Clss Notes 9: Power Series /) 8A Egieerig Mthetics
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
Fous Differetil Equtio - Power Series Air Eq. Chebch Eq. ) Herit Eq. Bessel Eq. t t t ) Euler Eq. b Legedre Eq. ) )
Power Series Defiitio A power series i - ) is the ifiite series of the for lso ow s power series cetered t
Eple - A Power Series Solutio - - - - - - - shift to the left b shift to the right b - - - - -
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.
Power Series Epsio Poit t A power series of epoet cetered t e!!!!
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 ) )
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 -
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
Rdius of Covergece
Itervl of Covergece The itervl of covergece is the set of ll rel ubers of for which the series coverges
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
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
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
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
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
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 =
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 )
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
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
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
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
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
Tlor Series Tlor Series: for poit f f f ) f f )!!! Mcluri Series: for poit f ) f f f ) f!!!
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)
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, ]
Arithetic of Power Series Multiplictio of Power Series Additio of power series Shiftig ide of sutio e e si 7 5 5 6 6 si II I ) ) )
Arithetic of Power Series Multiplictio e si e si 5 7 6 6 5 ) ) 6 6 6 5 5 sice e the se itervl d si both coverge o ) the product coverges o
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 )
Arithetic of Power Series Additio. Both series should strt with the se power II I ) ) ) ) )
Arithetic of Power Series Additio. Both idices of sutio should strt with the se uber ) ) ) ) ) ) se se
Arithetic of Power Series Shiftig Ide of Sutio ) ) fro strt - The ide of sutio i ifiite series is du preter.
Arithetic of Power Series Rewritig Geeric Ter ) ) ) ) ) ) Geeric ter to correspods
Arithetic of Power Series Rewritig Geeric Ter ) ) ) r r r r r r r r r itothesu put
Series Equlit - If two power series re equl ) b ) forechisoeopeitervlwith ceter) The b for,,,
Deteriig Coefficiets Assue - Wht this iplies bout the coefficiets - Rewritig both series with the se power of ) ) replce,,,, ) for
Deteriig Coefficiets!! 6 6 e
A Power Series Solutio Eple Step : clculte derivtive of the ssued solutio Step : substitute & ito the diff eq.
Step : shift idices of sutio ) ) Step : ) ) forllisoeitervl Becuse ' A Power Series Solutio Eple
) ) Step 5 : Defie the solutio e! ) e A Power Series Solutio Eple
A Power Series Solutio Eple si )! 5 5! 7 7! 9 9!!!
Fous Series Solutios ) ). '. ' ). '. ).. Eq s Legedre Eq s Euler t t t Eq s Bessel Eq Herite Eq Chebchev Eq Air