d. r. wilton ece dept. ece 6382 power series representations 8/24/10
TRANSCRIPT
D. R. WiltonECE Dept.
ECE 6382 ECE 6382
Power Series Representations
8/24/10
Geometric SeriesGeometric Series
Consider
2
0
2 1
1
1
11 1 1
2
0
1
1 1
1
1
0 1
1li 1
1
m S
NN n
Nn
NN
NN N N
N
N
i N NN N N
nN
N n
S z z z z
zS z z z ,
S zS z S z
zS
z
z r e r r z
z z zz
,
• Consider the sum
Noting that
we have that and hence
• Since iff
1
1
11 1
1
z
z
z
,
Geometric Series (G.S.)
• The above series converges inside, but diverges outside the unit circle. But there exists
another series representing that is valid outside the unit circle :
2 3 2 31
1 1 1 1 1 1 1 1 11 1 1
1 z
zz z z r zz z z z z
G.S.iff i.e.,
• The above series may or may not converge at points on the unit circle
• Note the interior infinite series is an expansion in (po z
z
sitive) powers of ; the exterior series
is an expansion in reciprocal powers of
x
y
1
1
1z
1z
Geometric Series, cont’dGeometric Series, cont’d
Consider 2 30 0 0 0
00
2 3
0 0 0 0 0 00
0
0
01 1 1
1 11
1 1 11 1
1
z zz z z z
zz z z z z z zz
z
z z z zz
z z z z z zzz
zz
z
• Note that if , i.e.
Similarly, if , i.e.
x
y
0z z
0z z
0z Radius of convergence
0z
Geometric Series, cont’dGeometric Series, cont’d
Consider
20 0
00 0
0
1 1 1 11
1
z .
z
z z z zz zz z z z z z z z z z z z
z zz z
:
• The above series were expanded about the origin, But we can also expand about another
point, say
30
00
2 3
0 0 0 0 0 00
0
0
1
1 1 1 11
1
1
z z
z z
z zz z z z
z z
z z z z z z
z z z z z z z z z z z z z zz zz z
z z
z zz
z z
if , i.e.
Similarly,
if , i.e. 0z z z
x
y
z z
0z z Radius of convergence
0z
z
z
Factor out the largest term!
Uniform ConvergenceUniform Convergence
Consider2 3
3 2 1
3
3
11
1
10 0 10 0 10 0
10 0
11 00 0 001 0 000001 0 000000001
1 10
z z zz
z i , i , i
z i
. . . .
.
:
• Consider the infinite geometric series,
Let's evaluate the series for some specific values, say
2
2
1 001001001001001
10 0
11 00 0 01 0 0001 0 000001
1 101 0101010101
.
z i
. . . .
.
:
Clearly, every additional term adds 3 more significant figures to the final result.
Here, however, each additional term a1
1
10 0
11 00 0 1 0 01 0 001
1 101 11111
z i
. . . .
.
:
dds only 2 more significant figures to the result.
And here each additional term adds only 1 more significant figure to the result.
In general, f z .| |or a given accuracy, the number of terms increases with
Uniform Convergence, cont’dUniform Convergence, cont’d
Consider2 31
11
0
S z z zz
z z
!
• For the infinite geometric series,
only the first term is needed to produce an exact result for But as increases
the number of terms needed to provide a fixed nu
12
111
rel
1 0
11
1
1
NN
N
NNN N
N N
z i .
zS z z z
z
z S SS S e S z z
z S
N
mber of significant figures increases,
approaching infinity as
• Since , the partial sum error is
; hence the relative error is
rel
rel
log1 ceil(n)
log
1
nz
z .
z R
( Note denotes )
Note the number of terms needed depends
on and The relationship is
plotted in the figure.
• On the other hand if we lim
it th
both
relrel
log1
logN ,
R
z
en
which depends on but
on (see next slide) not
Number of geometric series terms N vs. |z|
0
100
200
300
400
500
0 0.2 0.4 0.6 0.8 1
|z|
N
2 sig. digits
4 sig. digits
6 sig. digits
8 sig. digits
10 sig. digits
Uniform Convergence, cont’dUniform Convergence, cont’d
Consider 1
N
z
• As the figure shows, it is impossible to find a fixed value of which yields a specified accuracy
over the entire region , i.e., the series is in this region
• Note the
.
G
non -uniformly convergent
0
0
0
9
1
5
nn
z
z R .
f z
.
g z
,
R
.S. is uniformly
convergent, say, for ,
as shown, or for
region
• A series is in a region if corresponding to an
there exists a numbe
any
uniformly convergent
0
N
nn
N z N N
f z g z z .
,
R
r , dependent on but such that
implies for all in
independent of
050
100150200250300350400450500
0 0.2 0.4 0.6 0.8 1
N
|z|
Number of geometric series terms N vs. |z|
2 sig. digits
4 sig. digits
6 sig. digits
8 sig. digits
10 sig. digits
0.95
N1
N8
N6
N4
N2x
y
1
1 1z
0 95z .
Key Point: Term-by-term integration of a series is allowed over any region where it is uniformly convergent.
Taylor Series Expansion of an Analytic FunctionTaylor Series Expansion of an Analytic Function
0 0
00
0
0
0 00
1
2
1
2
1
21
1
2
1
2
C
C
C
n
nC
f zf z dz
i z z
f zdz
i z z z z
f zdz
i z zz z
z z
f z z zdz
i z z z z
zi
uniform convergence
• Write the Cauchy integral formula in the form
0 10 0
( )0 ( )
0 0 10 0
0 00
( )0
10
!
! 2
1
2 !
n
nn C
nn n
nn C
nn
n
n
n nC
f zz dz
z z
f z f znz z f z dz
n i z z
f z a z z f z z
f zf za dz
i nz z
( )
derivative formulas
recall
Taylor series expansion of about
where
(both forms are used!)
x
y
0z z
0z
zz
0z z
z z
C
sz
R
0 0z z z z
Taylor Series Expansion of an Analytic Function, Cont’dTaylor Series Expansion of an Analytic Function, Cont’d
0 0 0
0
0
0s
s
s
z z z z z z
z z
z z z z
;
• Note the construction is valid for any
where is the singularity nearest hence the region of convergence is
x
y
0z z
0z
z
0z z
z z
C
sz
R
0 0z z z z
z
The Laurent Series ExpansionThe Laurent Series Expansion
ConsiderThis generalizes the concept of a Taylor series, to include cases where the function is analytic in an annulus.
z0 a
b
0n
nn
f z a z z
0
0 1 0
1nn n n
n n
f z a z z bz z
or
n nb awhere
Converges for
0 0bz z b z z
Converges for
0 0
0
a
a
z z a z zz z
(we often have )
z
Key point: The point z0 about which the expansion is made is arbitrary, but determines the region of convergence of the Laurent or Taylor series.
za
zb
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
ConsiderExamples:
z0a
b 0
cos0 0
zf z z , a , b
z
0 1 01
zf z z , a , b
z
01
0 0f z z , a , bz
0 0 1 21 2
zf z z , a , b
z z
This is particularly useful for functions that have poles.
z
0 0 0a bz z a z z b z z
Converges in region
But the expansion point z0 does not have to be at a singularity, nor must the singularity be a simple pole:
022 3 4
2 1
zf z z , a , b
z z
y
x
0 2z
z
2 1 1 2
branch cut
pole
zbza
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
Consider
z0a
b
Theorem: The Laurent series expansion in the annulus region is unique.
(So it doesn’t matter how we get it; once we obtain it by valid steps, it must be correct.)
0
cos0 0
zf z z , a , b
z
0
2 4 611
2! 4! 6!
zz
z z zf z
z
analytic valid for for
3 51
02! 4! 6!
z z zf z , z
z Hence
Example:
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
ConsiderWe next develop a general method for constructing the coefficients of the Laurent series.
0n
nn
f z a z z
1
0
1
2n nC
f za dz
i z z
z0a
b
C
Note: If f (z) is analytic at z0, the integrand is analytic for negative values of n.
Hence, all coefficients for negative n become zero (by Cauchy’s theorem).
Final result:
(This is the same formula as for the Taylor series, but with negative n allowed.)
Consider
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
Pond, island, & bridge
Consider
11
1
2C c
f zf z dz
i z z
By Cauchy's Integral Formula,
2c
21
2
0 0
01
0 0 00 00
0
1
2 210 0
1 1
2 2
1 1 1
1
1 1
n
n
C
CC
n
f z f zdz dz
i z z i z z
z z z z
z z
z z z z z z z z z zz zz z
C
C ,z z z z
z
C C
z
, ,
,
where on
and on (note the convergence regions of overlap!)
0 01 1
0 0 0 10 0 00
0
1
1
1
n n ,n nn n
n nn n
z z z z
z z z z z z z z z zz zz z
x
y simply - connected regionR
1C2C
1c2cz
0z
1sz
2sz
z
z
• Contributions from the paths c1 and c2 cancel!
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
Pond, island, & bridge
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
Consider
1 1
1
2cC
f zf z dz
i z z
• Hence,
2c
1
2
2
0 10 0
0 11 0
0
01 20
1
1 2
1
2
1
2
1
2
nn
n
nn
n
n
n
n
C
n
n
C
C
C
C
f zz z dz
i z z
f zz z dz
i z z
f z a z z
f za dz C z
i z z
C C
C
.
, ,
uniformconvergence
where and encircles
Note we can deform to a s
2 1 1 2
10
0 0 0
1 2
n
s s s s
f zC z
z zz z z z z z z z
C C
,
,
,
ingle contour since is - independent
and analytic at least for where are the nearest
singularities to respectively.
x
y multiply - connected regionR
1C2C
z
0z
1sz
2szz
C
Examples of Taylor and Laurent Series Examples of Taylor and Laurent Series ExpansionsExpansions
Consider
0
1 2 20
1
1
0
1 1 1 1 11
2 2 21
1
2
nn
n
mn n n n
mC C C
f zz z
a z z
f za dz dz z dz , z
i i iz z z z
Obtain all expansions of about the origin :
The series will have the form (since )
where
( )
Example 1:
20
2 2
12 120 00 0
2 3
0 12 1
1
0 11 1 1 1
1 12 2
11 0 1
i in m
mC
i
n n mi n m i n mn mm m
, m n, m n
dz z re , dz ire di z
, nirea d d
, ni rr e e
f z z z z zz
; let
,
Examples of Taylor and Laurent Series Examples of Taylor and Laurent Series Expansions,cont’dExpansions,cont’d
Consider
1 2 31
30
30
0
1 1 1 1 1
2 2 21 1
1 1 11
2
1 1
2
1
2
n n n nC C C z
n mmC
i in m
mC
i
nm
f za dz dz dz
i i iz z z z
dz , zi z z
dz z re , dz ire di z
irea
i
On the other hand,
( )
; let
Example 1, cont'd
2 2
23 2300 0
2 3 4
0 22 2
0 21 1 1
1 22
1 1 11
n mi n m i n mn mm
, m n, m n
, nd d
, nrr e e
f z zz z z
,
In practice . To illustrate, we
the contour integral approach is rarely used
f z reconsider expanding as a partial fraction and using the geometric series.
Examples of Taylor and Laurent Series Expansions,cont’dExamples of Taylor and Laurent Series Expansions,cont’d
Consider
0 0
1
1
1
1 1
lim limz z
f zz z
A Bf z
z z z z
zA z f z
Expand about the origin (we use partial fractions and G.S.) :
;
Example 1, cont'd
z
1 1
11
1lim 1 limz z
z
zB z f z
1z z
2 3
1
1 1 1 1 1
1 1 1
11 0 1
1 1 1 1
1 1 1
1
f zz z z z z z
f z z z z zz
f zz z z z z
f zz
,
11
z
2 2 3 4
1 1 1 1 11z
z z z z z
,
Examples of Taylor and Laurent Series Expansions,cont’dExamples of Taylor and Laurent Series Expansions,cont’d
Consider
0
1
2 3
1
0 1 1
1 1 2
1 2
0 1 1
f zz z
z ,
z
z
z
z
f z
Expand in a Taylor / Laurent series
about valid in the annular regions
(1) ,
(2) ,
(3) .
For :
Using partial fraction expansion and G.S.,
Example 2
22
2
2 3
1 1 1
2 3 3 2
1 1 1 1
1 2 1 1 1 12 1 1 2
1 111 1 1 1
2 2 2
1 3 7 151 1 1 0 1 1
2 4 8 16
z z z z
z z zz
z zz z
f z z z z , z
(Taylor series)
y
1 2 3x
1 1 2z
z
1 2z
1 1z
Examples of Taylor and Laurent Series Expansions,cont’dExamples of Taylor and Laurent Series Expansions,cont’d
Consider
2
2 2
1 1 2
1 1 1 1
1 2 1 1 2 1 1 2 1 1 1 1
1 11 1 1 11 1
2 2 1 12 1
1 2
1 1 1 1
1 2 1 1 1 1 2 1
z
f zz z z z z
z zf z
z z z
z
f zz z z z z
For :
(Laurent series)
For :
Example 2,cont'd
1 1 1 1
11
1
z
z
2
2
2 2 11
1 11z zz
2
2 3 4
1 1
1 1
1 3 7
1 1 1
z z
f zz z z
(Laurent series)
Examples of Taylor and Laurent Series Expansions,cont’dExamples of Taylor and Laurent Series Expansions,cont’d
Consider
2
0
1 cos0
01 2 0
1 cos 1
z
z, z
f z zz, z
z
Find the series expansion about :
( is a "removable" singularity)
Example 3
1
2 4 6 2 4 6
2 4
2 4
2! 4! 6! 2! 4! 6!
1
2! 4! 6!
sin1
3! 5!
z z z z z z
z zf z z
zf z z
z,
z z
,
Similarly, we have
Examples of Taylor and Laurent Series Expansions,cont’dExamples of Taylor and Laurent Series Expansions,cont’d
Consider
3 5
sin
sin sin
sin cos cos sin sin
3! 5!
sin 0
z z
f z z z
z z z
z zf z z , z
f
Find the series for about :
Alternatively, use the derivative formula for Taylor series :
Example 4
3 5
cos 1
sin 0
cos 1
sin 0
cos 1
3! 5!
iv
v
f
f
f
f
f
z zf z z , z
Examples of Taylor and Laurent Series Expansions,cont’dExamples of Taylor and Laurent Series Expansions,cont’d
Consider
2
2
2 3 4
0
2 3 4
23 52
sin ln 1 0
11 1
1
1ln 1 1
1 2 3 4
ln 1 12 3 4
sin3! 5!
z
z z z
z z , zz
z z zdz z z , z
z
z z zz z , z
z zz z
Find the first few terms of the series for about :
Since then
Also
Example 5
42 6
4 2 3 42 2 6
43 5
2
3 45
2sin ln 1
3 45 2 3 4
0 12
zz z
z z z zz z z z z
zz z , z
Hence
(why?)
Summary of Methods for Generating Taylor and Laurent Summary of Methods for Generating Taylor and Laurent Series Expansions Series Expansions
Consider
0 0 0
0 0n n
n nn n
n n
z z , f z f z z z
f z a z z g z b z z
f z g z a b z
To expand about first write in the form , rearrange
and expand using known series or methods.
Note that if
t
,
he
n
0
00
0 !
n
n
nn
n nn
z
f zf z a z z a
n
Taylor ( Laurent) series, , can be generated using
Use partial fraction expansion and geometric series to generat
e serie
in their common region of convergence.
not
s for rational functions
(ratios of polynomials, degree of numerator less than degree of denominator).
Laurent / Taylor series can be integrated or differentiated term - by - term within their radius
o
0 00 0
0 00 0
n mn m
n m
n mn m
n m
f z a z z g z b z z
f z g z a z z b z z
f convergence
Two Taylor series can be multiplied term - by - term :
,
within their common region of convergence
00 0
nn
n n p n pn p
c z z c a b
wher e = .