Mutipy by r sin RT P to get sin R r r R + T sin (sin T )+ P P = (7) ffi So we hve P P ffi = m (8) choose m re so tht P is sinusoi. If we put this in b

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1 Topic 4: Lpce Eqution in Spheric Co-orintes n Mutipoe Expnsion Reing Assignment: Jckson Chpter Lpce Eqution in Spheric Coorintes Review of spheric por coorintes: x = r sin cos ffi y = r sin sin ffi z = r cos The three unit vectors er = (sin cos ffi; sin sin ffi; cos ) ; e = (cos cos ffi; sin sin ß; sin ) ; effi = ( sin ffi; cos ffi; )) form right hne coorinte system in the sense (r;;ffi). The unit of voume is where the eement of soi nge is the et function in spheric por coorintes is n Lpce's eqution is r ffi 3 (r r )= 3 x = r sin rffi=r r Ω Ω = (cos ) ffi () r sin ffi (r r ) ffi ( ) ffi (ffi ffi ) (3) + Φ + (4) r Cn the co-orintes be seprte? (not obvious) The nswer is yes, this is n importnt exmpe of seprbe coorinte system for the Lpcin. Let Φ=R(r)T()P(ffi) (5) so TP r r r R + RP sin r (sin T )+ RT r sin P = (6) ffi

2 Mutipy by r sin RT P to get sin R r r R + T sin (sin T )+ P P = (7) ffi So we hve P P ffi = m (8) choose m re so tht P is sinusoi. If we put this in bove n ivie by sin Choose R r r R + T sin (sin T ) m = (9) sin r R = const > () R r so tht R is not "wiggy" function. We cn imgine tht this is going to be ppicbe to spheric chrge istributions where the ri prt ies or grows with rius. As n nstz, tke so tht R r of c is given then + c =! r R = r R = r r R = r + r R = ( +)r r R = ( +)=const = c () = +p +4c = p +4c so + =. Thus we cou terntivey choose = s the constnt rther thn c n write the two soutions R = A r + B r () n r r R = A r + ( +)B r r r = A (+)r +(+)B r = (+)R=CR

3 So we finy hve the eqution for : or sin ( +)+ Tsin sin T (sin T ) m = (3) sin + ( +) m sin T = (4) The function T for the speci cse tht n m re integers (n speci initi conitions) is ce the ssocite Legenre function P m (cos ) : P m (x) stisfies x x x P m (x) + ( +) m P m (x)= (5) x This is the first time in the course tht we hve seen non-trivi exmpe of ifferenti eqution of the form x (p (x) f i)+q (i) f i = (6) where f i = f i (x), i cn be coection of inices, not necessriy integers, n both p (x) n q (i) (x) re given functions. If it is the cse tht p (x) n q (x) for rnge of x (or if both re negtive) then the soutions wi be oscitory in tht rnge since if f >, then x (pf ) < or pf is ecresing n since p>, f is ecresing. Eventuy f wi become negtive, f wi ecrese n pss through zero n its sign wi chnge, yieing the inverse of the bove rgument, so the gener behvior of f i when p> n q i >. The simpest exmpe is p = const; q = const so p> n q>wehve f k + k f k =!f k /e ±ikx (7) A cruciy importnt property of these oscitory functions is their orthogonity. We've seen this property for sines n cosines - now ook t it more genery. To fin the orthogonity conition, write for two ifferent i's syi; j Cross mutipy n subtrct to get But the first term is which is triviy integrbe Z b (pf i) + q (i) f i = pf j + q(j) f j = f j (pf i) f i pf j + q(i) q (j) fi f j = (8) x x x fj pf i f i pf jλ (9) Z b fj pf i f i pfjλ + q(i) q (j) fi f j x = () 3

4 Λ Λ Z b p (b) f j (b) f i (b) f i (b) f j (b) p () f j () f i () f i () f j () = q(j) q (i) fi f j x () If the points &b re chosen in such wy tht the eft hn sie of this eqution vnishes (e.g., in Legenre's eqution where p =x,choose = ;b= +) n if the right hn sie is proper integr, then we get Z b (q (j) q (i) )f i f j x = () Thus if q (j) q (i) 6=,wehve tht the functions f i n f j re orthogon with the weight function q (j) q (i) Strt with the m = cse which yies P (cos ) P (cos ) the Legenre function, or if is positive integer or zero, the Legenre poynomis. Then p =x ;q (i) =(+) (3) We therefore get the resut Z P P x = 6= (4) or Z P P x = ffi N () (5) More genery, ifm6=then q m = ( +) m x n so q m q m = ( +) ( +) (6) so Z where N m is normiztion ftor; P m (x) P m (x) x = ffi N ()m (7) N m = Z (P m ) x (8) n its vue is convention since the ifferenti eqution is iner n so ny mutipe of soution is soution. Note tht for sinusois in which p =;q = k > we cn rrnge the.h.s. to vnish by choosing &b suitby e.g. for k = integer= n we hve for exmpe Z ß sin nx sin mx x =if n 6= m (9) etc. In gener in soving the Sturm-Liouvie eqution, the points (if ny) t which p (x s )= re ce singur points since t them the n erivtive term vnishes from the eqution. If you were to ivie the eqution by p (x), i.e., write it s f i + P p f i + q (i) p f i = (3) 4

5 Then p = gives troube so x s where p (x s ) re the singur pointsof the eqution n except in speci cses, the soutions hve singurities t these points. Furthermore, if you mke series expnsion for to represent the function bout some point (tke Tyor series expnsion bout some point), the rius of convergence in the compex pne (exten the inepenent vribe x to the compex pne z = x + iy) is circe from the point of expnsion to the nerest singur point in the compex pne. For Legenre's eqution with m =,wehve x x P x + ( +)P = (3) One fins tht if is n integer tht the series soutions (for certin initi conitions) of this form re in fct simpe poynomis, ns so x = ± hve no probems. As usu, since this is n orer ifferentieqution, we nee two initi conitions; e.g. vue n sope t x =, to get prticur soution. As with sinusois we tke the "cosine-ike" function to hve c P (x =)= c P(x=)= (cosine-ike) (3) n s P (x =)= The bove nottion is by the wy not convention (more ter). To unerstn wht hppens, write Then if you pug this into x s P (x=)= (sine-ike) (33) P (x) = X n= n set ech coefficient ofpower of x to zero, you get n x n (34) x P + ( +)P = (35) m+ = m (m +)(+) m (36) (m+)(m+) For the cosine ike soution we hve o =; = n the eqution for the coefficients gives ; 4 ; 6 ::: 6=, ; 3 ; 5 ::: =. So c P (x) =+ (+) (+) 3(+) x + x 4 +::: (37) 3 4 Cery c P (x) = c P (x)n even function bout the origin n ifisneven number, the series termintes with the n = th term since the n =(+) th term vnishes n rger ones o so. So, if ) = even c P (x) is poynomi nit is usuy notte simpy by C P (x) 5

6 b) = o, c P (x) is n infinite series n it cn be shown to iverge ogrithmicy t x = ±. These functions re usuy ce D Q (x) (even function, o). For the sine-ike soutions we hve o =; = n the eqution gives ; 4 ; 6 ::: =, ; 3 ; 5 ::: 6=. So s P (x) =x+ (+) (+) 3 4(+) x 3 + x 5 +::: (38) Cery s P (x) = s P (x), n o function bout x = n if is n o number, the series termintes n we hve poynomi. So, if ) = o, s P (x) is poynomi n it is usuy simpy ce P (x) which is n o function when is o b) =even, s P (x) is n infinite series ivergent tx=±;usuy ce Q (x) (n o function where is even). Finy, ifis not n integer, the series re infinite n ivergent s x!±. So, in summry, with the convention normiztion constnts chosen to mke P (+) = P o (x) = P (x) = x P (x) = P 3 (x) = 3x3 5x3 3x R From the gener theory n the fct tht there re no ifficuties with the integr P P x we get Z P (x) P (x) = ()+ ffi (39) Jckson gives recursion retions n exmpes of expnsions. IMPORTANT FACT: The Legenre poynomis re the ony soutions owe in probems in which the points =n = ß re contine in the voume uner consiertion. For circumstnces which o not incue both these points, other soutions to Legenre's eqution show up we won't be concerne with these. 6