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

Transcription

1 Chapr VII Spcial Fucios Ocobr 7, CHAPTER VII SPECIAL FUNCTIONS Cos: Havisid sp fucio, filr fucio Dirac dla fucio, modlig of impuls procsss 3 Si igral fucio 4 Error fucio 5 Gamma fucio E Epoial igral fucio 6 Bssl fucios. Bssl quaio of ordr ν (BE). Sigular pois. Frobius mhod 3. Idicial quaio 4. Firs soluio Bssl fucio of h s kid 5. Scod soluio Bssl fucio of h d kid. Gral soluio of Bssl quaio 6. Bssl fucios of half ordrs sphrical Bssl fucios 7. Bssl fucio of h compl variabl Bssl fucio of h 3 rd kid (Hakl fucios) 8. Propris of Bssl fucios: - oscillaios - idiis - diffriaio - igraio - addiio horm 9. Graig fucios. Modifid Bssl quaio (MBE) - modifid Bssl fucios of h s ad h d kid. Equaios solvabl i rms of Bssl fucios - Airy quaio, Airy fucios. Orhogoaliy of Bssl fucios - slf-adjoi form of Bssl quaio - orhogoal ss i circular domai - orhogoal ss i aular fomai - Fourir-Bssl sris 7 Lgdr Fucios 8 Erciss

2 48 Chapr VII Spcial Fucios Ocobr 7, 7 VII. Havisid Fucio (ui sp fucio) Th Havisid sp fucio H( ) has oly wo valus: ad wih a jump a = whr fucio is o dfid: < H = () > Olivr Havisid ( 85-95) Graphically i ca b show as: > plo(havisid(),=-3..3); H Shifig of h sp fucio alog h -ais: < a H ( a) = () < a > plo(havisid(-),=-..4); H( ) filr fucio Th filr fucio ca b cosrucd i rms of h sp fucio: < a F(,a,b ) = H( a) H( b ) = a < < b > b (3) I cus h valus of fucios o zro ousid of h irval [ a,b] : > F(,,3):=Havisid(-)-Havisid(-3); > plo(g()*f(,,3),=-..5); F (,,3) g Th Havisid sp fucio is usd for h modlig of a sudd icras of som quaiy i h sysm (for ampl, a ui volag is suddly iroducd io a lcric circui) w call his sudd icras a spoaous sourc. Th filr fucio ca b usd for rprsaio of h pic-wis coiuous fucios.

3 Chapr VII Spcial Fucios Ocobr 7, 7 48 VII. Dirac Fucio (dla fucio) Th Dirac dla fucio δ is o a fucio i h radiioal ss i is rahr a disribuio a liar opraor dfid by wo propris. Th firs dscribs is valus o b zro vrywhr cp a = Paul Dirac ( 9-984) δ =, (4a) Th scod propry provids h ui ara udr h graph of h dla fucio: h h δ d = for ay h > (4b) Th dla fucio is vaishigly arrow a = bu vrhlss closs a fii ara. I is also kow as h ui impuls fucio. Th Dirac dla fucio ca b rad as h limi (i orm o poi by poi limi) of h squc of h followig fucios: a) rcagular fucios: ( + ) ( ) H h H h δ = lim Sh = lim h h h b) Gauss disribuio fucios: δ = lim Gσ = lim σ σ σ σ c) riagl fucios: δ = limδ, δ ( ) h h d) Cauchy dsiy (disribuio) fucios: δ = lim D = lim h ( + ) < h + h< < h h = + < < h h h > h ) si fucios: δ si = lim

4 48 Chapr VII Spcial Fucios Ocobr 7, 7 Propris ) Esio of h irval of igraio o all ral umbrs sill kps h ui ara udr h graph of h dla fucio: δ d = ) Th Dirac dla fucio is a gralizd drivaiv of h Havisid sp fucio: δ = dh d I ca b obaid from h cosidraio of h igral from h dfiiio of h dla fucio wih variabl uppr limi. I is obvious, ha < δ ( ) d = = H > Thrfor, h sp fucio is formally a aidrivaiv of h dla fucio which ow ca b irprd as a drivaiv of a discoiuous fucio. 3) Shifig i : δ ( a) = a+ c a c 4) Symmry: = a a δ a d =, c > δ δ ( ) = δ ( a) = δ ( a ) 5) Drivaivs: δ = δ Th drivaiv ca b dfid as a limi of riagl fucios ad irprd as a pur orqu i mchaics. Th highr ordr drivaivs of h dla fucio ar: ( k ) k k! δ = ( ) δ k =,,... k 6) Scalig: δ = for a a ( a) δ 7) Thr ar som impora propris of h dla fucio which rflc is applicaio o ohr fucios. If f is coiuous a = a, h δ ( ) = δ ( ) f a f a a c b δ ( ) = f a d f a b < a < c δ ( ) = f a d f a δ ( ) = ( ) f a d f a H a a

5 Chapr VII Spcial Fucios Ocobr 7, Applicaios Igraio wih drivaivs of h dla fucio (igraio by pars): f δ ( ) d = f δ f δ ( ) d = f f δ ( ) d = f δ f δ ( ) d = f = f 8) Laplac rasform: s { δ } δ L = d = s { δ } δ as a > L a = a d = 9) Fourir rasform: i iaω a > ω { δ } δ F a = a d = Th dla fucio is applid for modlig of impuls procsss. For ampl, h ui volumric ha sourc applid isaaously a im = is dscribd i h Ha Equaio by h dla fucio: u k u = δ ( ) If h ui impuls sourc is locad a h poi r = r ad rlass all rgy isaaously a im =, h h Ha Equaio has a sourc u k u = δ ( ) δ ( r r ) Impuls modls ar usd for calculaio of h Gr s fucio for o-homogous DE. Th ohr irpraio of h dla fucio δ ( ) as a forc applid isaaously a im = yildig a impuls of ui magiud. Eampl Cosidr IVP: ui impuls is imposd o a dyamical sysm iiially a rs a = 5 : y + 9 y = δ 5 iiial codiios: y = y = Soluio: Apply h Laplac rasform o h giv iiial valu problm (us h propry of h Laplac rasform): 5s s Y + 9Y = Solv h algbraic quaio fory : 5s Y = s + 9 Th ivrs Laplac rasform yilds a soluio of IVP: y( ) = H ( 5) si 3( 5) 3 Th graph of h soluio shows ha h sysm was a rs uil h im = 5, wh a impuls forc was applid yildig udumpd priodic oscillaios.

6 484 Chapr VII Spcial Fucios Ocobr 7, 7 VII.3 Si Igral Fucio Th si igral fucio is dfid by h formula: = si d Si ( ) (, ) (5) Th igrad ca b padd i Taylor sris ad h igrad rm by rm yildig a sris rprsaio of h si igral fucio: Graphically i ca b show as: = + ( ) ( + )( + ) Si = (6)! > plo(si(),=-5..5); Si Th limiig valus of h si igral fucio ar drmid by h Dirichl igral (impropr igral) siω dω = ω which ca b obaid as a paricular cas of h Fourir rasformaio of h sp fucio. Gibbs phoma i h Fourir sris approimaios of fucios wih jumps ar cocd o h propris of si igral fucio. sic fucio Th fucio sic is dfid as: si si c( ) = = I is kow as h sphrical Bssl fucio of zro ordr j ( ) (s Scio VII.6.6, p.498, Eq.(35).

7 Chapr VII Spcial Fucios Ocobr 7, VII.4 Error Fucio Th rror fucio is h igral of h Gauss dsiy fucio shadd ara rf ( ) d = rf ( ) = d ( ), (7) Gauss dsiy > plo(rf(),=-4..4); rf ( ) Th complimary rror fucio is dfid as rfc( ) = rf = d ( ), (8) > plo(rfc(),=-4..4); rfc( ) Powr sris pasio of h rror fucio: rf = = + ( )! ( + ) Drivaivs of h rror fucio: d d rf = d d rf 4 =

8 486 Chapr VII Spcial Fucios Ocobr 7, 7 VII.5 Gamma Fucio Dfiiio Th Gamma fucio appars i may igral or sris rprsaios of spcial fucios. Gamma fucio was iroducd by Loard Eulr i 79 who ivsigad h igral fucio p q ( ) d p,q which for aural valus p,q is qual o p!q! p q! ( + + ) Wih som rasformaio of his igral ad akig h limis, Eulr dd up wih h rsul ( l ) d =! Γ ( + ) Lar, h gamma fucio was dfid by h impropr igral which covrgs for all cp of ad gaiv igrs (Eulr, 78): Γ d (9) = > plo (GAMMA(), = -5..5); Γ Propris a) Γ ( ) = Γ + () Γ ( + ) = = = = ( + ) d d d + d lim = = = Γ d

9 Chapr VII Spcial Fucios Ocobr 7, b) Wh = ( ) (! ) is a aural umbr h Γ = =,,3,... ( + ) =! Γ =,,,... providd ha! = () Th gamma fucio is a gralizaio o ral umbrs of a facorial (which is dfid oly for o-gaiv igrs). Proof: Γ ( ) = h usig propry (a) Γ ( ) = Γ ( + ) = Γ = =! Γ ( 3 ) = Γ ( + ) = Γ = =! h by mahmaical iducio c) Th gamma fucio dos o is a zro ad gaiv igrs. digamma fucio d) Th gamma fucio is diffriabl vrywhr cp a =,,,.... I is a diffriabl sio of h facorial. Th drivaiv of h gamma fucio is calld h digamma fucio. I is dod by = Ψ Γ Ψ ) Sirlig formula (approimaio for larg, > 9 ) Γ + ( ) f) Calculaio of gamma fucio: Laczos approimaio i Forra or C++ Numrical rcips. g) Biomial cofficis: Γ ( z+ ) Γ z z! = = w w! ( z w )! Γ w+ z w+ ()

10 488 Chapr VII Spcial Fucios Ocobr 7, 7 E. Epoial igral fucios E = Th h ordr poial igral fucio E µ is dfid by quaio E = µ dµ =,,3,... (E-) or alraivly, by chag of variabl µ =, i is dfid as E = d =,,3,... (E-) I paricular, for =, h firs poial igral is rducd o o of h followig alraiv forms µ E = µ dµ E = d E = d Th h poial igral is dfid as E = (E-3) Th graphs of h firs hr poial igrals is show blow. E E E = E3 = E lim E E3 ( ) =, =,,,3,... lim E = Valus of poial igral a = ar E E = = E = =,3,... Ei =

11 Chapr VII Spcial Fucios Ocobr 7, Eulr s cosa fucio Ei( ) Numrical calculaio of h poial igrals is o so rivial. Diffr sris pasios, liarizaio, approimaios ad h rcurrc rlaioships ar usd i pracic: 3 E = γ l = γ l + ( ) (E-4)! 3 3! =! Ei( ) 3 E = + ( γ + l ) +...! 3! E = + O ( γ ) E O = + + (my simaio basd o E3 = + O( ) E3 = + γ + l ! 4! l ) whr γ = d is calld Eulr s cosa (s also VII.6.5, Eq.(7)) ad h supplmal poial igral fucio is dfid as 3 Ei = +... = d! 3 3! (E-5) Asympoic pasio for larg valus of (i my FORTRAN cod, for > 5 ) ( + ) ( + )( + ) E+ = (E-6) Diffriaio of poial igrals d E = = E d d E ( ) = E ( ) =,3,... d Igraio of h poial igrals E d = E + C + Rcurrc rlaioship E A algbraic rcurrc rlaioship bw poial igrals of coscuiv ordrs ca b obaid by applicaio of igraio by pars rul o dfiiio (E-) (rcis): + E =, =,,3,... (E-7)

12 49 Chapr VII Spcial Fucios Ocobr 7, 7 Epoial igrals dscribig radiaio i paricipaig mdium E3 E τ dcays vry fas i opically hick mdium τ dcays fas Moms of E ( ) Moms of poial igrals: E E d = E d = 3 d = E d = Algorihm Algorihm for umrical calculaio of h poial igrals E ( ) If 5 h asympoic pasio (Eq. E-6) is applid < h ) h supplmal poial igral If 5 If Ei is calculad firs usig h sris pasio (E-5): 3 Ei =...! + 3 3! ) h h s ordr poial igral is calculad usig quaio E-4: = h E = γ l + Ei( ) 3) h poial igrals, =,3,..., ar calculad usig h rcurrc rlaioship (E-7): = E E + E =. som vry big umbr E =, =,3,... FORTRAN Th FORTRAN subroui basd o his algorihm

13 Chapr VII Spcial Fucios Ocobr 7, 7 49 FORTRAN subroui for calculaio of h firs hr poial igrals VPS 3 Lodo, 5 Lyo INPUT: OUTPUT: E=Ei(,), E=Ei(,), E3=Ei(3,) SUBROUTINE Ei(,E,E,E3) IMPLICIT NONE DOUBLE PRECISION Eulr DOUBLE PRECISION,Eik,Ei,ps,ps DOUBLE PRECISION E,E,E3 DOUBLE PRECISION EB,Ek,EB,EB,EB3 INTEGER i,k,kb,n Eulr= d ps=.d-5 IF (.LT..d) THEN wri (*,*) ' is gaiv' END IF IF (.GT..d) THEN IF (.LT.5.d) THEN! sris pasio of Ei Ei= Eik= k= k=k+ Eik=-Eik*/k/k*(k-) IF (abs(eik).gt.ps) THEN Ei=Ei+Eik GO TO ELSE END IF E=-dlog()-Eulr+Ei! rcurrc rlaioship E=(dp(-)-*E) E3=(dp(-)-*E)/ ELSE! asympoic pasio for >5 ps=.5d! calculaio of Ei(,) N= k= EB= Ek= Ek=-Ek*(N+k)/ IF (abs(ek).gt.ps) THEN EB=EB+Ek k=k+ GO TO ELSE kb=k E=EB*p(-)/ END IF! rcurrc rlaioship E=(dp(-)-*E) E3=(dp(-)-*E)/ END IF ELSE E=.d E=.d E3=.5d END IF RETURN END

14 49 Chapr VII Spcial Fucios Ocobr 7, 7 Igro-Epoial Fucios E ( ) ad E ( ) E3 ( ) E. From h Nos Th Gralizd SLW Mhod 3 = E3 = E = ( ) + E E d E E d = = d E E 3 d E Aidrivaivs of poial igrals (hy ar usd i aalyical soluio of h Eac SLW modl): AE = E d = + ( ) E AE E3 3 AE = E d = + ( ) + E 3 3 = AE E 6 = (? Chck) AE AE3