Notes 10 Evaluation of Definite Integrals via the Residue Theorem

Transkript

1 ECE 638 Fall 7 David. Jackson Nots Evaluation of Dfinit ntgrals via th sidu Thorm Nots ar from D.. Wilton, Dpt. of ECE

2 Brif viw of Singular ntgrals ln Logarithmic singularitis ar ampls of intgrabl singularitis: lim lim sinc ε ln d ln d ln lim ln ε ε ε Not: Thr might b numrical troubl if on intgrats this function numrically!

3 Singular ntgrals (cont.) Singularitis lik / ar non-intgrabl. ε d lim d lim( ln ) ε ε ε ε 3

4 viw of Cauchy Principal Valu ntgrals Considr th following intgral: d d d ln ln + + A finit rsult is obtaind if th intgral intrprtd as d d d lim lim ln ln ε + + ε ε ε ( ε ) + ε ε ln lim ln ε ln + ln ln ε / ε ε Th infinit contributions from th two symmtrical shadd parts shown actly cancl in this limit. ntgrals valuatd in this way ar said to b (Cauchy) principal valu (PV) intgrals: Notation: or d PV d 4

5 Cauchy Principal Valu ntgrals (cont.) / singularitis ar ampls of singularitis intgrabl only in th principal valu (PV) sns. Principal valu intgrals must not start or nd at th singularity, but must pass through thm to prmit cancllation of infinitis 5

6 Singular ntgrals (cont.) Singularitis lik / ar non- intgrabl (vn in th PV sns). a ε ε b ε b d + d + + ε a ε ε b a but not that sgn, > <, has a PV intgral 6

7 Singular ntgrals (cont.) Summary of som rsults: ln is intgrabl at / α is intgrabl at for α < / α is non-intgrabl at for α f () sgn()/ α has a PV intgral at for α < if f () is continuous (Th abov rsults translat to singularitis at a point a via th transformation - a.) 7

8 Evaluating Cauchy Principal Valu ntgrals Considr th following intgral: C Eampl Evaluat: dz ρ ρ ρ lim, ( C z + z+ ) + ρ C C ( z + )( z+ ) dz + lim + + lim C C ( z + )( z+ ρ ρ ) ρ dz iρ iθ iφ iθ ( ρ ) iφ iφ dθ + + ρ idθ i dφ i i dφ i + + lim lim ( + )( + ) d iθ C : z+ ρ, ρ iθ dz i d iθ C ρ z + θ lim z z iφ C : z, i i dz i dφ i i φ d iφ iφ ( + )( + ) φ so C + i 8

9 Evaluating Cauchy Principal Valu ntgrals (cont.) d ( + )( + ) Nt, valuat th intgral C using th rsidu thorm: iθ C : z+ ρ, ρ C z iθ dz i d ρ θ z z iφ C : z, i i dz i i φ d φ C i[ s f( z i) + s f( z ) ] i lim z i ( z i) ( z i) + lim z i z i i i + i i i i( i) + 4i ( z + ) z + + ( z + ) ( z+ ) Thus C call i + i C + Hnc 9

10 Disprsion lations Assumptions: + ( z), z f z u iv f is analytic in th UHP (including th ral ais) in th UHP Lt i z ε θ UHP: Uppr Half Plan ε y ε CPV i f( ) i i θ dθ if z ε iθ f ( z) ε f f ( + ε ) dz lim + d + iθ, C ε + ε f d if if if ε i f v u f u + iv d d d i (From th rsidu thorm or th Cauchy intgral thorm.)

11 Disprsion lations (cont.) i f v u f u + iv d d d i Equat th ral and imaginary parts: u( ) v( ) v u d d Nt, rlabl:,

12 Disprsion lations (cont.) Hnc, w hav u v ( ) v ( ) u d d Hlbrt u, v ar Hilbrt Transforms of on anothr Assumptions : +, f z u iv is analytic in th UHP f z z in th UHP (including th ral ais)

13 Disprsion lations: Circuit Thory + Vin ( ω ) H( ω) H ( ω) + ih ( ω) H( ω) V ω / V ω out in V + ( ω) out mω Assumptions ( p( iωt) tim convntion) H ( ω) is analytic in th LHP H ( ω), for ω in th LHP : LHP: Lowr Half Plan ω Not : A pol in th LHP would corrsponds to a nonphysical growing rspons. ω ω + ω r i t i rt it i ω ω ω i Not : Th systm is assumd to b unabl to rspond to a signal at vry high frquncy. Symmtry proprty: * ( ω) ( ω) ( ω) ( ω) ; ( ω) ( ω) H H H H H H (s Appndi) 3

14 Disprsion lations: Circuit Thory C Us th path shown blow: ω ε iθ H( ω ) H( ω ) H( ω+ ε ) i dω lim + dω + iε θ dθ ih( ω), iθ ω ω ω ω ε ω+ ε ε CPV ih( ω) H ( ω ) dω ih( ω) ω ω m H ( ω ) H( ω) dω i ω ω ω ε ω C 4

15 Disprsion lations: Circuit Thory (cont.) H ( ω) H ( ω ) dω i ω ω ( ω ) i H ( ω ) H H ( ω) H ( ω) + ih ( ω) dω dω + ω ω ω ω W thus hav H H ( ω) ( ω) ( ω ) H dω ω ω ( ω ) H dω ω ω Th ral and imaginary parts of th transfr function ar Hilbrt transforms of ach othr. 5

16 Disprsion lations: Circuit Thory (cont.) W can also driv an altrnativ form: Similarly, w hav ( ω ) ( ω ) ( ω ) H H H H ( ω) dω dω dω ω ω ω ω ω ω ( ω ) H ( ω )( ω) H H ( ω) dω dω ω ω ω ω ( ω ) H ( ω ) H dω dω ω ω ω ω ( ω ) H ( ω ) H dω dω ω + ω ω ω ( ω ) ( ω ω) + ( ω + ω) ( + )( ) ( ω )( ω ) H ω ω ω ω H ω ω ( ) dω dω First on : Us ω ω First on : Us H This intgral starts at zro. ( ω) H ( ω) 6

17 Disprsion lation: Circuit Thory (cont.) Summarizing, w hav ( ω ) H ( ω )( ω ) H H ( ω) dω dω ω ω ω ω ( ω ) H ( ω )( ω) H H ( ω) dω dω ω ω ω ω Assumptions: H ( ω) is analytic in th LHP H ( ω), for ω in th LHP 7

18 Kramrs-Kronig lations Assumption: Th rlativ prmittivity ε ( ω) is analytic in th LHP and ε ( ω), ω in LHP ( ε ω ) r ( ε ω ) m r r Similar to th transfr - function analysis, on obtains th Kronig - Kramrs disprson rlations : ( ε ω ) ω m ( r ) ω ω ( ε ω ) ω r ω ω r dω dω Matrial paramtrs : ε r χ ( ω) ε ( ω) rlativ prmittivity χ lctric suscptibility r P εχ E (polarization pr unit volum) χ ω as ω in LHP 8

19 Kramrs-Kronig lations Dnot ε ( ω) ε ( ω) jε ω ( using jinstad of i) r r r Th final form of th Kramrs-Kronig rlations is thn: ( ω) ωε r ε r ( ω) + dω ω ω ( r ( ω ) ) ω ε ε r ( ω) dω ω ω Kramrs Not: This shows that if thr is loss, thn thr will usually b disprsion (th ral part of th prmittivity varis with frquncy). 9

20 Laplac Transform Th Laplac transform is dfind as: st F s f t dt Assum m ( s) < A γ t f t Thn F(s) is analytic in th rgion s > γ γ Analytic ( s) This is bcaus w hav uniform convrgnc of th intgral in this rgion. Not : st F s t f t dt

21 Laplac Transform (cont.) Dfin a function g(t) (dfind to b zro for t < ): γ γ γ t f() t g t, > Th Fourir transform of g(t) ists. (Th function g (t) stays finit along th ntir ral ais and tnds to zro.) W thn hav, from th invrs Fourir transform, iωt g( t) g( ω) dω, t > Thn γt iωt f ( t) g( ω) dω, t > m ( s) γ C ( s) Lt s γ + iω γ + i st f ( t) g ( i ( s γ )) ds, t > i γ i

22 Laplac Transform (cont.) γ + i st f ( t) g ( i ( s γ )) ds, t > i γ i W hav for th intgrand trm: ( ( γ )) ( ( γ )) i is t g i s g t dt g t g t ( ( γ )) i is t ( s γ ) t dt dt m ( s) γ C ( s) f t F s st dt

23 Laplac Transform (cont.) Summary γ + i st f ( t) F ( s) ds, t > i γ i Bromwich intgral nvrs Laplac transform m ( s) C Th Bromwich contour C is chosn to th right of all singularitis of th function F(s). γ ( s) 3

24 Laplac Transform (cont.) Evaluation of th Bromwich ntgral i f t Cas : C t < Clos th contour to th right. Only th vrtical path C contributs as C ight : st Top and bottom : F s f ( t ) ( assumption) st F s ds : i f t m ( s) γ + i γ i γ st F s ds Th intgrand is analytic insid C. C C ( s) (by Cauchy s thorm) 4

25 Laplac Transform (cont.) Evaluation of th Bromwich ntgral Cas : t > Clos th contour to th lft. f t γ + i i γ i st F s ds Only th vrtical path C contributs as C Lft : st Top and bottom : i f t F s ( assumption) C L st F s ds : f ( t ) rsidus at pols to th lft of C L C L m ( s) γ C ( s) (This assums that thr ar only pol singularitis.) 5

26 Laplac Transform (cont.) Evaluation of th Bromwich ntgral Summary m ( s) γ + i st f ( t), t > F ( s) ds i γ i, t < γ C ( s) Not: This invrs Laplac transform intgral givs us zro for t <, no mattr how th original function f (t) was dfind for t <. 6

27 Laplac Transform (cont.) Eampl at f t at st F ( s) dt dt s a ( s at ) s at F ( s ), ( s ) > a s a m ( s) For t > : γ + i st f ( t) ds, γ a i > s a γ i st f ( t) is i s a s a C L a γ ( s) at, f t t > 7

28 ntgrals of th Form f ( sin,cos ) Assumptions: θ θ dθ f is finit within th intrval. f is a rational function (ratio of polynomials) of sinθ, cosθ. Lt θ θ z, dz i dθ dθ i z unit circl i i dz z z z+ z sin θ, cosθ i + f ( sin θ,cosθ) dθ i f, i z z z z z dz z z z+ z sidus of f, insid th unit circl z i z Not : z f z z, z + z i will b a rational function of z 8

29 ntgral Evaluation (cont.) Eampl: dθ sinθ dθ dz i z z z + sinθ z 4 i 4i 4dz ( idz) z + 5iz z + i z + i z z Multiply top and bottom by 4i. ( z+ i) ( z+ i) ( z+ i) dz i lim ( z+ i)( z+ i) z i z 8 3 y z i z i z dθ 8 + sinθ

30 ntgrals of th Form Assumptions: f is analytic in th UHP cpt for a finit numbr of pols (can b tndd to handl pols on th ral ais via PV intgrals). o( z ) f is /, i.. lim zf z in th UHP. z f d y iθ C : z, dz iθ i d θ On th larg smicircl w thn hav Hnc iθ iθ lim f ( z) dz lim f ( ) i dθ C Not: f th function is analytic in th LHP cpt for pols, thn w would clos th contour in th LHP. f ( z) dz f d i rsidus of f ( z) in th UHP C 3

31 Eampl: ntgral Evaluation (cont.) y d ( + 9)( + 4) z d z d z + 9 z + 4 ( z 9)( z 4) + + f( z) z 4 i s f ( 3 i) + s f ( i) ( z 3i) z d z i z i lim + lim z 3i z i ( z+ 3i) ( z 3i) ( z + 4) dz ( z + 9) ( z i) ( z+ i) 3 i z + 9 z+ i z z z z+ i + z+ i z + 9 i + lim 4 5 z i ( z + 9) ( z+ i ) 3i 3i i 5 d ( + 9)( + 4) m d m s f( z) ( z z ) f( z) m ( m! ) dz z z z 3i z i (Not th doubl pol at z i!) 3

32 ntgrals of th Form Assum Assumptions: f is analytic in th UHP cpt for a finit numbr of pols (can asily b tndd to handl pols on th ral ais via PV intgrals), lim f( z), arg z ( z in UHP) z (Fourir ntgrals) a > (clos th path in th UHP) i a f d y iθ C : z, dz iθ i d θ Choosing th contour shown, th contribution from th smicircular arc vanishs as by Jordan s lmma. (S nt slid.) Fourir 3

33 Fourir ntgrals (cont.) Jordan s lmma Assum lim f( z), arg z ( z in UHP) z Thn C iaz f ( z) dz y iθ C : z, dz iθ i d θ Proof: C / iaz iθ iacosθ asinθ iθ iθ asinθ f ( z) dz f ( ) i dθ ma f ( ) dθ / aθ iθ i θ f a ma f ( ) d ma ( θ ) a θ / : sin θ θ asinθ aθ θ sinθ θ 33

34 Fourir ntgrals (cont.) i a f d a > y iθ C : z, dz iθ i d θ W thn hav ia iaz iaz rsidus of in th UHP C f d f z dz i f z Qustion: What would chang if a <? 34

35 Eampl: Fourir ntgrals (cont.) y cosλ d, ( a, λ ) >. + a z ia Not : iλ cos λ+ isin λ Us th symmtris of cosλ and sinλ and th Eulr formula, w hav: iλ + a d (imaginary part vanishs by symmtry!) iλz iλz iλz ( z ia) s lim z + a z + a z ia ( z + ia ) ( z ia ) dz i i i z ia a i a λ aλ a cosλ + aλ a a 35

36 Eponntial ntgrals Thr is no gnral rul for choosing th contour of intgration; if th intgral can b don by contour intgration and th rsidu thorm, th contour is usually spcific to th problm. y Eampl: a d, < a < + Considr th contour intgral ovr th path shown in th figur: az az C dz dz z z + C + γ γ γ3 γ4 γ 4 γ 3 γ z 3 i z i γ Th intgrand has simpl pols at z (n+ ) i, n, ±, ±, z i Th contribution from ach contour sgmnt in th limit must b sparatly valuatd (nt slid). 36

37 Eponntial ntgrals (cont.) γ :,, z dz d 3 az lim z + γ lim γ 3 dz γ : z + i, dz d, a ia ia az dz lim d z + + γ 4 γ 3 γ y z 3 i z i z i γ γ : z + iy, dz idy + iy γ az a iay dz i dy + + z iy ( a ) a dy dy, a < + iy Smallst for : y y 37

38 Eponntial ntgrals (cont.) y γ : z + iy, dz idy, 4 γ Hnc 4 az a iay dz i dy + + z iy a dy, a > ia ( ) i s f ( z i) ilim ( z i) z i az ilim ( z i) ilim z i z i z i i lim z i ( z i ) az ( z i ) ( z i ) az + z az ( z i ) ( z i) ( z i) ia i γ 4 gz Altrnativly, f( z), h( z ) hz g( z ) s f( z i) lim lim h ai γ 3 γ ai i az ( z ) d ( + ) z z z i z a i z i z i dz γ 38

39 Eponntial ntgrals (cont.) W thus hav ia ( ) i ia Thrfor a ia ia i i d ia ia ia + ia, < a < sin a Hnc a d, < a < + sin a 39

40 ntgration Around a Branch Cut A givn contour of intgration is chosn: usually problm spcific, usually must not cross a branch cut. W tak advantag of th fact that th intgrand changs across th branch cut. Usually an valuation of th contribution from th branch point is rquird. Eampl: y k d, < k < + Assum th branch θ < ( k positiv ral) z Not: W choos th branch cut to th positiv ral ais (th ais of intgration). 4

41 ntgration Around a Branch Cut (cont.) k d, < k < + First, not th intgral ists sinc th intgral of th asymptotic forms of th intgrand at both limits ists: k + k + k which is intgrabl at, k < k which is intgrabl at k >, 4

42 ntgration Around a Branch Cut k d, < k < + y W ll valuat th intgral using th contour shown. < θ < C z L L ε ε ε ε C ln k ln k( ln z+ iarg z) k( ln r+ iθ ) k z k z k ikθ z r,< θ < 4

43 ntgration Around a Branch Cut (cont.) Now considr th various contributions to th contour intgral: y L+ L+ C+ C whr f ( ) f z dz i s f, ( z) k z z + C z L L ε ε ε ε C : z r, dz ir dθ, z r iθ iθ k k ikθ C C ε k ikθ iθ r ir dθ k iθ ikθ r, iθ r + r ε ε f z dz lim lim ir dθ C : z, dz i dφ, z iφ iφ k k ikφ C ε k ikφ iφ i dφ k, iφ + ε ε ikφ f z dz lim lim i dφ 43

44 ntgration Around a Branch Cut (cont.) For th path L : f ( z) k z z + y L :,, z r dz dr z r L iε iε k k ikε, ε, r k k i( ε kε) iε r r r r dr r dr f ( z) dz lim + + C z L L ε ε ε ε For th path L : C L ( ε) iε iε k k ik( ε) : i L,, z r r dz dr z r, ε, r r [ ] k k i ε+ k ε r dr ik r dr ik iε r + r + f z dz lim 44

45 ntgration Around a Branch Cut (cont.) Hnc i k ( ) i s f ( z ) z ( z ) i lim + i i i i ik k k k z z + Not: Arg (-) hr. Thrfor, w hav k ik ik i i d ik + ik i k ( ) ik ( ), < k < sin k k d, < k < + sin k 45

46 Proof of symmtry proprty: Appndi H * ( ω) H ( ω) + Vin ( ω ) H( ω) H ( ω) + ih ( ω) V + ( ω) out mpuls rspons: so iωt ht H( ω) dω Us ral ω ω * * iωt * iω t * iω t * iωt h t h t H d H d H d H d ( ) ( ) ( ) Hnc: H( ω) V ω / V ω out ω ω ω ω ω ω ω ω in iωt ( ω) H h t dt labl ω ω iωt * iωt H( ω) dω H ( ω) dω ( ω) ( ω) F H * F H * H ( ω) H( ω) 46