

At a simple zero, must have the form

where is analytic and non- vanishing at . Hence the

,

logarithmic

derivative,

1

0

)

( )

1

( ) (0) 1 1

( ) (0)

( ) ( ) (0)ln ( ) ln (0) ln

( ) (0)

n

n n n

z

g z

z a

f z f

f z f z a a

f z f z fdz f z f z

f z f f

has a simple pole at with residue ! By the Mittag-Leffler

theorem, which, on integrating, yields

,

,

1

1

(0) (0) (0)ln 1 ln 1(0) (0) (0)

1 1

ln(0)

( ) (0) (0) (0) 1n n nn n

n

n n n

z z z zz f z f z fa a a af f f

n n n

z a z

a a

zf z f e e f e e e f e

a

Upon exponentiating, we obtain the desired result,

.

n

z

ae

.

Useful Product ExpansionsUseful Product Expansions

2 2

2 2 2 21 1

2 2

2 22 21 1

1 1

2 2

2 2 2 2

2 2

2 22 24 4

2 1 2 1

sin 1 sinh 1

4 4cos 1 cosh 1

2 1 2 1

1 1tan tanh

1 1

n n

n n

n n

z zn n

z z

n n

z zz z z z

n n

z zz z

n n

z z z z

• Product expansions generalize for entire functions the factorization of the numerator and denominator polynomials of a rational function into products of their roots

( )

( ) ( )

( ) 0

0

;n

n n

M

n

n n

n

f z M z a

f z z a g z

g z z a M

M

Assume has a pole zero of order ( ) at we write

non - vanishin

where is analytic and at . Note for

poles and

g

or multiplicity

,

( )ln ( ) ln ln ( )

( )

( )

( )

1ln ( )

)

2

( 0

n n

n

n

n n

nnC

d f z d df z M z a g z

dz f z dz dz

M g z

z a g z

z a M

df z dz M

i dz

for zeros. Hence the logarithmic derivative

has a simple pole at with residue . Therefore,

where the sum

,

( )f z C is over the poles and zeros of enclosed by .

The Argument PrincipleThe Argument Principle

The Argument Principle (cont.)The Argument Principle (cont.)

zC

2a

11z a

23a

4a

3

4

1ln ( )

2

1 1ln ( ) ln ( )

2 2

C

C

df z dz N P

i dz

NC

P

df z dz f z

i dz i

number of zeros,where encircled by , counting multiplicities.

number of poles

Since the integrand is an exact differential, we also ha

ve

endpoint of

beginning point of

endpoint of endpoint of

beginning point of beginning point of

0

1 arg ( )ln ( )

2 2

1change in arg ( ) as goes around

2

C

C

CC

CC

f zf z

i

f z z C

This is the result from wh

ich the "argument principle" gets its name.

Summation of SeriesSummation of Series

The residue theorem is also frequently used to sum series. Some of the

most important results are obtained from integrals of the form over

the contour shown below and are summarized as followC

dz

C

2 12

2

( ) ( ) cot ( )

1 ( ) ( ) csc ( )

( ) tan ( )

1

s :

(sum of residues of at poles of )


(sum of residues

of at poles of )

n

n

n

n

n

n

n

f n f z z f z

f n f z z f z

f f z z f z

f

12 ( )sec ( )

( )


where all poles

of are used.n

f z z f z

f z

x

y

1 2 3 …… -3 -2 -1 0

C

Summation of Series, cont’dSummation of Series, cont’d

2

2

1.

1

( ) ( ( ) cot ( ) )

1 1( ) ( ) ,

1 ( )( )

Res ( )cot( ) lim ( )

Example :

Evaluate the sum

From

sum of residues of at poles

has poles a

of

wit

h t

n

n

z i

n

f n f z z f z

f n f z z i in n i n i

f i i z i

cot( )

( ) ( )

i

z i z i

2

cos( ) cosh( ) coth( )

2 sin( ) 2sinh( ) 2

1coth( )

1

for residues

n

i

i i

n

both

D. R. Wilton ECE Dept. ECE 6382 Pole and Product Expansions, Series Summation 8/24/10.

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Transcript of D. R. Wilton ECE Dept. ECE 6382 Pole and Product Expansions, Series Summation 8/24/10.