D. R. Wilton ECE Dept. ECE 6382 Power Series Representations 8/24/10.
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Transcript of D. R. Wilton ECE Dept. ECE 6382 Power Series Representations 8/24/10.
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D. R. WiltonECE Dept.
ECE 6382 ECE 6382
Power Series Representations
8/24/10
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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
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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
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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!
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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
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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
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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.
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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
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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
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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
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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
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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:
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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.)
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Consider
The Laurent Series Expansion, cont’dThe Laurent Series Expansion, cont’d
Pond, island, & bridge
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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
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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
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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
,
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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.
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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
,
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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
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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)
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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
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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
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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?)
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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 = .