ECE 638 Fall 7 David R. Jackso Notes 6 Power Series Represetatios Notes are from D. R. Wilto, Dept. of ECE
Geometric Series the sum N + S + + + + N Notig that N N + we have that S S S S N S + + +, N + N+ N+ iθ ( N+ N+ N Sice r e r iff r <, N N N N N ad hece < > lim SN + + + N, < Geometric Series (G.S. The above series coverges iside, but diverges outside the uit circle. But there eists aother series represetig that is valid outside the uit circle : G.S. + + + iff < i.e., > r 3 3 The above two series ma or ma ot coverge at poits o the uit circle Note the iterior ifiite series is a epasio i (positive powers of ; the eterior series is a epasio i reciprocal powers of
Geometric Series (cot. Geeralie: ( p : 3 Note that + + + + if <, i.e. p p p p p p p p < p p < p > p Radius of covergece for series above : R p R 3 p p p p Similarl, + + + + if <, i.e. > p p p 3
Geometric Series (cot. The above series were epaded about the origi,. poit, sa : But we ca also epad about aother 3 + + p ( ( p ( p p + + p p p p if <, i.e. < p p Decide whether p - or - is larger (i.e., if is iside or outside the circle at right, ad factor out the term with largest magitude! p p R Radius of covergece : p R Similarl, 3 p p p + p ( ( p p + + + ( ( p if <, i.e. > p 4
Geometric Series (cot. Summar ( p 3 + + + + if < p p p p p Coverges iside circle p Coverges outside circle 3 p p p + + + + if > p ( p 5
3 Uiform Covergece the ifiite geometric series, 3 + + + + 3 Let's evaluate the series for some specific values, sa + i, + i, + i. + i :. +. +. +. + 3. Clearl, ever additioal term adds 3 more sigificat figures to the fial result. + i :. +. +. +. +. Here, however, each additioal term adds ol more sigificat figures to the result. + i :. +. +. +. +. Ad here each additioal term adds ol more sigificat figure to the result. I geeral, f or a give accurac, the umber of terms eeded icreases with. The series is said to coverge o - uiforml. 6
Uiform Covergece (cot. A series f g is uiforml coverget i a regio if for a ε >, there eists a umber N, depedet o ε but idepedet of, such that N > N N implies f g < ε for all i ε. ε 3 + + + + The series coverges slower ad slower as approaches. < < R< R No-uiform covergece Uiform covergece Ke Poit: Term-b-term itegratio of a series is allowed over a regio where it is uiforml coverget. We use this propert etesivel later! 7
Uiform Covergece (cot. S + + + + N 5 N 3 S + + + + N + N+ N+ The partial sum error is S SN en S S SN N + The relative error is εrel S Note: A relative error of -p meas p sigificat Number of geometric series terms N vs. figures of accurac. N+ N+ 4 3 N sig. digits 4 sig. digits 6 sig. digits 8 sig. digits sig. digits The closer gets to the boudar of the circle, the more terms we eed to get the same level of accurac (o-uiform covergece...4.6.8 8
Uiform Covergece (cot. Assume : The relative error is ε rel < R< S S N S N + ε rel < N R + For eample: N 5 45 4 35 3 5 5 5 Number of geometric series terms N vs. N N 8 N 6 N 4 N..4.6.8.95 sig. digits 4 sig. digits 6 sig. digits 8 sig. digits sig. digits R. 95 ε rel N + < R. 95 N R + < R< Usig N 35 will give 8 sigificat figures everwhere iside the regio. 9
The Talor Series Epasio This epasio assumes we have a fuctio that is aaltic i a disk. f a s Aaltic a f( ( + π i C d R c a f ( ( derivative formula Here s is the closest sigularit.! Note: Both forms are useful. The path C is a couterclockwise closed path that ecircles the poit. R c radius of covergece of the Talor series The Talor series will coverge withi the radius of covergece, ad diverge outside.
The Talor Series Epasio (cot. Talor's theorem is amed after the mathematicia Brook Talor, who stated a versio of it i 7. Yet a eplicit epressio of the error was ot provided util much later o b Joseph- Louis Lagrage. A earlier versio of the result was alread metioed i 67 b James Gregor. Brook Talor (685-73 From Wikipedia
Talor Series Epasio of a Aaltic Fuctio (cot. Write the Cauch itegral formula i the form where ( f f ( d π i C C f ( ( ( f ( π i d π i ( > C ( ( f π i C uiform covergece d d f ( ( ( + C Talor series epasio of f ( ( ( ( ( f (! f ( recall (! πi + C d π i ( f d f a f about a derivative formulas π i C + d f ( (! (both forms are useful! f is aaltic iside s C < < ( sigularit s
Talor Series Epasio of a Aaltic Fuctio (cot. s Note that i the result for a, the itegrad is aaltic awa from, ad hece the path is ow arbitrar, as log as it ecircles. C a f ( ( + π i C d Note that the costructio is valid for a s < s where is the sigularit earest ; hece the series will coverge if < s Note: It ca also be show that the series will diverge for > s 3
Talor Series Epasio of a Aaltic Fuctio (cot. The radius of covergece of a Talor series is the distace out to the closest sigularit. R c s Ke poit: The poit about which the epasio is made is arbitrar, but It determies the regio of covergece of the Talor series. Coverges for : < R c s 4
Talor Series Epasio of a Aaltic Fuctio (cot. Properties of Talor Series R c radius of covergece distace to closest sigularit A Talor series will coverge for - < R c (i.e., iside the radius of covergece. A Talor series will diverge for - > R c (i.e., outside the radius of covergece. A Talor series ma be differetiated or itegrated term-b-term withi the radius of covergece. This does ot chage the radius of covergece. A Talor series coverges absolutel iside the radius of covergece (i.e., the series of absolute values coverges. A Talor series coverges uiforml for - R < R c. Whe a Talor series coverges, the resultig fuctio is a aaltic fuctio. Withi the commo regio of covergece, we ca add ad multipl Talor series, collectig terms to fid the resultig Talor series. J. W. Brow ad R. V. Churchill, Comple Variables ad Applicatios, 9 th Ed., McGraw-Hill, 3. 5
The Lauret Series Epasio This geeralies the cocept of a Talor series to iclude cases where the fuctio is aaltic i a aulus. or f a ( f a + b b Aaltic a a b where b a a f( ( + π i C d (This is the same formula as for the Talor series, but with egative allowed. (derived later Here a ad b are two sigularities. Note: The poit b ma be at ifiit. The path C is a couterclockwise closed path that stas iside the aulus a ecircles the poit. Note: We o loger have the derivative formula as we do for a Talor series. 6
The Lauret Series Epasio Lauret series: f a The Lauret series coverges iside the regio a < < b b Aaltic a a b The Lauret series diverges outside this regio if there are sigularities at a, b 7
The Lauret Series Epasio (cot. The Lauret series was amed after ad first published b Pierre Alphose Lauret i 843. Karl Weierstrass ma have discovered it first i a paper writte i 84, but it was ot published util after his death. Pierre Alphose Lauret (83-854 From Wikipedia 8
The Lauret Series Epasio (cot. This is particularl useful for fuctios that have poles. Eamples of fuctios with poles, ad how we ca choose a Lauret series: f( ( Choose : a,b< f : a,b< f : a,b ( Choose < f( : a,b ( ( ( ( Choose ( Choose f( : a,b < ( Choose b Aaltic a a b 9
The Lauret Series Epasio (cot. The sigularit does ot have to be a simple pole: f( a,b ( ( Choose : 3 4 Brach cut Pole
The Lauret Series Epasio (cot. Theorem: The Lauret series epasio i the aulus regio is uique. (So it does t matter how we get it; oce we obtai it b a series of valid steps, it is correct! This is justified b our Lauret series epasio formula, derived later. Eample: cos( f(,a,b< f ( aaltic for > valid for < 4 6 + +! 4! 6! X a b Hece 3 5 f( +, < <! 4! 6!
The Lauret Series Epasio (cot. A Talor series is a special case of a Lauret series. f a C a f( ( + π i C d b a a,,, 3 Here f is assumed to be aaltic withi C. If f ( is aaltic withi C, the itegrad is aaltic for egative values of. Hece, all coefficiets a for egative become ero (b Cauch s theorem.
The Lauret Series Epasio (cot. Derivatio of Lauret Series We use the bridge priciple agai Pod, islad, & bridge Pod: Domai of aalticit Islad: Regio cotaiig sigularities Bridge: Regio coectig islad ad boudar of pod 3
The Lauret Series Epasio (cot. Cotributios from the paths c ad c cacel! B Cauch's Itegral Formula, f C+ c c C ( f d πi ( f( f d d πi πi C C simpl - coected regio s s c c C C Pod, islad, & bridge where o C, >, ( ( ad o C, < (ote the covergece regios for C,C ( (, ( ( + + ( ( ( ( overlap! + 4
The Lauret Series Epasio (cot. f Hece, C+ c c C ( f d πi uiform covergece f( ( f( ( ( + C d π i ( + C + d π i Let ( call them path C C C. multipl - coected regio s s C C We thus have f( ( π i + C ( f( + ( π i ( f d d + C 5
The Lauret Series Epasio (cot. Because the itegrad for the coefficiet is aaltic with, the path C is arbitrar as log as it stas withi. multipl - coected regio f( ( π i + C ( f( + ( π i ( f d d + C s s C We thus have ( f a The path C is ow arbitrar, as log as it stas i the aaltic (blue regio. where a f( ( + π i C d 6
Eamples of Talor ad Lauret Series Epasios Eample : Obtai all epasios of f( ( about the origi. Use the itegral formula for the a coefficiets. a f( ( + π i C d The path C ca be iside the ellow regio or outside of it (parts (a ad (b. 7
Eamples of Talor ad Lauret Series Epasios a Lauret series with a,b f m a d d d, ( < ( + + + πi πi πi C C C m Hece π i, m + d d πi πi πi,m + ( for a m+ m+ C m m C m +, From uiform covergece 3 f, < < f ( From previous eample i Notes 3 C The path C is iside the ellow regio. 8
Eamples of Talor ad Lauret Series Epasios (cot. b Lauret series with a,b ( f a d d d ( ( + + + 3 πi i i C π C π C C 3 π i + m m d, ( > π i, m d d πi πi πi,m + m+ 3 + m+ 3 C m m C ( for m From uiform covergece From previous eample i Notes 3 a, f ( Hece + + +, > 3 4 f The path C is outside the ellow regio. C 9
Eamples of Talor ad Lauret Series Epasios (cot. Summar of results for the eample: f ( 3 f, < < + + +, > 3 4 f 3
Eamples of Talor ad Lauret Series Epasios (cot. Note: Ofte it is easier to directl use the geometric series (GS formula together with some algebra, istead of the cotour itegral approach, to determie the coefficiets of the Lauret epaso. This is illustrated et (usig the same eample as i Eample. 3
Eamples of Talor ad Lauret Series Epasios (cot. Eample Epad about the origi (we use partial fractios ad GS : f( f ( ( A B + A lim f lim B lim f lim f ( + + + ( + Hece 3 f, < < 3
Eamples of Talor ad Lauret Series Epasios (cot. Alterative epasio: f + + ( f + + + + Hece f ( + + +, > 3 4 33
Eamples of Talor ad Lauret Series Epasios (cot. Eample Epad i a Talor / Lauret series f( ( ( 3 about, valid followig i the aular regios : (a <, (b < <, (c >. a For < : Usig partial fractio epasio ad GS, f 3 3 + ( ( ( ( ( ( + + + + + ( + ( + Hece 3 7 5 3 f( + ( + ( + ( +, < 4 8 6 3 < < < > (Talor series 34
Eamples of Talor ad Lauret Series Epasios (cot. f ( ( 3 b For < < : f so f ( ( ( ( ( ( ( + + + + + + ( ( ( (Lauret series 3 < < < > 35
Eamples of Talor ad Lauret Series Epasios (cot. f ( ( 3 c For > : f ( ( ( ( ( ( ( + + + ( ( + + ( ( + so f 3 7 + + + ( ( ( 3 4 (Lauret series 3 < < < > 36
Eamples of Talor ad Lauret Series Epasios (cot. Summar of results for eample f ( ( 3 3 7 5 3 f( + ( + ( + ( +, < 4 8 6 f ( ( + + + + + +, < < ( ( ( 3 7 f + + +, > ( ( ( 3 4 3 < < < > 37
Eamples of Talor ad Lauret Series Epasios (cot. Eample 3 Fid the series epasio about : cos, ( is a "removable" sigularit, f Hece cos f 4 6 4 6 + + +! 4! 6!! 4! 6! 4 +, <! 4! 6! Similarl, we have f 4 si + < 3! 5! Note : 3 5 si + < 3! 5! 38
Eamples of Talor ad Lauret Series Epasios (cot. Eample 4 Fid the series for si about π : si si ( π + π si ( π cosπ + cos( π siπ si ( π f Alterativel, directl use the derivative formula for Talor series : f ( π ( π ( π ( π siπ f cosπ f ( iv ( π ( v ( π siπ f cosπ + f f ( π siπ cosπ ( π ( π 3 5 Note : f + +, < 3! 5! a 3 5 si + < 3! 5! ( f a f ( ( + π i C ( ( f d! ( π ( π ( π 3 5 f + +, < 3! 5! 39
Eamples of Talor ad Lauret Series Epasios (cot. Eample 5 Fid the first three terms of the series for si l about. Sice Also + + +, < 3 4 3 4 d l ( + + + +, < 3 4 l (, < 3 4 Hece si the 3 5 3 5 + + 3! 5! 3! 5! 4 6 + + 3 45 4 3 4 6 si l ( + + + + + + 3 45 3 4 4 3 5 + +, < The brach cut is chose awa from the ellow regio. 4
Summar of Methods for Geeratig Talor ad Lauret Series Epasios Summar of Methods Talor ( ot Lauret series, f a, ca be geerated usig a f ( (! f( ( Talor ad Lauret series, f a, ca be geerated usig a d + π i C To epad about, first write f i the form f +, rearrage ad epad usig geometric series or other methods. Use partial fractio epasio ad geometric series to geerate series for ratioal fuctios (ratios of polomials, degree of umerator less tha degree of deomiator. Lauret / Talor series ca be itegrated or differetiated term - b - term withi their regio of covergece. 4
Summar of Methods for Geeratig Talor ad Lauret Series Epasios (cot. Summar of Methods Note that for two Talor or Lauret series, (, ( f a g b the + ( + ( f g a b i their commo regio of covergece. Two Talor series ca be multiplied term -b- term withi their commo regio of covergece : m m m p m p m p p p where m p (, ( f a g b ( ( ( f g a b c c a b 4