2 Dirac delta function, modeling of impulse processes. 3 Sine integral function. Exponential integral function

Transcription

1 Chpter VII Speil Futios Otober 7, CHAPTER VII SPECIA FUNCTIONS Cotets: Heviside step futio, filter futio Dir delt futio, modelig of impulse proesses Sie itegrl futio 4 Error futio 5 Gmm futio E Epoetil itegrl futio 6 Bessel futios. Bessel equtio of order (BE). Sigulr poits. Frobeius method. Idiil equtio 4. First solutio Bessel futio of the st kid 5. Seod solutio Bessel futio of the d kid. Geerl solutio of Bessel equtio 6. Bessel futios of hlf orders spheril Bessel futios 7. Bessel futio of the omple vrible Bessel futio of the rd kid (Hkel futios) 8. Properties of Bessel futios: - osilltios - idetities - differetitio - itegrtio - dditio theorem 9. Geertig futios. Modified Bessel equtio (MBE) - modified Bessel futios of the st d the d kid. Equtios solvble i terms of Bessel futios - Air equtio, Air futios. Orthogolit of Bessel futios - self-djoit form of Bessel equtio - orthogol sets i irulr domi - orthogol sets i ulr fomi - Fourier-Bessel series 7 egedre Futios 8 Eerises

2 48 Chpter VII Speil Futios Otober 7, 8 VII. Heviside Futio (uit step futio) The Heviside step futio H( ) hs ol two vlues: d with jump t where futio is ot defied: < H () > Oliver Heviside ( 85-95) Grphill it be show s: > plot(heviside(),-..); H Shiftig of the step futio log the -is: < H ( ) () < > plot(heviside(-),-..4); H( ) filter futio The filter futio be ostruted i terms of the step futio: < F(,,b ) H H( b ) < < b > b () H H( ) It uts the vlues of futios to zero outside of the itervl [,b] > F(,,):Heviside(-)-Heviside(-); > plot(g()*f(,,),-..5); F (,,) g : The Heviside step futio is used for the modelig of sudde irese of some qutit i the sstem (for emple, uit voltge is suddel itrodued ito eletri iruit) we ll this sudde irese spoteous soure. The filter futio be used for represettio of the piee-wise otiuous futios.

3 Chpter VII Speil Futios Otober 7, 8 48 VII. Dir Futio (delt futio) The Dir delt futio δ is ot futio i the trditiol sese it is rther distributio lier opertor defied b two properties. The first desribes its vlues to be zero everwhere eept t Pul Dir ( 9-984) δ, (4) The seod propert provides the uit re uder the grph of the delt futio: h h δ d for h > (4b) The delt futio is vishigl rrow t but evertheless eloses fiite re. It is lso kow s the uit impulse futio. The Dir delt futio be treted s the limit (i orm ot poit b poit limit) of the sequee of the followig futios: ) retgulr futios: ( ) ( ) H h H h δ lim Sh ( ) lim h h h b) Guss distributio futios: δ lim Gσ lim e σ σ σ σ π ) trigle futios: δ limδ, δ ( ) h h d) Cuh desit (distributio) futios: δ lim D lim h ( ) π <h h< < h h < < h h h > h e) sie futios: δ si lim π

4 48 Chpter VII Speil Futios Otober 7, 8 Properties ) Etesio of the itervl of itegrtio to ll rel umbers still keeps the uit re uder the grph of the delt futio: δ d ) The Dir delt futio is geerlized derivtive of the Heviside step futio: δ dh d It be obtied from the osidertio of the itegrl from the defiitio of the delt futio with vrible upper limit. It is obvious, tht < δ ( t ) dt H H δ > Therefore, the step futio is formll tiderivtive of the delt futio whih ow be iterpreted s derivtive of disotiuous futio. δ ( ) ) Shiftig i : δ ( ) 4) Smmetr: δ d, > δ δ ( ) δ ( ) δ ( ) 5) Derivtives: δ δ The derivtive be defied s limit of trigle futios d iterpreted s pure torque i mehis. The higher order derivtives of the delt futio re: ( k ) k k! δ ( ) δ k,,... k 6) Slig: δ for δ 7) There re some importt properties of the delt futio whih reflet its pplitio to other futios. If f is otiuous t, the δ ( ) δ ( ) f f b δ ( ) f d f b < < δ ( ) f d f δ ( ) ( ) f t t dt f H

5 Chpter VII Speil Futios Otober 7, 8 48 Applitios Itegrtio with derivtives of the delt futio (itegrtio b prts): f δ ( ) d f δ f δ ( ) d f f δ ( ) d f δ f δ ( ) d f f 8) ple trsform: s { δ } δ e d s { δ } δ s > e d e 9) Fourier trsform: i iω > ω { δ } δ F e d e The delt futio is pplied for modelig of impulse proesses. For emple, the uit volumetri het soure pplied istteousl t time t is desribed i the Het Equtio b the delt futio: u k u δ ( t) t If the uit impulse soure is loted t the poit r r d releses ll eerg istteousl t time t t, the the Het Equtio hs soure u k u δ ( t t ) δ ( r r ) t Impulse models re used for lultio of the Gree s futio for o-homogeeous DE. The other iterprettio of the delt futio δ ( t t ) s fore pplied istteousl t time t t ieldig impulse of uit mgitude. Emple Cosider IVP: uit impulse is imposed o dmil sstem iitill t rest t t 5 : 9 δ ( t 5) Iitil oditios:,. Solutio: Appl the ple trsform to the give iitil vlue problem (use the propert of the ple trsform): 5s s 9 e Solve the lgebri equtio for : 5s e s 9 The iverse ple trsform ields solutio of IVP: ( t) H ( t 5) si ( t 5) The grph of the solutio shows tht the sstem ws t rest util the time t 5, whe impulse fore ws pplied ieldig udmped periodi osilltios.

6 484 Chpter VII Speil Futios Otober 7, 8 VII. Sie Itegrl Futio The sie itegrl futio is defied b the formul: sit dt t Si ( ) (, ) (5) The itegrd be epded i Tlor series d the itegrted term b term ieldig series represettio of the sie itegrl futio: Si() ( ) ( )( ) Si (6)! Grphill it be show s: > plot(si(),-5..5); Si The limitig vlues of the sie itegrl futio re determied b the Dirihlet itegrl (improper itegrl) siω π dω ω whih be obtied s prtiulr se of the Fourier trsformtio of the step futio. Ci() os t Ci( ) dt (6b) t! Gibbs pheome i the Fourier series pproimtios of futios with jumps re oeted to the properties of sie itegrl futio. si futio The futio si is defied s: siπt t si ( ) πt t j ( ) or si It is kow s the spheril Bessel futio of zero order j ( ) (see Setio VII.6.6, p.498, Eq.(5), see lso p.546.

7 Chpter VII Speil Futios Otober 7, VII.4 Error Futio The error futio is the itegrl of the Guss desit futio shded re π t erf ( ) e dt erf ( ) π t e dt (, ) (7) Guss desit e t π erf ( ) erf ( ) > plot(erf(),-4..4); t erf ( ) The omplimetr error futio is defied s erf( ) erf π e t dt ( ), (8) > plot(erf(),-4..4); erf( ) Power series epsio of the error futio: erf π ( )! ( ) Derivtives of the error futio: The me " error futio" d its bbrevitio erf were proposed b. W.. Glisheri i 87 d d d d erf erf e π 4 e π

8 486 Chpter VII Speil Futios Otober 7, 8 VII.5 Gmm Futio Defiitio The Gmm futio ppers i m itegrl or series represettios of speil futios. Gmm futio ws itrodued b eord Euler i 79 who ivestigted the itegrl futio p q ( ) d p,q whih for turl vlues p,q is equl to p!q! ( p q! ) With some trsformtio of this itegrl d tkig the limits, Euler eded up with the result ( l ) d! Γ ( ) ter, the gmm futio ws defied b the improper itegrl whih overges for ll eept of d egtive itegers (Euler, 78): t Γ e t dt (9) > plot (GAMMA(), -5..5); Γ Properties ) Γ ( ) Γ () Γ ( ) t ( ) e t dt e t t t dt de t t t t t e e dt lim t e t t e t dt Γ

9 Chpter VII Speil Futios Otober 7, b) Whe is turl umber the (! ) Γ,,,... Γ ( )!,,,... ( )! Γ,,,... provided tht! () digmm futio Ψ Γ The gmm futio is geerliztio to rel umbers of ftoril (whih is defied ol for o-egtive itegers). Proof: Γ ( ) the usig propert () Γ ( ) Γ ( ) Γ! Γ ( ) Γ ( ) Γ! the b mthemtil idutio ) The gmm futio does ot eist t zero d egtive itegers. d) The gmm futio is differetible everwhere eept t,,,.... It is differetible etesio of the ftoril. The derivtive of the gmm futio is lled the digmm futio. It is deoted b Ψ e) Stirlig formul (pproimtio for lrge, > 9 ) Γ ( ) π e f) Clultio of gmm futio: zos pproimtio i Fortr or C Numeril reipes. g) Biomil oeffiiets: Γ ( z ) Γ z z! w w! ( z w )! Γ w z w ()

10 488 Chpter VII Speil Futios Otober 7, 8 VII.5.E Epoetil itegrl futios The th order epoetil itegrl futio E µ is defied b equtio E µ e dµ,,,... (E-) or ltertivel, b hge of vrible t µ t, it is defied s E t e dt,,,... (E-) I prtiulr, for, the first epoetil itegrl is redued to oe of the followig ltertive forms µ E µ e dµ t e E dt t t e E dt t The th epoetil itegrl is defied s E e (E-) The grphs of the first three epoetil itegrls is show below. E E E E E E lim E E ( ),,,,,... lim E Vlues of epoetil itegrl t re E E E,,... Ei

11 Chpter VII Speil Futios Otober 7, Numeril lultio of the epoetil itegrls is ot so trivil. Differet series epsios, lieriztio, pproimtios d the reurree reltioships re used i prtie: E γ l... γ l (E-4)!!! Ei( ) E ( γ l )...!! E O γ (m estimtio bsed o E O E O( ) 4 E γ l..., where! 4! l ) Euler s ostt t e γ dt t futio Ei( ) is lled Euler s ostt (see lso VII.6.5, Eq.(7)) d the supplemetl epoetil itegrl futio is defied s t e Ei... dt!! (E-5) t Asmptoti epsio for lrge vlues of (i m FORTRAN ode, for > 5 ) ( ) ( )( ) e E... (E-6) Differetitio of epoetil itegrls d E e E d d E ( ) E ( ),,... d Itegrtio of the epoetil itegrls E d E C Reurree reltioship E A lgebri reurree reltioship betwee epoetil itegrls of oseutive orders be obtied b pplitio of itegrtio b prts rule to defiitio (E-) (eerise): e E,,,,... (E-7)

12 49 Chpter VII Speil Futios Otober 7, 8 Epoetil itegrls desribig rditio i prtiiptig medium E E τ des ver fst i optill thik medium τ des fst Momets of E ( ) Momets of epoetil itegrls: E E d E d d E d Algorithm Algorithm for umeril lultio of the epoetil itegrls E ( ) If 5 the smptoti epsio (Eq. E-6) is pplied < the ) the supplemetl epoetil itegrl If 5 If Ei is lulted first usig the series epsio (E-5): Ei...!! ) the the st order epoetil itegrl is lulted usig equtio E-4: the E γ l Ei( ) ) the et epoetil itegrls,,,..., re lulted usig the reurree reltioship (E-7): E e E E. some ver big umber E,,,... FORTRAN The FORTRAN subroutie bsed o this lgorithm:

13 Chpter VII Speil Futios Otober 7, 8 49 FORTRAN subroutie for lultio of the first three epoetil itegrls VPS odo, 5 o INPUT: OUTPUT: EEi(,), EEi(,), EEi(,) SUBROUTINE Ei(,E,E,E) IMPICIT NONE DOUBE PRECISION Euler DOUBE PRECISION,Eik,Ei,eps,eps DOUBE PRECISION E,E,E,EB,Ek,EB,EB,EB INTEGER i,k,kb,n Euler d eps.d-5 IF (.T..d) THEN write (*,*) ' is egtive' END IF IF (.GT..d) THEN IF (.T.5.d) THEN! series epsio of Ei Ei Eik k kk Eik-Eik*/k/k*(k-) IF (bs(eik).gt.eps) THEN EiEiEik GO TO ESE END IF E-dlog()-EulerEi! reurree reltioship E(dep(-)-*E) E(dep(-)-*E)/ ESE! smptoti epsio for >5 eps.5d! lultio of Ei(,) N k EB Ek Ek-Ek*(Nk)/ IF (bs(ek).gt.eps) THEN EBEBEk kk GO TO ESE kbk EEB*ep(-)/ END IF! reurree reltioship E(dep(-)-*E) E(dep(-)-*E)/ END IF ESE E.d E.d E.5d END IF RETURN END

14 49 Chpter VII Speil Futios Otober 7, 8 Itegro-Epoetil Futios E ( ) d E ( ) E ( ) E. From the Notes The Geerlized SW Method E e E ( ) E e E d E E d d E E d e E Atiderivtives of epoetil itegrls (the re used i ltil solutio of the Et SW model): AE E d e E AE E AE E d ( ) e E AE E 6 (? Chek) 4 AE AE

15 Chpter VII Speil Futios Otober 7, 8 49 VII.6 BESSE FUNCTIONS VII.6.. Bessel s Equtio Friedrih Bessel ( ) I the method of seprtio of vribles pplied to PDE i lidril oordites, the equtio of the followig form ppers: ( r) rr ( r) ( r ) R( r) r R r > This equtio is the seod order lier ordir differetil equtio with vrible oeffiiets. It iludes two prmeters d. It is ot of the Euler-Cuh tpe. C be solved b the Frobeius method. Simplif equtio b the hge of vribles to elimite prmeter : R( r) r dr d d dr d dr d R d dr d d d d d dr dr dr dr d d d dr The the differetil equtio beomes Bessel Equtio of order (4) Now the equtio is writte i trditiol vribles, d it iludes ol oe prmeter. This equtio is lled Bessel Equtio of order. Appl power-series solutio to this equtio. VII.6.. Sigulr Poit Sigulr poits of the differetil equtio with vrible oeffiiets re the poits t whih the first oeffiiet beomes zero: is the ol sigulr poit of the Bessel Equtio. Therefore, if we fid power-series solutio roud this poit, it will be overget for ll rel umbers. Determie the tpe of the sigulr poit. Divide the equtio b to rewrite it i the orml form: Idetif oeffiiets of the equtio i orml form: P d Q. Chek if the sigulrit is removble: P is lti p Q is lti q Therefore, is regulr sigulr poit, d the Frobeius theorem be used for solutio of the Bessel Equtio. VII.6.. Idiil Equtio Substitute oeffiiets p d q ito the idiil equtio: p r q r r There re two roots of this equtio: r r (hoose for oveiee, lter we bdo this ssumptio). The Frobeius pproh depeds o the form of the differee of roots of the idiil equtio: r r

16 494 Chpter VII Speil Futios Otober 7, 8 Two ses of the Frobeius theorem m be ivolved: ) r r iteger b) r r iteger this se iludes, N (positive itegers d zero) d (hlf of the odd iteger) I both ses, the first solutio, followig the Frobeius theorem, hs to be foud i the form: r, > (5) Proeed to this solutio, d the we will lze how it hdles the bovemetioed ses. VII.6.4. First Solutio Usig ssumed form of solutio (), lulte the derivtives ( ) ( ) d substitute them ito the Bessel Equtio (): ( ) Divide the equtio b ( ) Reme idies: i the first sum m m ( ) m m m m d ollet the terms m m m ; i the seod sum m Combie both series: m ( ) m( m ) m m (6) m Applig the Idetit Theorem to the term with summtio, we obti reurree reltioship: m for m,,... m m m ( ) Use this reltioship d the first two terms of the equtio (6): m rbitrr m ( ) (b ssumptio, ) m ( ) ( ) ( ) m ( ) m ( 4 ) ( )( ) ( )( ) m

17 m 6 Chpter VII Speil Futios Otober 7, ( 6 ) ( ) Bessel futio of the st kid! 6 4 All oeffiiets with odd idies re equl to zero. Reogizig the ptter, we determie the oeffiiets with eve idies: k k k,,,... k k! ( )( ) ( k ) This epressio m be writte i more elegt form if the gmm Γ : futio is used. Multipl d divide the epressio b k Γ ( ) k k k! Γ ( )( )( ) ( k ) Repetedl usig the propert () of the gmm futio, we squeeze the produt i the deomitor: Γ ( )( )( ) ( k ) Γ ( )( )( ) ( k ) Γ ( k ) The the epressio for the oeffiiets beomes: k k ( ) Γ ( ) k k! Γ ( k ) Choose the vlue for the rbitrr oeffiiet (, the Γ ) k ( ) k k k! Γ ( k ) The the solutio beomes k k k k ( ) k k! Γ k k! Γ k k k This series solutio overges bsolutel for ll beuse there re o other sigulr poits. The futio represeted b this powerseries solutio is lled the Bessel futio of the st kid of order d it is deoted b k k ( ) k! Γ ( k ) k (7) This formul is vlid for rel (iludig itegers d hlf of the odd itegers ). If is iteger (let ), the the gmm futio is repled b the ftoril Γ ( k ) ( k)! d the solutio simplifies to: k k ( ) k! ( k) k,,,... (8)! This is Bessel futio of the st kid of iteger order (iludig zero).

18 496 Chpter VII Speil Futios Otober 7, 8 VII.6.5. Seod Solutio Cse ) r it eger iludig ( ) Beuse ppers squred i the Bessel equtio, the seod solutio be obtied from the first b replemet of b i (8): k k k (9) k! Γ Futios d ( k ) re lierl idepedet. It be show tht the Wroski of Bessel futios (7) d (9) is: siπ W ( ), ( ) π () If is ot iteger, the the Wroski is ot zero d the Bessel futios d re lierl idepedet. The the geerl solutio of the Bessel Equtio m be writte s geerl solutio for it eger () Cse ) Whe is iteger, the Wroski () is equl to zero for >, therefore, Bessel futios of iteger orders d re lierl depedet. We show tht i this se futios d re just multiples of eh other. Ideed, write Bessel futio of egtive iteger order replig b i equtio (9): k k () k k! Γ( k ) d hge the ide of summtio k b s to mke substitutio i the epoetitio k s the k s, d equtio () beomes s s ( ) ( ) () s ( s )! Γ ( s ) Cosider ftor i the deomitor Γ( s ) : whe s (opositive iteger), the gmm futio is ubouded, therefore, the first terms from s to s i the summtio () re equl to zero, the tkig ito out tht for itegers, Γ ( s ) s!, we obti or s s s!s! s!s! s s ( ) s s ( ) So, futio is the futio (4) up to the sig.

19 Chpter VII Speil Futios Otober 7, Therefore, we eed to fid the seod lierl idepedet solutio. Aordig to the Frobeius Theorem., it be foud i the form: k d k k l or we use the redutio formul (setio V..9, p.59, Eq.()) to fid the seod solutio: d where log divisio d the Cuh produt should be used (whih is tedious but mgeble). Bessel futio of the d kid Trditioll, the seod idepedet solutio is itrodued b the defiitio of the Bessel futio of the d kid of order : d for itegers, s the limit osπ for ot iteger (5) siπ lim (6) whih pper to eist for ll, ±, ±,... (or Z ). The followig epressio be derived (see lso p.489): l γ... π (7) γ lim l m m m Euler s ostt Futios hve logrithmi sigulrit t zero, while futios re fiite t zero, tht leds to their lier idepedee. It be show tht the Wroski for these futios is give b W ( ), ( ) π (8) Geerl solutio of Bessel equtio Futios d re lierl idepedet for ll (iludig itegers), d be used for ostrutio of the geerl solutio of the Bessel equtio: for ll (9) Whe the order of the Bessel equtio is ot iteger, the omplete solutio m be lso give ol i the terms of Bessel futios of the first kid: it eger () The seod solutio ws derived mostl for iteger roots, so, we emphsize it b the followig sttemet: the omplete solutio of the Bessel equtio of iteger order is give b: iteger ()

20 498 Chpter VII Speil Futios Otober 7, 8 VII.6.6. Bessel futios of hlf orders It hppes tht futios of orders ± be epressed i terms of elemetr futios. Show it for ±. Cosider BE (4). Use substitutio u u u du d 5 du u 4 d the the equtio beomes 4 u u For ±, this equtio redues to lier ODE with ostt oeffiiets u u the geerl solutio of whih is give b Appl the bk substitutio os si os du d u, the the solutio beomes si If we hoose for ostts π to be osistet with defiitio (7), we obti tht Bessel futios of hlf orders re os π si π It be verified tht Bessel futios of ± re: () si os π os si π (5) spheril Bessel futios j j Other Bessel futios of hlf of odd iteger orders lso be epressed i terms of elemetr futios. These futios re used for ostrutio of spheril Bessel futios j ( ) ( ) π π d si d d os d (6) (7) whih re solutios of the Bessel equtio [ ( ) ], ±, ±,... (8) This equtio ppers s oe of the ODE i the seprtio of vribles of the pli i spheril oordites [Abrmowitz d Stegu].

21 Chpter VII Speil Futios Otober 7, VII.6.7. Bessel futios of the rd kid A lier ombitio i the geerl solutio (9) ssumes tht oeffiiets d re rel umbers. We lso osidered the vrible to be rel umber too. But the obtied equtios d futios re vlid lso for omple umbers. Two prtiulr ombitios of Bessel futios v ( z) d ( z) with omple oeffiiets led to the itrodutio of the omple versio of Bessel futios, whih lso re the solutios of the Bessel equtio but i the field of omple umbers z Z : We defie two Bessel futios of the rd kid of order (the re lso lled Hkel futios) s H H ( ) ( z) ( z) i ( z) ( ) ( z) ( z) i ( z) (9) (4) or we epress them i terms of the Bessel futio ( z) ol if futio ( z) is repled i this defiitio b omple versio of equtio (5): H H ( ) ( z) ( ) ( z) Defiitios (4) d (4) re for πi ( z) e ( z) (4) isiπ πi ( z) e ( z) (4) i siπ v iteger. For,,,,..., we tke the limits: H H ( ) ( z) ( ) ( z) πi e lim (4) isiπ lim πi e i siπ (44) Beuse Hkel futios H z d H z re lier ombitios of Bessel futios of the st d the d tpe, the hve the similr properties. The Wroski of Hkel futios ( ) ( [ ( z), H ) ( z) ] W H H 4i πz ( z) d ( z) H is therefore futios H z d H z form fudmetl set for the Bessel equtio; d the geerl solutio of the Bessel equtio be writte s ( ) ( H ( z) H ) ( z) for order of the Bessel equtio (iludig itegers).

22 5 Chpter VII Speil Futios Otober 7, 8 VII.6.8. Properties of Bessel futios Futios d m roots for > : 4 re both osilltor; the hve ifiitel. The sme properties hold both for d Idetities (45) (46) d d [ ] Differetil idetities ( ) ( ) (47) d d [ ] (48) (49) (5) Itegrl idetities d d (5) (5) k k Additio theorem ( ) ( ) (5) k t VII.6.9. Geertig futios t (55) e t os 4 6 si (Wht if?) 5 7 Epsios of 4 k 5 ( k ) 5 k ( ) 4 4 ( ) 9 6 ( ) k k ( ) 8

23 Chpter VII Speil Futios Otober 7, 8 5 VII.6.. Modified Bessel equtio The modified Bessel equtio is give b ( ) (56) whih be writte i the form of Bessel equtio (4) with the seod prmeter i : ( i ) (57) whih hs geerl solutio give b equtio (9) with repled b i: i i (58) Equtio (58) provides solutio of the modified Bessel equtio (9) i omple form. But it is desirble to hve rel form of solutio. Cosider k k k i i i (59) k k! Γ k k k! Γ( k ) beuse: k k k k k k k k k ( i) ( i) i ( i ) i i i i. The defie futio whih is lled the modified Bessel futio of the st kid of order : k k I i ( i) (6) k! Γ ( k ) whih is rel futio d whih is solutio of the modified Bessel equtio (56). Nottio I for this futio reflets the method of its defiitio, d it mes the futio of imgir rgumet. For egtive vlues of prmeter, defie seod solutio of the modified Bessel equtio s I i ( i) k (6) k k! Γ ( k ) The Wroski of futios I d W I [ I, I ] be lulted s siπ for v iteger π therefore, futios I d I re lierl idepedet d form fudmetl set, the the geerl solutio of the modified Bessel equtio of o-iteger order is give b I I iteger v (6)

24 5 Chpter VII Speil Futios Otober 7, 8 I the se of iteger orders, futio I is the sme s futio I. Ideed, whe is hged for i equtio (6) I [ ] i ( i) i ( ) ( i) i i i ( i) ( i ) I ( ) I I For iteger orders,,,,,, the seod solutio of the modified Bessel equtio is defied with the help of the modified Bessel futio of the d kid of order : K I π I siπ (6) s the limit K K lim (64) Geerl solutio i se v I K v iteger Futios I d I I I K re ot osilltor I ( ) I ( ) I K ( ) K ( ) K

25 Chpter VII Speil Futios Otober 7, 8 5 VII.6.. Equtios solvble i terms of Bessel futios Cosider some geerliztios of the Bessel equtio whih lso be solved i terms of the Bessel futio. ) The geerlized Bessel differetil equtio is writte i the form m α p p If α α ( m ) m p is solutio of the Bessel equtio, the the futio e m α p ( ) is solutio of the geerlized equtio. For iste, for rel (iludig itegers), the geerl solutio be writte s e m α p [ ( ) ( )] p or for o-iteger orders, geerl solutio be writte s p [ ( ) ( )] p m α e Proof of this sttemet be mde b the pproprite hge of vribles d b redutio of the differetil equtio to the Bessel equtio. Emple Chek tht the modified Bessel equtio ( ) is prtiulr se of the geerlized equtio. Rewrite it i the form of the geerlized equtio: from whih we idetif m, α, p, d d, therefore, solutios of the modified Bessel equtio should ilude futios ( i), ( i), d ( i) wht we kow from Setio. Air equtio Emple Cosider the Air equtio whih is the simplest se of the lier d order ODE with vrible oeffiiets. This equtio hs pplitios i dmis (osilltio of gig sprig), qutum mehis d optis. Rewrite the Air equtio i the form of the geerlized equtio Goerge Biddell Air ( 8-89)

26 54 Chpter VII Speil Futios Otober 7, 8 Difrtio o irulr perture is disribed b Air futios from whih we idetif 4 m, α, p,, d 9 9 from the lst equtio we determie the order of the equtio ± The solutios of the Air equtio be writte s i i ( ) If we rewrite Bessel futios of the st kid of omple rgumets i terms of modified Bessel futios usig equtio (95), the the solutios beome I I These two lierl idepedet solutios (ote, tht order of modified Bessel futios is ot iteger) m be used for ostrutio of the trditiol form of solutios Ai I I Bi I I whih re lled Air futios. The et plot shows the grph of Air futios. plot({airai(),airbi()},..); Bi Ai

27 Chpter VII Speil Futios Otober 7, 8 55 b) The et equtio is prtiulr se of the Air equtio, but it is more oveiet for pplitios i simpler ses: p solutios of this equtio hve the forms p p p p p p Solutios of the Air equtio be obtied i this se muh fster. ) Equtio ( e p ) hs solutios ( ) p ( e ) ( ) ( e ) p ( ) p ( e ) Air futios of the omple rgumet

28 56 Chpter VII Speil Futios Otober 7, 8. Orthogolit of Bessel futios We kow from Sturm-iouville theor tht solutios of the self-djoit differetil equtio stisfig homogeeous boudr oditios form omplete set of futios orthogol with some weight futio (Sturm-iouville theorem). Cosider pplitio of this theor to solutios of BVP for BE. Self-djoit form of BE The Bessel equtio of order with prmeter ( ) (65) be redued to self-djoit form with the help of multiplig ftor µ e d e d e After multiplitio of (65) b it be redued to self-djoit form p l where the weight futio be idetified s p. The, the Sturm-iouville Problem i the itervl [, ] produes ifiitel m vlues of the prmeter (eigevlues) for whih there eist o-trivil solutios (eigefutios). Aordig to the Sturm-iouville theorem, the obtied eigefutios re orthogol with the weight futio p : m d for m Sigulr Sturm-iouville Problem We studied regulr Sturm-iouville Problem i whih the ordir differetil equtio is set i the fiite itervl d both boudr oditios do ot vish. I sigulr Sturm-iouville problem ot ll of these oditios hold. Usull, the itervl is ot fiite, d oe or both boudr oditios re missig. Isted of boudr oditios, whe the solutio m ot eist t the boudries, the eigefutios should stisf some limitig oditios. Oe of suh requiremets be the followig: et d be eigefutios orrespodig to two distit eigevlues d, orrespodigl. The the hve to stisf the followig oditio: [ ] lim p [ ] lim p I the other ses the bsee of boudr oditios is beuse of the periodil or led domi, whe we demd tht the solutio should be otiuous d smooth ( ) d ( ) I this se, it is still possible to hve the orthogol set of solutios,. { } o [ ]

29 Chpter VII Speil Futios Otober 7, 8 57 Orthogol sets for irulr domi Cosider BE i the fiite irle. The geerl solutio is give b The phsil sese of solutio of lssil PDE requires fiite vlue of solutio i ll poits of [,]. Bessel futios of the seod kid re ubouded t, therefore, to stisf this oditio we hve to put rbitrr ostt equl to zero. The solutio of BE ( ) Boudr oditios Cosider the homogeeous boudr oditios t : I Dirihlet II Neum III Robi [ H ] Equtios for eigevlues I We re lookig for the vlues of the prmeter whih provide otrivil solutios stisfig boudr oditios I-III. These vlues be foud b substitutio of the solutio ito the boudr oditios I-III s the positive roots of the followig equtios:, d whe II d d hi rule H III Proof: H d H H d H H I the prtiulr se, whe the Bessel futio is of zero order,, equtios for eigevlues re: I ( ) II d whe III H

30 58 Chpter VII Speil Futios Otober 7, 8 Orthogolit Obtied equtios geerte ifiitel m eigevlues,,,,... For whih the orrespodig eigefutios re: { } ( ) The orrespodig set of solutios { ( ) } respet to the weight futio p : is orthogol with ( ) ( ) m d N m m Norm of eigefutios where the squred orm of eigefutios is determied s [Ozisik N. Het Trsfer, p.; Mhl Bessel Futios for Egieers, p.]: N, ( ) d ( ) ( ) or itegrtig with Mple: N, ( ) d ( ) ( ) ( ) The derivtive be epressed s (use hi rule d idetit (5)) d ( ) ( ) d ( ) ( ) or if we use the other idetit for lower order the d ( ) ( ) d The tkig ito out tht eigevlues stisf equtios I-III (tht simplifies epressios), the squred orm for speifi boudr oditios is give: I II III N ( ) or ( ),, N, H N N,

31 Chpter VII Speil Futios Otober 7, 8 59 SOID CINDER se of Equtio obtied b seprtio of vribles i the Het Equtio: R R µ R rr rr rr µ rr Boudr Coditios: R < R( r ) Sturm-iouville Theorem: bouded solutio r r rr r R self-djoit form ( µ ) Eigevlues: Weight Futio: p( r) µ,,... r For solutio, rewrite eq i the form of the Bessel Equtio of order : r R rr r R Geerl Solutio: R ( r) ( r) ( r),, Bouded Solutio: R ( r) ( r) I II Dirihlet problem R( r ) Neum problem R r Eigefutios: R ( r) ( r) Eigevlues: re positive roots of hrteristi equtio: r r Squred Norm: R r ( r) rdr ( r) Eigefutios: R R r r Eigevlues: re positive roots of hrteristi equtio: r r Squred Norm: R r r rdr r r R r r rdr r ( ) ( ) III Robi problem R r HR r Eigefutios: R ( r) ( r) Eigevlues: re positive roots of hrteristi equtio: r H r r H r Squred Norm: R r ( r) rdr ( r)

32 5 Chpter VII Speil Futios Otober 7, 8 Bessel-Fourier Series The obtied orthogol sstems be used for ostrutig the futio epsio i geerlized Fourier series ( ) f where oeffiiets re determied from the equtio ( ) f d ( ) d ( ) f d N, Emple (I Dirihlet boudr oditio) Cosider orthogol set obtied s solutio of Dirihlet problem with Bessel futios where ( { ) ( )} eigevlues re positive roots of equtio ( ) The squred orm of eigefutios be lulted s N, ( ) The epsio of futio f i Fourier-Bessel series hs the form ( ) f d f, where (it is lso kow s the Hkel-Fourier series (869)). Cosider ow epsio of the futio f H ( ), [,] Cse i the Hkel series of order. Coeffiiets re ( ) d N, N, d the epsio beomes f ( ) ( ) ( ) 9 ( ) ( ) This emple be illustrted with Mple presettio (SF-.mws)

33 Chpter VII Speil Futios Otober 7, 8 5 SF-.mws Emple Fourier-Bessel series > u:; > :; I Dirihlet boudr oditio order of Bessel Futios : : > f():-heviside(-); f( ) : Heviside ( ) Chrteristi equtio > w():bessel(u,*); w( ) : Bessel (, ) > plot(w(),..); Eigevlues > d:.5; d :.5 > ::for m from to 4 do :fsolve(w(),m*d..(m)*d):if tpe (,flot) the lmbd[]:: : fi od: > for i to do lmbd[i] od; Eigefutios Squred Norm of eigefutios > N:-;:'':i:'i':m:'m'::''::'': N : 4 > []:Bessel(u,lmbd[]*); : Bessel (, ) >N[]:it(*[]^,..): N[]:subs(Bessel(u,*lmbd[]),N[]): Fourier-Bessel oeffiiets Fourier-Bessel series > []:it(*[]*f(),..)/n[]; : 9 Bessel (, ) Bessel (, ) > u():sum([]*[],..n); 4 u( ) : Bessel (, ) Bessel (, ) 9 Bessel (, ) > u():sum([]*[],..n): > plot({f(),u()},..);

34 5 Chpter VII Speil Futios Otober 7, 8 Emple 4 (II Neum boudr oditio) Cosider orthogol set obtied s solutio of the Neum where eigevlues problem with Bessel futios re positive roots of the equtio ( is lso eigevlue for ) The squred orm of eigefutios be lulted s N, d N, Fourier-Bessel series: ( f ), where > f, where ( ) f d N, f d ( ) f d N, Cse Cosider ow epsio of the futio f H ( ), [,] d 9 N, ( ) d ( ) ( ) N, 9 Mple solutios: SF---.mws SF---.mws

35 Chpter VII Speil Futios Otober 7, 8 5 Emple 5 (III Robi boudr oditio) H Cosider orthogol set obtied s solutio of the Robi problem with Bessel futios ( ) ( ) positive roots of equtio H where eigevlues re The squred orm of eigefutios be lulted s H N, ( ) Mple solutio (for ) SF--.mws Cse Fourier-Bessel series: f ( ) where ( ) f d N,, H Cosider epsio of futio f H ( ), [,]

36 54 Chpter VII Speil Futios Otober 7, 8 ANNUAR DOMAIN Equtio obtied b seprtio of vribles i the Het Equtio: R R µ R rr [ ] r r [ ] r r R R Sturm-iouville Theorem: ( rr ) ( µ ) r R (self-djoit form) r r r Eigevlues: Weight Futio: p( r) µ,,... r For solutio, rewrite it i the form of Bessel Equtio of order : r R rr r R Geerl Solutio: R ( r) ( r) ( r),, I Dirihlet-Dirihlet Problem: R( r ) R( r ) Eigefutios: R ( r) Eigevlues: ( ) ( ) ( ) ( ) r r r r re positive roots of hrteristi equtio: ( ) ( ) ( ) ( ) r r r r Norm: r r R r R r rdr

37 Chpter VII Speil Futios Otober 7, 8 55 STURM-IOUVIE PROBEM Bessel Equtio of order : r R rr r R Cosider BE i the ulr domi ( ), with homogeeous boudr oditios: d h k h d H k k d h d, h H k The geerl solutio is give b The derivtive of the geerl solutio: A BVP for BE i the fiite domi ordig to the Sturm-iouville theorem geertes ifiite set of eigevlues d orrespodig eigefutios orthogol with the weight futio p. A prtiulr form of the orthogol set depeds o the tpe of boudr oditios. I I Dirihlet-Dirihlet [ ] [ ] d II I Neum-Dirihlet d [ ] I II Dirihlet-Neum [ ] d d d 4 II II Neum-Neum d d d 5 I III Dirihlet-Robi [ ] k h d d d 6 II III Neum-Robi d d 7 III I Robi-Dirihlet k h d d k h d [ ] d 8 III II Robi-Neum k h d d d d 9 III III Robi-Robi k h d d k h d

38 56 Chpter VII Speil Futios Otober 7, 8 I - I Dirihlet-Dirihlet Cosider BE i the ulr domi ( ), (, ) Boudr oditios: (Dirihlet) (Dirihlet) Appl boudr oditios to the geerl solutio of BE: ( ) ( ) ( ) ( ) This is homogeeous sstem of two lier lgebri equtios for d. Rewrite it i the mtri form ( ) ( ) We re lookig for o-trivil solutio of BVP, i.e. both oeffiiets i geerl solutio ot be zero A homogeeous lier sstem hs o-trivil solutio ol if the determit of the sstem mtri is equl to zero: Equtio for eigevlues : det ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The roots of this equtio ield the eigevlues for whih BVP hs o-trivil solutios (eigefutios). Osilltor propert of Bessel futios provides ifiite set of eigevlues d orrespodig eigefutios re,, Determie ow the oeffiiets, d, from sstem where eigevlues re substituted ( ) ( ),, Beuse lier sstem hs sigulr mtri, solutios for, d, re lierl depedet d be determied just from oe equtio, let it be the seod oe, ( ), ( ) oe of the ukows i this equtio is free prmeter, hoose,, the, The eigefutios hve the form: Eigefutios ( ) ( )

39 Chpter VII Speil Futios Otober 7, 8 57 The orm of eigefutios is give b: N, d ( ) ( ) ( ) ( ) d d d ( ) ( ) ( )... epress i terms of, ( ) ( ) ( ) d Summr For ulr domi with boudr oditios: Eigevlues re positive roots of the hrteristi equtio ( ) ( ) ( ) ( ) The eigefutios re ( ) ( ) Fourier-Bessel series: f where f d d ( ) f N, d Mple emples: SF-AD--.mws SF-AD--.mws, 5 f H ( )

40 58 Chpter VII Speil Futios Otober 7, 8 II - I Neumt-Dirihlet [igu Wei] Cosider BE i the ulr domi ( ) Boudr oditios: d d, (, ) The geerl solutio is give b (Neum) (Dirihlet) The derivtive of the geerl solutio (use hi rule d differetil idetities) d ( ) ( ) ( ) ( ) ( ) d Appl boudr oditios to the geerl solutio of BE: Deote: ( ) ( ) ( ) ( ) ( ) ( ) The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio: Equtio for eigevlues ( ) ( ) ( )

41 Chpter VII Speil Futios Otober 7, 8 59 The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: Eigefutios: ( ) ( ),, ( ) ( ) ( ) ( ) The orm of the eigefutios is determied b the itegrl N, d, Fourier-Bessel Series: N, d ( ) ( ) ( ) ( ) d f where f d f d N, d

42 5 Chpter VII Speil Futios Otober 7, 8 I -II Dirihlet-Neum [Crig Peterso] Cosider BE i the ulr domi ( ), (, ) Boudr oditios: (Dirihlet) d (Neum) d The derivtive of the geerl solutio (use hi rule d differetil idetities) d ( ) ( ) ( ) ( ) ( ) d The geerl solutio is give b Equtio for eigevlues : Appl boudr oditios to the geerl solutio of BE: Deote: ( ) ( ) ( ) ( ) ( ) The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio: ( ) ( ) ( ) ( ) ( ) ( )

43 Chpter VII Speil Futios Otober 7, 8 5 The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the first oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: Eigefutios: ( ) ( ),, ( ) ( ) ( ) ( ) The orm of eigefutios is give b: N, d ( ) ( ) ( ) ( ) ( ) d d ( ) ( ) ( ) ( ) ( ) d ( ) d Summr: For ulr domi with boudr oditios: (Dirihlet) d (Neum) d Eigevlues re positive roots of the hrteristi equtio [ ( )] ( ) ( ) The eigefutios re ( ) ( ) [ ] ( ) ( ) ( ) ( )

44 5 Chpter VII Speil Futios Otober 7, 8 4 II - II Neum-Neum Cosider BE i the ulr domi ( ), (, ) with homogeeous boudr oditios: d d d d The geerl solutio is give b (Neum) (Neum) The derivtive of the geerl solutio (use hi rule d differetil idetities) d ( ) ( ) ( ) ( ) ( ) d Substitute ito boudr oditios: ( ) ( ) ( ) ( ) Deote: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio:

45 Chpter VII Speil Futios Otober 7, 8 5 Equtio for eigevlues ( ) ( ) Eigefutios: The positive roots of this equtio provide ifiite set of eigevlues (ote for Neum boudr oditios, is lso eigevlue for ). The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: for ( ) ( ),, ( ) ( ) The orm of the eigefutios is determied b the itegrl N, d, N, d d Fourier-Bessel series: f where d d f for f d f d f d N, d d f d f d N, d for

46 54 Chpter VII Speil Futios Otober 7, 8 5 I - III Dirihlet-Robi [Bri ieht] Cosider BE i the ulr domi ( ), (, ) with homogeeous boudr oditios: [ ] (Dirihlet) d k h d k The geerl solutio is give b h H (Robi) The derivtive of the geerl solutio (use hi rule d differetil idetities) d ( ) ( ) ( ) ( ) ( ) d Substitute ito boudr oditios: ( ) ( ) ( ) ( ) ( ) ( ) H ( ) H ( ) Collet terms ( ) H ( ) H ( ) Deote: ( ) ( ) ( ) H ( ) H ( ) The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio:

47 Chpter VII Speil Futios Otober 7, 8 55 Equtio for eigevlues ( ) ( ) H ( ) ( ) H ( ) The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: Eigefutios: ( ) ( ) The orm of the eigefutios is determied b the itegrl N, d Fourier-Bessel series: f where f d d f N, d Mple emple: SF-AD-5-.mws SF-AD-5-.mws, 5 H f H

48 56 Chpter VII Speil Futios Otober 7, 8 6 II - III Neum-Robi [ur Hse] Cosider BE i the ulr domi ( ), (, ) with homogeeous boudr oditios: d d d k h d k The geerl solutio is give b (Neum) h H (Robi) The derivtive of the geerl solutio (use hi rule d differetil idetities) d ( ) ( ) ( ) ( ) ( ) d Substitute ito boudr oditios: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) H ( ) H ( ) Collet terms H H Deote: for : ( ) ( ) ( ) ( ) ( ) ( ) H ( ) H H ( ) H The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det

49 Chpter VII Speil Futios Otober 7, 8 57 This ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio: Equtio for eigevlues ( ) H ( ) ( ) H ( ) The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form:, Eigefutios: ( ) ( ),, ( ) ( ) ( ) H ( ) H The orm of the eigefutios is determied b the itegrl N, d Fourier-Bessel series: f where f d d f N, d

50 58 Chpter VII Speil Futios Otober 7, 8 7 III - I Robi-Dirihlet [Adrew Eldredge] Cosider BE i the ulr domi ( ), (, ) with homogeeous boudr oditios: d k h d [ ] k h H (Robi) (Dirihlet) The geerl solutio is give b The derivtive of the geerl solutio (use hi rule d differetil idetities) d ( ) ( ) ( ) ( ) ( ) d Substitute ito boudr oditios: ( ) ( ) ( ) ( ) H ( ) H ( ) Collet terms ( ) H ( ) ( ) H ( ) Deote: ( ) H ( ) ( ) ( ) ( ) H ( ) The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio:

51 Chpter VII Speil Futios Otober 7, 8 59 Equtio for eigevlues H ( ) ( ) H ( ) The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: Eigefutios: ( ) ( ),, ( ) ( ) ( ) ( ) The orm of the eigefutios is determied b the itegrl N, d Fourier-Bessel series: f where f d d f N, d Mple emple: SF-AD-7-.mws SF-AD-7-.mws, 5 H,

52 Chpter VII Speil Futios Otober 7, III - II Robi-Neum [so Thoms & Tim Pollok] Cosider BE i the ulr domi,, with homogeeous boudr oditios: h d d k k h H (Robi) d d (Neum) The geerl solutio is give b The derivtive of the geerl solutio (use hi rule d differetil idetities) d d Substitute ito boudr oditios: H H Collet terms H H Deote: H H The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio:

53 Chpter VII Speil Futios Otober 7, 8 5 Equtio for eigevlues H ( ) H ( ) Eigefutios: The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: ( ) ( ),, ( ) ( ) ( ) ( ) ( ) The orm of the eigefutios is determied b the itegrl N, d Fourier-Bessel series: f where f d d f N, Mple emple: SF-AD-8-.mws SF-AD-8-.mws, 5 H, H f H d

54 Chpter VII Speil Futios Otober 7, III - III Robi-Robi Cosider BE i the ulr domi,, with homogeeous boudr oditios: h d d k k h H (Robi) h d d k k h H (Robi) The geerl solutio is give b The derivtive of the geerl solutio (use hi rule d differetil idetities) d d Substitute ito boudr oditios: H H H H H H H H Deote: H H H H The sstem for oeffiiets hs the followig mtri form: A eessr oditio for sstem to hve o-trivil solutio is det it ields hrteristi equtio for vlues of the prmeter for whih the BVP hs o-trivil solutio:

55 Chpter VII Speil Futios Otober 7, 8 5 Equtio for eigevlues H ( ) ( ) H ( ) H ( ) ( ) H ( ) The positive roots of this equtio provide ifiite set of eigevlues. The for the determied eigevlues, oeffiiets, d, be foud from oe of the equtios of the sstem (hoose the seod oe): Oe of the oeffiiets be tke s free prmeter, hoose, the With determied oeffiiets, solutios of the BVP (eigefutios) hve the form: Eigefutios: ( ) ( ),, ( ) H ( ) ( ) ( ) H ( ) The orm of the eigefutios is determied b the itegrl N, d Fourier-Bessel series: f where f d d f N, d Mple emple: SF-AD-9-.mws SF-AD-9-.mws, 5 H, H f H

56 54 Chpter VII Speil Futios Otober 7, 8 VII.7 egedre Polomils egedre Equtio Adrie-Mrie egedre ( 75 8) Seprtio of vribles of the pli i spheril oordite sstem ields group of ODEs oe of whih hs the form m [( ) ] ( ) where m d re seprtio ostts. This equtio is lled egedre s ssoited differetil equtio. Solutio of this equtio iludes egedre s ssoited futios of degree d of order m P m Q m. of the st d the d kid d Whe m (i se whe the pli does ot deped o the vrible φ ), equtio is lled the egedre differetil equtio Solutio of this equtio iludes egedre s futios of degree of Q. the st d the d kid P d Solutio of egedre Equtio Cosider the egedre differetil equtio rewritte i stdrd form ( ) ( ) R This equtio hs two sigulr poits ±, ll other poits re ordir poits. We will ppl power-series solutio method roud the ordir poit. The itervl of overgee for this solutio is gurteed to be (, ). Assume tht the solutio is represeted b power series hge k b k k k k the derivtives of the solutio re k k k k k k ( k ) k k Substitute them ito equtio k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k i the first term k k k k k ( ) ( ) { [ k( k ) k ( ) ] ( k )( k ) k } { } k k ( ) ( ) { [ ( ) k( k ) ] ( k )( k ) k } k k

57 Chpter VII Speil Futios Otober 7, 8 55 Usig the ompriso theorem, determie the reltio for oeffiiets: ( ) ( ) ( )( ) k ( k ) ( ) ( k ) ( k ) ( k)( k ) k ( k ) ( k ) k k k,,... Coeffiiets d re rbitrr, osider them to be the prmeters for the geerl solutio d ollet the terms orrespodig to these oeffiiets. The the power series solutio of the egedre Equtio beomes ( ) ( )( )( ) ( )( )( )( )( ) 4 5! 4! 6! ( )( ) ( )( )( 4) ( )( 5)( )( 4)( 6)! 5! 7! et us deote the power series solutio s,, Choose sequee of o-egtive vlues of,,,... Note, tht i the solutio ll terms eept for fiite umber ltertigl dispper: if k is eve the i the first series ll terms with multiple ( k) dispper, if k is odd the i the seod series ll terms with multiple ( k ) dispper, d the beome the fiite polomils. Write these terms epliitl:,, ( ), ubouded t 4 5,..., 5,! 5! [ see Asmr, p.] ( )! (! )!! Choose ( ), ( )!!! The egedre futios of the st kid for differet vlues of prmeter geerte the followig set of polomils

58 56 Chpter VII Speil Futios Otober 7, 8 egedre Polomils P P P P P P P4 P5 P 5 P P whih re lled egedre polomils. Beuse egedre polomils re solutios of the seprted ple equtio i spheril oordites, the re lso lled spheril hrmois (d the method of solutio i terms of egedre futios is lled orrespodigl the Method of Spheril Hrmois). Rell tht this sstem of polomils up to slr multiple ws lso obtied from orthogoliztio of the lier idepedet set of,,,,... o the itervl [,] mooms { }. Reurree formul ( ) P ( ) P P d! d Rodrigues formul P Orthogolit of egedre polomils egedre polomils re orthogol i the itervl [,] with the weight futio p m Pm P d m Fourier-egedre series egedre polomils be used for epsio of the futio f, [, ] i the Fourier-egedre series: f P where epsio oeffiiets re f P P d d f P d

59 Chpter VII Speil Futios Otober 7, 8 57 Itegrl trsform The egedre itegrl trsform is bsed o the Fourier-egedre epsio f f K d with iverse trsform f f K where the kerel of the itegrl trsform K ormlized egedre futio K P is defied s Emple 8 (epsio i Fourier-egedre series (spheril hrmois)) f H [, ] f P dp SF-8.mws Emple 8 Fourier-egedre Series > restrt; > with(orthopol); [ G, H,, P, T, U ] > for from to 6 do P(,) od;

60 58 Chpter VII Speil Futios Otober 7, 8 > f():heviside(); f( ) : Heviside ( ) Fourier-egedre oeffiiets > []:(/)*it(f()*p(,),-..); : P (, ) d Fourier-egedre series: > u():sum([]*p(,),..); 895 u( ) : > plot({f(),u()},-..); 9 > u():sum([]*p(,),..): > plot({f(),u()},-..); oseph Fourier ( 768 8)

61 Chpter VII Speil Futios Otober 7, 8 59 The Best Approimtio b Polomils Cosider vetor spe of squre itegrble futios [,] ll polomils of order is subspe of [,] π. et f [,] d let f P k be the i [,].. Cll it. The sp of th prtil sum of the Fourier-egedre epsio of the futio f ( ) The f ( ) provides the best pproimtio of the futio f ( ) b the th order polomils. Tht mes tht futio f ( ) is the losest to the futio f ( ) mog the futios i π i the sese tht it miimizes the diste ( ) ( ) f p f p,f p f p d for ll f f f p p π

62 54 Chpter VII Speil Futios Otober 7, 8 Grphs of egedre polomils egedre-.mws egedre polomils > restrt; > with(orthopol): > plot({p(,),p(,),p(,)},-..); P P P > plot({p(,),p(4,),p(5,)},-..); P P 5 P 4 > plot({p(6,),p(7,),p(8,)},-..); P 6 P7 P8

63 Chpter VII Speil Futios Otober 7, 8 54 SHIFTED EGENDRE PONOMIAS. Shifted egedre Polomils defied o the itervl [,] re obtied b orthogoliztio (Grm-Shmidt proess) of the power futios (mooms):,,,... (Grm determits) k k k P b,,,... ( k)! ( ) k b k k k! k!,,,...,! k k! ( k)! P P P 6 6 P P P P P ( m) P, P if m orthogol, P P P egedre-fourier series: f ( ) P, k k k ( f, P ) P, P f P d k k ( k ) f P k ( ) d P k d

64 54 Chpter VII Speil Futios Otober 7, 8. Normlized Shifted egedre Polomils defied o the itervl [,] k k k P b,,,... b k ( k) ( )! k! k! k k,,,..., P ( ) P P P P P ( ) P P P ( P, P ) P d (, ) P P δ orthoorml m m egedre-fourier series: f ( ) P, k k k (, ) f P f P d k k k

65 Chpter VII Speil Futios Otober 7, 8 54 Guss-egedre Qudrtures of Itegrtio See [Chihr, p., Theorem 6.], [Guthi], [otes o momets], [otes SW Optimiztio ] grge iterpoltio polomil polomil of degree t most - whose grph psses through the presribed poits (ti,i). Roots of egedre polomils, their properties b [ f ] f w d k f ( k ) [ f ] qudrture k If the formul is et wheever f is polomil of degree m, the qudrture is of preisio m. Defiitio: Guss qudrture is qudrture of degree of preisio -, i whih k re zeroes of the pproprite orthogol polomils (i ft, uique qudrture with odes whih hs degree of preisio -). [ f ] [ f ]

66 544 Chpter VII Speil Futios Otober 7, 8 VII. 8 EXERCISES: ) Show δ ( π) si si δ π δ ( π) Hit: multipl both sides b rbitrr differetible futio d itegrte. ) Solve the IVP d sketh the solutio urves (use Mple d ple trsform): ) ( t ) H ( t ) b) δ ( t ) ) Show δ ( 6) f d f ( ) Prove tht, i geerl, f ( ) δ g ( ) f ( ) d, where g( ) g ( ) 4) Sig futio is defied s sg > < ) Epress sg i terms of Heviside step futio H b) Epress Heviside step futio H i terms of sg d d ) Clulte sg d) Sketh the grph of sg( ) 5) Usig mthemtil idutio (Setio I.4, p.49), prove Equtio (), p.487: Γ ( )! 6) Use term-b-term differetitio (wh we do it?) to show d d 7) Solve i terms of Bessel futios (see VII.6., p.5) ( ) 8) Fiish Emple 5 i setio 6., p.5 for the se

67 Chpter VII Speil Futios Otober 7, ) Use mthemtil softwre of our hoie to sketh the grph of the futio d fid the first five positive roots: ) w( t) 4 ( t) ( t) b), t t t w t os si, t Ptheo, Pris

68 546 Chpter VII Speil Futios Otober 7, 8 The first spheril Bessel futio j () is lso kow s the (uormlized) si futio. Pul Dir