82A Engineering Mathematics

Transcription

1 Clss Notes 9: Power Series /) 8A Egieerig Mthetics

2 Secod Order Differetil Equtios Series Solutio Solutio Ato Differetil Equtio =, Hoogeeous =gt), No-hoogeeous Solutio: = c + p Hoogeeous No-hoogeeous Fudetl solutios D.E. with costt coefficiets D.E. with vrible coefficiets Eleetr fuctio Power Series

3 Fous Differetil Equtio - Power Series Air Eq. Chebch Eq. ) Herit Eq. Bessel Eq. t t t ) Euler Eq. b Legedre Eq. ) )

4 Power Series Defiitio A power series i - ) is the ifiite series of the for lso ow s power series cetered t

5 Eple - A Power Series Solutio shift to the left b shift to the right b

6 Power Series Epsio Poit t A power series of Si cetered t si )! 5 5! 7 7! ) )! As the degree of the Tlor poloil rises, it pproches the correct fuctio. This ige shows si d its Tlor pproitios, poloils of degree,, 5, 7, 9, d.

7 Power Series Epsio Poit t A power series of epoet cetered t e!!!!

8 Power Series Epsio Poit t - Eples A power series developed roud = ) A power series developed roud = =) Eple - Power series cetered roud - Eple - Power series cetered roud ) )

9 Itervl / Rdius of Covergece & Error si )! 5 5! 7 7! The sie fuctio blue) is closel pproited b its Tlor poloil of degree 7 pi) for full period cetered t the origi. Error - for < <, the error is less th. Rdius of Covergece -

10 Itervl / Rdius of Covergece & Error Error - The Tlor poloils for log + ) ol provide ccurte pproitios i the rge <. Note tht, for >, the Tlor poloils of higher degree re worse pproitios. Rdius of Covergece - ) ) log

11 Rdius of Covergece

12 Itervl of Covergece The itervl of covergece is the set of ll rel ubers of for which the series coverges

13 Rdius of Covergece The rdius ρ of the itervl of covergece of power series is clled its rdius of covergece If ρ > - The power series coverges diverges or If ρ = - The power series coverges ol t If ρ = - The power series coverges for ll

14 Covergece A power series is coverget t specified vlue of if its sequece of prtil su s ) coverges li s ) li ) eist does'teist coverge diverge

15 Absolute Covergece Of Power Series Absolute Covergece of Power Series A power series is sid to coverge bsolutel t poit if coverges Rdius Of Covergece Of A Power Series PS) If power series bout - coverges for ll vlues of i The ρ is sid to be rdius of covergece of the PS

16 Deterie The Rdius Of Covergece ρ) For A Give Power Series Rtio Test) If If for fied vlue of The the power series t tht vlue of ) Coverges if ) Diverges if ) Icoclusive if L L li li ) ) li L L

17 Deterie The Rdius Of Covergece ρ) For A Give Power Series Rtio Test) Eple Fid which vlues of does power series coverges li ) ) Thus li ot fuctioof li ) ) ) coverges

18 Deterie The Rdius Of Covergece ρ) For A Give Power Series Rtio Test) Eple Cotiue) For = For = Diverges Diverges The rdius of covergece is ρ =

19 Power Series of Give Fuctio If for give the liit eist li The the series is sid to be power series epsio of f) The series coverges for = It coverge for ll It coverge for soe vlue of d ot for others f )

20 Power Series of Give Fuctio A power series defies fuctio tht is f ) whose doi is i the itervl of covergece of the series If the rdius of covergece is R> R= f ) The o the itervls is Differetible Cotiuous Itegrble Covergece t ed poit be lost b differetitio gi b itegrtio

21 Alticl Fuctios & Power Series Alticl Fuctio Defiitio A fuctio f) is sid to be ltic t = if f) c be differetited t uber of ties. For ltic fuctio d d f ) eists bouded for ll

22 Alticl Fuctios & Power Series 6 ) Note tht for = the first ter is. Strt suig fro Note tht for =,= the first d the secod ters re. Strt suig fro

23 Power Series PS) Represettio of Altic Fuctio A ltic fuctio f) hs power series represettio withi the doi of covergece f) c be writte s d f ) d f ) withi the doi of covergece The epsio poit of the PS

24 Tlor Series Suppose tht coverges to f) for The the vlue of is give b f )! If d the series is clled the Tlor Series for f bout = f ) ft) is cotiuous f )! Hs derivtive of ll orders o the itervl of covergece The derivtives of f c be coputed b differetitig the relevt series ter b ter

25 Tlor Series Tlor Series: for poit f f f ) f f )!!! Mcluri Series: for poit f ) f f f ) f!!!

26 6 7 5 ) ) l!!!! )! ) 6!!! ) cos )! ) 7! 5!! ) si e Itervlofcovergece These results c be used to obti power series represettios of other fuctio ).. e e g replce 6!!!! e Itervl of covergece PS Epsios of Alticl Fuctio Mclurie Series)

27 PS Epsios of Alticl Fuctio Mclurie Series) To obti Tlor series represettio of l cetered t Replce l l )) ) ) ) ) ) ) Itervl of covergece is shifted b fro -, ] to, ]

28 Arithetic of Power Series Multiplictio of Power Series Additio of power series Shiftig ide of sutio e e si si II I ) ) )

29 Arithetic of Power Series Multiplictio e si e si ) ) sice e the se itervl d si both coverge o ) the product coverges o

30 Arithetic of Power Series Additio II I ). Both series should strt with the se power. Both idices of sutio should strt with the se uber II I )

31 Arithetic of Power Series Additio. Both series should strt with the se power II I ) ) ) ) )

32 Arithetic of Power Series Additio. Both idices of sutio should strt with the se uber ) ) ) ) ) ) se se

33 Arithetic of Power Series Shiftig Ide of Sutio ) ) fro strt - The ide of sutio i ifiite series is du preter.

34 Arithetic of Power Series Rewritig Geeric Ter ) ) ) ) ) ) Geeric ter to correspods

35 Arithetic of Power Series Rewritig Geeric Ter ) ) ) r r r r r r r r r itothesu put

36 Series Equlit - If two power series re equl ) b ) forechisoeopeitervlwith ceter) The b for,,,

37 Deteriig Coefficiets Assue - Wht this iplies bout the coefficiets - Rewritig both series with the se power of ) ) replce,,,, ) for

38 Deteriig Coefficiets!! 6 6 e

39 A Power Series Solutio Eple Step : clculte derivtive of the ssued solutio Step : substitute & ito the diff eq.

40 Step : shift idices of sutio ) ) Step : ) ) forllisoeitervl Becuse ' A Power Series Solutio Eple

41 ) ) Step 5 : Defie the solutio e! ) e A Power Series Solutio Eple

42 A Power Series Solutio Eple si )! 5 5! 7 7! 9 9!!!

43 Fous Series Solutios ) ). '. ' ). '. ).. Eq s Legedre Eq s Euler t t t Eq s Bessel Eq Herite Eq Chebchev Eq Air