Notes 10 Evaluation of Definite Integrals via the Residue Theorem

Transkript

1 ECE 638 Fall 6 David. Jackson Nots Evaluation of Dfinit ntgrals via th sidu Thorm Nots ar from D.. Wilton, Dpt. of ECE

2 viw of Cauchy Principal Valu ntgrals Considr th following intgral: / d d d ln ln + + ε ε 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 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

3 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 3

4 Brif viw of Singular ntgrals ln Logarithmic singularitis ar ampls of intgrabl singularitis: ln d ln sinc lim ln 4

5 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 5

6 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 if f () is continuous (Th abov rsults translat to singularitis at a point a via th transformation - a.) 6

7 Evaluating Cauchy Principal Valu ntgrals Considr th following intgral: C Eampl Evaluat: d ρ ρ ρ lim, ( C + + ) + ρ C C ( + )( + ) d + lim + + lim C C ( + )( + ρ ρ ) ρ d iρ iφ ( ρ ) iφ iφ dθ + + ρ idθ i dφ i i dφ i + + lim lim ( + )( + ) d C : + ρ, ρ d i d ρ + θ lim iφ C :, i i d i dφ i i φ d iφ iφ ( + )( + ) φ so C + i 7

8 Evaluating Cauchy Principal Valu ntgrals (cont.) d ( + )( + ) Nt, valuat th intgral C using th rsid thorm: C : + ρ, ρ d i d ρ θ iφ C :, i i d i i φ d φ C i[ s f( i) + s f( ) ] i lim i ( i) ( i) + lim i i i i + i i i i( i) + 4i ( + ) + + ( + ) ( + ) Thus C call i + i C + Hnc 8

9 Disprsion lations Assumptions: + ( ), f u iv f is analytic in UHP (including th ral ais) in UHP y Lt i θ ε ε CPV i f( ) i i θ dθ if ε f ( ) ε f f ( + ε ) d lim + d +, C ε + ε f d if if if ε i f v u f u + iv d d d i (rsidu thorm) 9

10 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, rplac:,

11 Disprsion lations (cont.) Hnc, w hav u v ( ) v ( ) u d d Hlbrt u, v ar Hilbrt Transforms of on anothr

12 Disprsion lations: Circuit Thory + Vin ( ω ) H( ω) H ( ω) + ih ( ω) H( ω) V ω / V ω out in V + ( ω) out m ω Assumptions (p(iωt) tim convntion): H ( ω) is analytic in LHP H ( ω), ω () in LHP () Not : A pol in th LHP would corrsponds to a nonphysical 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)

13 Disprsion lations: Circuit Thory C Us th path shown blow: ω ε H( ω ) H( ω ) H( ω+ ε ) i dω lim + dω + iε θ dθ ih( ω), ω ω ω ω ε ω+ ε ε ih( ω) H ( ω ) dω ih( ω) ω ω m H ( ω ) H( ω) dω i ω ω ω ε ω C 3

14 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. 4

15 Disprsion lations: Circuit Thory (cont.) W can also driv an altrnativ form: ( ω ) ( ω ) ( ω ) H H H H ( ω) dω dω dω ω ω ω ω ω ω Similarly, w hav ( 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 ( ω ) H ( ω )( ω) dω H H ( ω) dω dω ω ω ω ω First on : Us ω ω This intgral starts at ro. 5

16 Disprsion lation: Circuit Thory (cont.) Summariing, 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 6

17 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 (polariation pr unit volum) χ ω as ω in LHP 7

18 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). 8

19 Laplac Transform Th Laplac transform is dfind as: st F s f t dt Assum m ( s) t < A γ 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. 9

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

21 Laplac Transform (cont.) γ + i st f ( t) g ( i ( s γ )) ds, t > i γ i W hav ( ( 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

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

23 Laplac Transform (cont.) Evaluation of th Bromwich ntgral t < i f t t > Clos th contour to th right. C st F s ds Clos th contour to th lft. i f t f ( t ) C L st F s ds Th intgrand is analytic insid C. f t i C L m ( s) γ + i γ i st F s ds γ C C ( s) f ( t ) rsidus at pols to th lft of C (This assums that thr ar only pol singularitis.) 3

24 Laplac Transform (cont.) Eampl at f t at st F ( s) dt dt s a F ( s ), ( s ) > a s a ( s at ) s at m ( s) γ + i st f ( t) ds, γ a i > s a γ i For t > : st f ( t) is i s a s a C L a γ ( s) at f t 4

25 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, d i dθ dθ i Unit circl i i d + sin θ, cosθ i + f ( sin θ,cosθ) dθ i f, i d + sidus of f, insid th unit circl i Not : f, + i will b a rational function of 5

26 Eampl: ntgral Evaluation (cont.) dθ sinθ dθ d i + sinθ i Multiply top and bottom by 4i. 4i 4d ( id) + 5i + i + i ( + i) ( + i) ( + i) d i lim ( + i)( + i) i iy 8 3 i i dθ 8 + sinθ

27 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( ) f is /, i.. lim f in th UHP. f d y C :, d i d θ On th larg smicircl w hav Hnc lim f ( ) d lim f ( ) i dθ C f ( ) d f d i rsidus of f ( ) in th UHP C 7

28 Eampl: ntgral Evaluation (cont.) y d ( + 9)( + 4) d d ( 9)( 4) + + f( ) 4 i s f ( 3 i) + s f ( i) ( 3i) d i i lim + lim 3i i ( + 3i) ( 3i) ( + 4) d ( 9) ( i) + i 3 i i + i + + i + 9 i + lim 4 5 i ( + 9) ( + i ) 3i 3i i 5 d + ( + 9)( + 4) m d m s f( ) ( ) f( ) m ( m! ) d 3i i (Not th doubl pol at i!) 8

29 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( ), arg ( in UHP) (Fourir ntgrals) a > (clos th path in th UHP) i a f d y C :, d i d θ Choosing th contour shown, th contribution from th smicircular arc vanishs as by Jordan s lmma. (S nt slid.) Fourir 9

30 Fourir ntgrals (cont.) Jordan s lmma Assum lim f( ), arg ( in UHP) Thn C ia f ( ) d y C :, d i d θ Proof: C / ia ia cosθ asinθ asinθ f ( ) d f ( ) i dθ ma f ( ) dθ / aθ i θ f a ma f ( ) d ma ( θ ) a θ / : sin θ θ asinθ aθ θ sinθ θ 3

31 Fourir ntgrals (cont.) i a f d a > y C :, d i d θ W thn hav ia ia ia rsidus of in th UHP C f d f d i f Qustion: What would chang if a <? 3

32 Eampl: Fourir ntgrals (cont.) y cosλ d, ( a, λ ) >. + a ia cos λ sin λ Us th symmtris of and and th Eulr formula, iλ cos λ+ isin λ iλ + a d (imag. part vanishs by symmtry!) iλ iλ iλ ( ia) s lim + a + a ia ( + ia ) ( ia ) d i i i ia a i a λ aλ a cosλ + aλ a a 3

33 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: γ 3 i a d, < a < + γ 4 γ i γ Considr th contour intgral ovr th path shown in th figur: a a C d d + C + γ γ γ3 γ4 Th intgrand has simpl pols at (n+ ) i, n, ±, ±, Th contribution from ach contour sgmnt in th limit must b sparatly valuatd (nt slid). 33

34 Eponntial ntgrals (cont.) γ :,, d d 3 a lim + γ lim γ 3 d γ : + i, d d, a ia ia a d lim d + + γ 4 γ 3 γ y i i γ γ : + iy, d idy + iy γ a a iay d i dy + + iy ( a ) a dy dy, a < + iy Smallst for : y y 34

35 4 Hnc γ Eponntial ntgrals (cont.) γ : + iy, d idy, 4 a a iay d i dy + + iy a dy, a > ia ( ) i s f ( i) ilim ( i) i a ilim ( i) ilim i i i i lim i ( i ) a ( i ) ( i ) γ 4 a + a ( i ) ( i) ( i) ia i γ 3 γ y i i γ g Altrnativly, f( ) so, h a s f( i) lim i ( d + ) d ai ai a i i 35

36 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 36

37 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: Choos th branch cut to li along th positiv ral ais. y k d, < k < + Assum th branch θ < ( k positiv ral) C C L L ε ε ε ε W ll valuat th intgral using th contour shown ln k ln k( ln + iarg ) k( ln r+ ) k k k ikθ r,< θ < 37

38 ntgration Around a Branch Cut (cont.) y k d, < k < + First, not th intgral ists sinc th intgral of th asymptotic forms of th intgrand at both limits ists: C C L L ε ε ε ε k + k + k which is intgrabl at, k < k which is intgrabl at k >, 38

39 ntgration Around a Branch Cut (cont.) Now considr th various contributions to th contour intgral: y L+ L+ C+ C whr f ( ) f d i s f, ( ) k + C L L ε ε ε ε C : r, d ir dθ, r k k ikθ C C ε k ikθ r ir dθ k ikθ r, r + r ε ε f d lim lim ir dθ C :, d i dφ, iφ iφ k k ikφ C ε k ikφ iφ i dφ k, iφ + ε ε ikφ f d lim lim i dφ 39

40 ntgration Around a Branch Cut (cont.) For th path L : f ( ) k + y L :,, r d dr r L iε iε k k ikε, ε, r k k i( ε kε) iε r r r r dr r dr f ( ) d lim + + C L L ε ε ε ε For th path L : C L ( ε) iε iε k k ik( ε) : i L,, r r d dr r, ε, r r [ ] k k i ε+ k ε r dr ik r dr ik iε r + r + f d lim 4

41 ntgration Around a Branch Cut (cont.) Hnc i k ( ) i s f ( ) ( ) i lim + i i i i ik k k k + 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 4

42 Proof of symmtry proprty: Appndi H * ( ω) H ( ω) + Vin ( ω ) H( ω) H ( ω) + ih ( ω) V + ( ω) out mpuls rspons: iωt ht H( ω) dω ral H( ω) V ω / V ω out in iωt ( ω) H h t dt so Us ω ω * * iωt * iω t * iω t * iωt h t h t H d H d H d H d ( ) ( ) ( ) ω ω ω ω ω ω ω ω Hnc iωt * iωt H( ω) dω H ( ω) dω * H ( ω) H( ω) 4