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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
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.
Example: Pole Expansion of cot Example: Pole Expansion of cot z (cont.)z (cont.)
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
Example: Pole Expansion of cot Example: Pole Expansion of cot z (cont.)z (cont.)
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) ―
Example: Pole Expansion of cot Example: Pole Expansion of cot z (cont.)z (cont.)
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 )
(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 ( )
( )
(sum of residues of at poles of )
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