An Introduction to Asymptotic Expansions

Transcription

1 A Itroductio to Asmptotic Expasios R. Shaar Subramaia Asmptotic xpasios ar usd i aalsis to dscrib th bhavior of a fuctio i a limitig situatio. Wh a fuctio ( x, dpds o a small paramtr, ad th solutio of th govrig quatio for this fuctio is ow wh =, a prturbatio mthod ma prov usful i obtaiig a solutio for small valus of. Such a approach is particularl attractiv wh th govrig quatio is oliar ad o gral tchiqus ar availabl for xact solutio. If appars as a multiplicativ factor i a trm i th govrig quatio, th stadard approach is to tr a powr sris solutio of th followig form: ( ( ( ( x, = x + x + x +... ( whr th smbol stads for highr ordr trms. Th sris is isrtd ito th govrig quatio ad boudar coditios, ad cofficits of li powrs of ar th groupd to obtai a sris of quatios for th cofficit fuctios j ( x, which ar th solvd i a squtial mar. Th rsultig sris d ot covrg for a valu of ; vrthlss, th solutio ca b usful i approximatig th fuctio ( x, wh is small. Covrgt ad Asmptotic Sris Computatioall, a covrgt sris is ot alwas usful, bcaus covrgc is a cocpt rlatig to th bhavior of th trms i th sris at th tail d, that is, as j. That a sris covrgs sas othig about how rapidl th trms will dcras i magitud. O th othr had, i a asmptotic sris, th trms will usuall dcras rapidl with j at first for sufficitl small. Somtims, th ma bgi to icras with icrasig j at som poit aftr dcrasig iitiall. Wh th trms ar dcrasig rapidl, if w sum just th first fw trms ad w ow that th rror icurrd is of th ordr of th xt trm, w ca gt a good stimat of th sum. This is wh asmptotic sris, v wh divrgt, ar practicall usful. Th mai problm with asmptotic sris is that o vr ows how accurat th aswr is. Th rsults must b validatd b compariso with som othr rprstatio of th xpctd aswr. Nvrthlss,

2 asmptotic sris ma b th ol mas of obtaiig a aaltical solutio of a difficult problm, ad ar usd commol for this purpos. To illustrat th idas rgardig computatioal utilit, writ a computr program to sum th first trms i th Talor sris for si θ giv blow, first for θ =.5, ad th for θ = 4. + θ si θ = ( ( ( +! = Th sris is ow to b uiforml covrgt for all valus of θ. Th 4 rsult for θ = rflcts th fact that th prcisio of th machi computatio is fiit. Errors itroducd b th limitd prcisio lad to a absurd rsult wh th sum is calculatd. You ca s a dmostratio of this b pritig ach trm ad th sum as succssiv trms ar addd. Now, cosidr th followig sris for th complmtar rror fuctio. x rfc( x~ + x π = 35 ( ( (3 ( x This sris divrgs for all valus of x. It is a asmptotic sris that rprsts th fuctio i th limit as x. I spit of its divrgc, it is usful for computig th complmtar rror fuctio for larg valus of x, bcaus th trms i th sris dcras rapidl with icrasig for small valus of ad th rror icurrd b trucatig th sris at a crtai trm is of th ordr of magitud of th xt trm, which is much smallr tha th trm rtaid as log as x is larg ad w us ol a small umbr of trms. Tr calculatig rsults from this sris for x = 5,, ad ad chc th sum aftr addig ach trm agaist th xact rsult. Also, s if ou ca dmostrat to ourslf that this is a divrgt sris. Som Basic Cocpts Som basic cocpts i usig asmptotic sris ar dscribd xt. Two smbols ar commol usd to dscrib th bhavior of a fuctio f ( i th limit as. Th ar " O " ad " o" ad ar trmd big oh ad littl oh. If w hav two diffrt fuctios of, aml f ( ad g (, w sa that

3 ( ( ( ( if lim f f O g g ( I words, this is statd as follows: f ( is of th ordr of g (. If th limit is zro, th th smbol o is usd. = < (4 f ( o( g( ( ( f = if lim = g (5 I th abov, th fuctio g ( is trmd a gaug fuctio. A commo st of gaug fuctios is th st of powrs of psilo,,,.... Ths powrs ( ar oft usd to dscrib th bhavior of som othr fuctio of psilo. For xampl, w ma writ si (6 which should b rad as si psilo is asmptoticall qual to psilo. Ev though th phras as approachs zro is omittd, it is implid. Of cours, ta (7 at ladig ordr, so that w s that diffrt fuctios ca hav idtical asmptotic rprstatios. Powr sris ar just o tp of asmptotic sris. A mor gral asmptotic sris for a fuctio ( x, is of th form N (, ( ( x = f = (8 Not that w hav trmiatd th sris at a fiit valu of th idx. Thrfor, covrgc is ot a issu hr. Th fuctios f ( must satisf f+ ( lim =, =,,,... (9 f ( 3

4 This mas that ach mmbr of th st of fuctios approachs zro mor rapidl tha th prvious mmbr as. W call th st of fuctios { f ( } a asmptotic squc if th mmbrs satisf th coditio giv i Equatio (9. Not that th st of powrs of psilo is idd a asmptotic squc. Th cofficit fuctios j ( x ca b rmid uiqul from th proprt of th mmbrs of a asmptotic squc otd abov. First, b dividig both sids of Equatio (8 b f ( ad taig th limit as, w obtai th followig rsult for th ladig ordr cofficit ( x. ( ( lim x, x = ( f ( Now, subtract f( ( x from both sids of Equatio (8, divid b f (, ad ta th limit as. This ilds (, ( ( ( lim x f x = ( f ( Usig this procdur, it is straightforward to show that th cofficit fuctio ( x i th asmptotic sris ca b writt as j j ( j (, ( ( x f x = j = lim, =,,3,... f j ( Th cofficits i th asmptotic sris for a giv fuctio dpd o th choic of th squc; oc th squc is dfid, th cofficits ar uiqul rmid b Equatios ( ad (. I a giv problm, w usuall do ot ow th dpdc of ( x, o so that th rsults i ths quatios should b rgardd ol as formal dfiitios of th cofficit fuctios. Nxt, w dmostrat how ths cofficits ar rmid i xampl cass. ( A Itgral Cosidr th itgral I ( dfid as show blow. 4

5 I( = (3 + t Procdig to itgrat b parts, w obtai I( = { } { } + t ( + = ( + ( { } { } = 3 + t ( + ( ( =! +! ( + Cotiuig to itgrat b parts i this mar, w ca show that As a ifiit sris, I ( = (! + (!... + ( ([ ]! ( (! + 3 ( + + ( ( I = (! = (4 (5 (6 is divrgt for all valus of. But, for rlativl small valus of, th sris i Equatio (6, trucatd aftr a small umbr of trms, provids a good approximatio of th itgral. A Diffrtial Equatio Cosidr th diffrtial quatio alog with th iitial coditio W ow that th solutio is + = ( = = x (7 (8 (9 5

6 Lt us s how a asmptotic xpasio ca b dvlopd for ( x,. Writ = = ( x Substitut this xpasio ito th govrig quatio (7, ildig + = = Rarrag this quatio to writ it as = + ( = + = ( ( ( with th covtio that =. W ca s that b taig th limit, w obtai = (3 ad b subtractig this rsult from Equatio (, dividig both sids b, ad taig th limit agai, w gt = (4 Rpatig th procss as ma tims as dd lads to =, =,,,... (5 W could also hav writt Equatio (5 b formall sttig th cofficit of to zro i Equatio ( for ach valu of. B isrtig th asmptotic xpasio giv i Equatio ( ito th iitial coditio, w obtai which ilds = = ( = δ (6 whr δ = wh i = j ad othrwis. It is ow as th Krocr dlta. ij Th solutio of alog with = is =. Usig this, w ca solv th quatio for, which is, alog with = to ild = x. B cotiuig th procss, w fid ( x = x /!, 3 = x 3 /3!, ad so o. Th solutio for ( x, ca b writt as ( = ( = = ( (7 6

7 (, ( ( x = (8 x =! which is th Talor sris for th xpotial fuctio = x. This sris happs to covrg uiforml for all valus of ad x. I this xampl, our attmpt to fid a powr sris xpasio i has ld to a covrgt sris, v though w caot xpct th sam i othr problms. Cocludig Rmars W hav s how a usful approximatio to th solutio of problms ivolvig a small paramtr ca b obtaid b xpadig i a asmptotic sris i that paramtr. This mthod is ow as prturbatio. It ca b show that th simpl tchiqu illustratd hr fails if th small paramtr multiplis th highst ordr drivativ i a diffrtial quatio. This is bcaus th ordr of th diffrtial quatio is rducd wh th small paramtr is st qual to zro. This lads to qualitativ diffrcs i th solutio, ad i boudar valu problms, th iabilit to satisf th complt st of boudar coditios o th problm. Also, a simpl prturbatio mthod ca fail v wh th small paramtr ol multiplis a low ordr drivativ if th domai is uboudd, as ca occur i idalizd mathmatical problms. Ths problms ar hadld b usig sigular prturbatio tchiqus. You ca lar mor about prturbatio mthods from a of th followig rfrcs. Rfrcs. A.H. Nafh, Prturbatio Mthods, Joh Wil & Sos, Nw Yor (973.. M. Va D, Prturbatio Mthods i Fluid Mchaics, Parabolic Prss, Staford, Califoria ( J. Kvoria ad J.D. Col, Prturbatio Mthods i Applid Mathmatics, Sprigr-Vrlag, Nw Yor ( E.J. Hich, Prturbatio Mthods, Cambridg Uivrsit Prss, Cambridg, UK (99. 7