Notes 12 Asymptotic Series
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1 ECE 6382 Fall 207 David R. Jackso otes 2 Asymptotic Series
2 Asymptotic Series A asymptotic series (as ) is of the form a ( ) f as = 0 or f a + a a + + ( ) ote the asymptotically equal to sig. The asymptotic series shows how the fuctio behaves as gets large i magitude. Importat poit: A asymptotic series does ot have to be a covergig series. (This is why we do ot use a equal sig.) 2
3 Asymptotic Series (cot.) Properties of a asymptotic series: f a + a a + + ( ) For a fixed umber of terms i the series, the series get more accurate as the magitude of icreases. For a fixed value of, the series does ot ecessarily get more accurate as the umber of terms icreases. The series does ot ecessarily coverge as we icrease the umber of terms, for a fixed value of. 3
4 Big O ad small o otatio This otatio is helpful for defiig ad discussig asymptotic series. Big O otatio: O( ) f = g ( ) ( ) Qualitatively, this meas that f behaves like g as gets large (or possibly goes to ero eve faster). Defiitio: There exists a costat k ad a radius R such that f( ) < kg( ) For all > R 4
5 Big O ad small o otatio (cot.) Examples: 0 = O 2 + = O 2 si = O ote : si = ! 5! = ! 5! 5
6 Big O ad small o otatio (cot.) Small o otatio: ( ) f = o g ( ) ( ) Qualitatively, this meas that f is smaller tha g as gets large. Defiitio: For ay ε there exists a radius R (which depeds o ε) such that f ( ) < ε g( ) For all > R 6
7 Big O ad small o otatio (cot.) Examples: = o = o 2 3 π π e = o m, m =, 2, 3, < arg ( ) < 2 2 ote : e = e e = e e x iy r cosθ ir siθ 7
8 Defiitio of Asymptotic Series a f ( ) as = 0 Defiitio of asymptotic series: I order for this to be a asymptotic series we require the followig: a f ( ) = o for ay = 0 As gets large, the error i stoppig at term = is smaller tha this last term. Example: a a a ( ) f a a a2 f ( ) a0 + + o 2 = 2 8
9 Defiitio of Asymptotic Series Theorem If The a f ( ) as = 0 a f ( ) = O 0 + for ay = ote: This theorem gives us a good way to estimate the error as gets large! 9
10 Defiitio of Asymptotic Series Proof of theorem Assume a f ( ) as = 0 a f ( ) o + = 0 = + (from defiitio of asymptotic series) a a+ f ( ) o = = 0 a f ( ) = O 0 + = 0
11 Summig Asymptotic Series Oe must be careful whe summig a asymptotic series, sice it may diverge: it is ot clear what the optimum umber of terms is, for a give value of = 0. a f ( 0 ) What is the optimum? = 0 0 Geeral rule of thumb : Pick so that the th term i the series is the smallest.
12 Geeratio of Asymptotic Series Various method ca be used to geerate a asymptotic series expasio of a fuctio. Asymptotic methods such as the method of steepest descet Itegratio by parts Watso s Lemma Other specialied techiques 2
13 Example The expoetial itegral fuctio: t e E ( ) dt, Path C does ot cross the real axis. t Im ( t) Im ( ) X Re( t) Brach cut Re( ) ote: E () is discotiuous (by 2πi) across the egative real axis. 3
14 Example (cot.) Use itegratio by parts: b a dv b du u dt = uv v dt a dt dt b a t e ( ) t E dt = e dt t t dv t t = ( e ) 2 ( e ) dt t t e 2 t e t 2 t = 2 ( e ) 3 ( e ) dt t t e e 2 t = + e dt t u e t dt dt 2 3 ote: It is very importat which of the two fuctios is chose to be u ad which oe is chose to be v. 4
15 Usig itegratio by parts times: Example (cot.) e e e t ( ) = ( ) (! ) + ( )! + t E e dt or ( ) ( )! t E ( ) = e ( )! e dt + + t Questio: Is this a valid asymptotic series: ee ( ) = ( ) ( )!??? ote: a 0 = 0 here. 5
16 Examie the differece term: Example (cot.) ( ) ( )! t e E ( ) + + e 2 ( )! e dt = + t < = t e! e dt e! e + + or <! + ote : e e = ee = so = O o + = 6
17 Example (cot.) Hece ee ( ) = ( ) ( )! Questio: Is this a covergig series? Use the d Alembert ratio test: lim L < : L > : a a + = L coverges diverges! L = lim + (! ) = lim ( ) = The series diverges! 7
18 Example (cot.) x ( ) ( ) F x ee x S ( x) = ( ) ( )! x a S ( x) a = = ( ) ( )! x a x = 5 = odd Exact value = = eve Usig = 5 or 6 is optimum for x = 5. This is also where A is the smallest. 8
19 Example (cot.) As x gets large, the error i stoppig with terms is approximately give by the first term that is omitted. To see this, use: ( ) ( ) + x! ee( x) = o + = x x (from defiitio of asymptotic series) Therefore, we have Hece ( ) ( ) ( ) x!! ee( x) = + + o + + = x x x Error = x ( ) ee( x) ( ) ( ) ( )!! + = x x 9
20 Example (cot.) ee ( x) ( ) (! ) Error ( ) T( x) = x ( )! x + T( x) = = 2 x 20
21 ote o Covergig Series Assume that a series coverges for all 0, so that f ( ) a = = 0 The it must also be a valid asymptotic series: Proof: f ( ) f ( ) a = 0 This is a covergig series that approaches a costat as gets large. (It is also a covergig Taylor series if we let w = /.) a a a a+ + = + = + + = 0 = + = 0 = 0 Let = + + The O o + = 2
22 ote o Covergig Series (cot.) Example: e / = ! 3! The poit = 0 is a isolated essetial sigularity, ad there are o other sigularities out to ifiity. This Lauret series coverges for all 0. This is a valid asymptotic series. 22
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