Учебник - Методы решения основных задач уравнений математической физики - Колесникова, страница 8

ª.−Φ′′ (ϕ)| ®¤­®-¬¥à­ë© ®¯¥à â®à ˜âãଠ{‹¨ã¢¨««ï. ®«ã稫 áì §­ ª®¬ ï§ ¤ ç  (á¬. á. 47):½Φ′′ (ϕ) + µ2 Φ(ϕ) = 0,⇐⇒Φ(ϕ + 2π) = Φ(ϕ)⇐⇒ Φ0 (ϕ) = 1, Φn (ϕ) = An cos nϕ + Bn sin nϕ,‘Φ(ϕ)n ∈ N.®¯à¥¤¥«¨«¨áì. ’ ª ª ª ¬ë ¤¥«¨«¨ ¯¥à¥¬¥­­ë¥ ¢ýத­®¬þ ãà ¢­¥­¨¨,   ­¥ ¢ ᮮ⢥âáâ¢ãî饬, â® à¥è¨¬ ãà ¢­¥­¨¥ ¨ ¤«ïr(rR′ (r))′R(r)R:= n2 ⇐⇒ r2 R′′ (r) + rR′ (r) − n2 R(r) = 0 ⇒⇒⇒DnRn (r) = Cn rn + n , n = 1, 2, . . .r(â. ª. íâ® ãà ¢­¥­¨¥ ©«¥à )⇒ R0 (r) = C0 + D0 ln r,Žâá á«¥¤ã¥â, çâ® ¬ë ¯®«ã稫¨ à¥è¥­¨ï ãà ¢­¥­¨ï ‹ ¯« á  ¢ ¢¨¤¥v0 (r, ϕ) = R0 (r) = C0 + D0 ln r, n = 0,µ¶Dnvn (r, ϕ) = Rn (r)Φn (ϕ) = Cn rn + n (An cos nϕ + Bn sin nϕ),rn = 1, 2, .

. .III.’¥¯¥àì, ¢ § ¢¨á¨¬®á⨠®â ⮣®, ª ªãî ¨§ âàñå § ¤ ç à¥-è ¥¬, ¡ã¤¥¬ ¨áª âì à¥è¥­¨¥ ¢ ¢¨¤¥ ä®à¬ «ì­®£® à鸞, 㤮¢«¥â¢®àïî饣® ᮮ⢥âáâ¢ãî騬 ãá«®¢¨ï¬ ¯à¨r=0¨«¨= ∞.1) …᫨ § ¤ ç  ¢­ãâਠªà㣠, â®u(r, ϕ) = C0 +∞X156rn (An cos nϕ + Bn sin nϕ).r=2) …᫨ § ¤ ç  ¢­¥ ªà㣠, â®u(r, ϕ) =∞X1(An cos nϕ + Bn sin nϕ) + C0 .rn13) …᫨ § ¤ ç  ¢ ª®«ìæ¥, â®u(r, ϕ) = C0 + D0 ln r +¶∞ µXDnCn rn + n cos nϕ+r1¶∞ µXBnnAn r + n sin nϕ.+r16.2.

Š ª ­ ©â¨ ç áâ­®¥ à¥è¥­¨¥, ¥á«¨ ᢮¡®¤­ë© ç«¥­f (r, ϕ) = ar n cos mϕ(n + 2)2 6= m2 ?¨«¨f (r, ϕ) = ar n sin mϕ,◮ ‚ í⮬ á«ãç ¥ à¥è¥­¨¥ ¬®¦­® ¨áª âì ¢ ¢¨¤¥ uç áâ­ == brn+2 cos mϕ, â. ¥. 㢥«¨ç¨¢ á⥯¥­ì r ­  2. ®¤áâ ¢«ï¥¬11¢ ãà ¢­¥­¨¥ urr + r ur + 2 uϕϕ = ar n cos mϕ:rb(n + 2)(n + 1)r cos mϕ + b(n + 2)rn cos mϕ − bm2 rn cos mϕ =a= arn cos mϕ ⇐⇒ b =.(n + 2)2 − m2n‚¨¤­®, çâ® â ª ¬®¦­® ¤¥« âì, ¥á«¨(n + 2)2 6= m2 .◭6.3. Š ª ­ ©â¨ ç áâ­®¥ à¥è¥­¨¥, ¥á«¨ ᢮¡®¤­ë© ç«¥­f (r, ϕ) = ar n cos mϕ(n + 2)2 = m2 ?◮‚ í⮬ á«ãç ¥¨«¨f (r, ϕ) = ar n sin mϕ,rn+2 cos(n + 2)ϕï¥âáï à¥è¥­¨¥¬ ãà ¢­¥-­¨ï ‹ ¯« á . ¥è¥­¨¥ ¨é¥âáï ¢ ¢¨¤¥¯à¨¬¥à 18).à¨¬¥à 17.◮uç áâ­ = ϕ(r) cos mϕ(á¬.◭½y2,∆u =r < 2,u|r=2 = sin ϕ + 34 cos ϕ − 23 cos 2ϕ.ˆé¥¬ ç áâ­®¥ à¥è¥­¨¥ ¢ ¢¨¤¥uç áâ­ = ar4 + br4 cos 2ϕ ⇒⇒ 12ar2 + 4ar2 + 12br2 cos 2ϕ + 4br2 cos 2ϕ − 4br4 cos 2ϕ =57=r4r4r2 r2 cos 2ϕ−⇒ uç áâ­ =−cos 2ϕ2232 24„¥« ¥¬ ᤢ¨£:r4r4v =u−+cos 2ϕ ⇒32 24½∆v = 0, r < 2,v|r=2 = − 21 + sin ϕ + 43 cos ϕ.’ ª ª ª íâ® § ¤ ç  „¨à¨å«¥ ­  ªà㣥, â® à¥è¥­¨¥ ­ å®¤¨âáïýãáâ­®þ:4 ³r´1 ³r´sin ϕ +cos ϕ.v=− +223 2Žâ¢¥â.r4 − r4 cos 2ϕ − 1 + r sin ϕ + 2r cos ϕ.u = 32223( 24cos 3ϕ, r > 2,∆u =5à¨¬¥à 18.◮◭ru|r=2 = sin 2ϕ + cos ϕ.‚ í⮬ ¯à¨¬¥à¥, ¥á«¨ 㢥«¨ç¨âì á⥯¥­ì ­  2, â®cos 3ϕr3ï¢-«ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨ï ‹ ¯« á .

®í⮬ã ç áâ­®¥ à¥è¥­¨¥ ­ ¤® ¨áª âì ¢ ®¡é¥¬ ¢¨¤¥:uç áâ­ = f (r) cos 3ϕ ⇒ r2 f ′′ (r) + rf ′ (r) − 9f (r) = r−3 ,Bf®¤­®à (r) = Ar3 + 3 .r‚¨¤­®, çâ® ¨¬¥¥â ¬¥áâ®à¥§®­ ­á :fç áâ­ = br−3 ln r ⇒⇒ 12br−3 ln r − 7br−3 − 3br−3 ln r + br−3 − 9br−3 ln r = r−3 ⇐⇒1ln r⇐⇒ b = − ⇒ fç áâ­ = − 3 .66r‚ ª ç¥á⢥ ç áâ­®£® à¥è¥­¨ï ¬®¦­® ¢§ïâì− ln r3 cos 3ϕ.6r„¥« ¥¬ ᤢ¨£:v =u+58ln rln rcos 3ϕ ⇐⇒ u = v − 3 cos 3ϕ ⇒36r½ 6r∆v = 0, r > 2,⇒ v|ln 2r=2 = sin 2ϕ + cos ϕ + 48 cos 3ϕ.®«ã祭­ ï § ¤ ç  „¨à¨å«¥ à¥è ¥âáï ãáâ­®:µ ¶µ ¶2µ ¶3222ln 2v=cos 3ϕ.cos ϕ +sin 2ϕ +rrr48(ln 2 − ln r) cos 3ϕϕ.u = 2 cos+ 4 sin2 2ϕ +rr6r322 ∆u = 12(x − y ), 1 < r < 2,ur |r=1 = −6 cos 2ϕ + 16 sin 4ϕ,à¨¬¥à 19.u |1r r=2 = 28 cos 2ϕ + 2 sin 4ϕ.◭Žâ¢¥â.◮‡ ¬¥â¨¬, çâ®12(x2 −y 2 ) = 12r2 cos 2ϕ.®í⮬㠨饬 ç áâ-­®¥ à¥è¥­¨¥ ¨ ¤¥« ¥¬ ᤢ¨£:uç áâ­ = r4 cos 2ϕ ⇒ v = u − r4 cos 2ϕ ⇒ ∆v = 0,⇒ vr |r=1 = −10 cos 2ϕ + 16 sin 4ϕ,1v |r r=2 = −4 cos 2ϕ + 2 sin 4ϕ.’¥¯¥àì ¨é¥¬ à¥è¥­¨¥ § ¤ ç¨:µ¶µ¶bd24v = cos 2ϕ ar + 2 + sin 4ϕ cr + 4 + C.rr®¤áâ ¢«ï¥¬ ¢ ªà ¥¢ë¥ ãá«®¢¨ï:−10 cos 2ϕ + 16 sin 4ϕ == cos 2ϕ(2a − 2b) + sin 4ϕ(4c − 4d),⇐⇒−4 cos 2ϕ + 12 sin 4ϕ =´³´³4d= cos 2ϕ 4a − 2b8 + sin 4ϕ 32c − 32 .½64a = − 1115 , b = 4 15 ,⇐⇒c = 0, d = −4.´³cos 2ϕ42 + 64 − 4 sin 4ϕ+Žâ¢¥â.

u(r, ϕ) = r cos 2ϕ+−11r2415rr+ C.à¨¬¥ç ­¨¥. Ž¡à â¨â¥ ¢­¨¬ ­¨¥ ­  â®, çâ® ¯à¨à¥è¥­¨¨ § ¤ ç¨ ¥©¬ ­  ¬ë ­¥ ®¯à¥¤¥«¨«¨ ç¨á«®C.â® ¯à -¢¨«ì­®, ¯®â®¬ã çâ® à¥è¥­¨¥ § ¤ ç¨ ¥©¬ ­  ¤«ï ãà ¢­¥­¨ï‹ ¯« á  ®¯à¥¤¥«¥­® á â®ç­®áâìî ¤® ¯à®¨§¢®«ì­®© ¯®áâ®ï­­®©.◭59§ 7.Œ¥â®¤ ”ãàì¥ á ¯à¨¬¥­¥­¨¥¬ áä¥à¨ç¥áª¨åä㭪権7.1. ‘奬  à¥è¥­¨ï’¥¯¥àì ¡ã¤¥¬ à¥è âì ªà ¥¢ë¥ § ¤ ç¨ ¤«ï ¢­ãâ७­®áâ¨è à , ¢­¥è­®á⨠è à  ¨ è à®¢®£® á«®ï.‡ ¤ ç  14.‡ ¤ ç  15.‡ ¤ ç  16.I. ∆u = f (r, ϕ, θ), r < R0 ,(αu + βur )|r=R0 = u0 (ϕ, θ), u(r, ϕ + 2π, θ) = u(r, ϕ, θ). ∆u = f (r, ϕ, θ), r > R0 ,(αu + βur )|r=R0 = u0 (ϕ, θ), u(r, ϕ + 2π, θ) = u(r, ϕ, θ).∆u = f (r, ϕ, θ), R1 < r < R2 ,(α1 u + β1 ur )|r=R1 = u0 (ϕ, θ),(α u + β2 ur )|r=R2 = u1 (ϕ, θ), 2u(r, ϕ + 2π, θ) = u(r, ϕ, θ).Š ª ¢á¥£¤  ¯à¨ à¥è¥­¨¨ ãà ¢­¥­¨ï ã áá®­  á­ ç «  ­ -室¨¬ ç áâ­®¥ à¥è¥­¨¥ ¨ ᢮¤¨¬ á ¯®¬®éìî ᤢ¨£  ª à¥è¥­¨îãà ¢­¥­¨ï ‹ ¯« á .II.®í⮬ãáࠧ㭠ç­ñ¬á®¢â®à®£®¯ã­ªâ :è âì ãà ¢­¥­¨¥ ‹ ¯« á  | ¨é¥¬ à¥è¥­¨¥ ¢ ¢¨¤¥¡ã¤¥¬à¥-u(r, ϕ, θ) == R(r)V (ϕ, θ):µ¶d2 dR(r)V (ϕ, θ)r+drdrµ¶∂V (ϕ, θ)R(r) ∂ 2 V (ϕ, θ)R(r) ∂= 0 ⇐⇒sin θ++sin θ ∂θ∂θ∂ϕ2sin2 赶1 d2 dR(r)⇐⇒r=R(r) drdr¶µµ¶∂V (ϕ, θ)1 ∂ 2 V (ϕ, θ)1 ∂1= λ.sin θ+=−V (ϕ, θ) sin θ ∂θ∂θ∂ϕ2sin2 赶³´1 ∂ sin θ ∂ + 1∂2Ž¯¥à â®à −­ §ë¢ ¥âáï22sin θ ∂θ∂θsin θ ∂ϕ®¯¥à â®à®¬ ¥«ìâà ¬¨.

 ©¤ñ¬ ᮡá⢥­­ë¥ ä㭪樨 ¨ ᮡáâ-60¢¥­­ë¥ §­ ç¥­¨ï § ¤ ç¨−µ∂V (ϕ, θ)1 ∂sin θ ∂θ sin θ∂θ³V (ϕ + 2π, θ) = V (ϕ, θ).Ž¯ïâìà §¤¥«¨¬´∂ 2 V (ϕ, θ)+ 12∂ϕ2sin θ¯¥à¥¬¥­­ë¥:¶= λV (ϕ, θ),V (ϕ, θ) = Φ(ϕ)Θ(θ) ⇒′′sin θ (sin θ · Θ′ (θ))′ − λ sin2 θ = Φ (ϕ) = −µ2 .⇒ − Θ(θ)Φ(ϕ)®ï¢¨« áì §­ ª®¬ ï § ¤ ç  ˜âãଠ{‹¨ã¢¨««ï:½Φ′′ (ϕ) + µ2 Φ(ϕ) = 0,⇐⇒Φ(ϕ + 2π) = Φ(ϕ)⇔ µ2 = k 2 ⇒ Φ0 (ϕ) = 1, Φk (ϕ) = Ak cos kϕ + Bk sin kϕ, k ∈ N,¨ ­¥§­ ª®¬®¥ ãà ¢­¥­¨¥−1 dsin θ dθµsin θdΘ(θ)d趑¤¥« ¥¬ § ¬¥­ã ¯¥à¥¬¥­­ëå:+ k2Θ(θ)= λΘ(θ).sin2 θξ = cos θ, ξ ∈ [−1; 1].“à ¢­¥-­¨¥ ¯à¨¬¥â ¢¨¤ ãà ¢­¥­¨ï ˜âãଠ{‹¨ã¢¨««ï:d−dk22 dΘ(ξ)(1 − ξ )+Θ(ξ) = λΘ(ξ),dξ1 − ξ2ã ª®â®à®£®, ­  ¯¥à¢ë© ¢§£«ï¤, ­¥â ­¨ª ª¨å ªà ¥¢ëå ãá«®¢¨©.θ ∈ [0; π], â.

¥.ξ = 1 ¨ ξ = −1.‡ ¬¥â¨¬, çâ® ¯à¨ à¥è¥­¨¨ ­ è¨å § ¤ ç¥áâìθ=0à¨¨θ = π,¨«¨ ¢ ­®¢ëå ¯¥à¥¬¥­­ëåí⮬ ¢ íâ¨å â®çª åp(ξ) = 0.â®, ¢ ᨫ㠮âáâ㯫¥­¨ï ­  á. 48®§­ ç ¥â, çâ® ¬®¦¥â áãé¥á⢮¢ âì ­¥ ¡®«¥¥ ®¤­®£® à¥è¥­¨ï,[−1; 1].[−1; 1] ¨£à ¥â ஫쮣࠭¨ç¥­­®£® ¢¬¥áâ¥ á ¯à®¨§¢®¤­®© ­  ®â१ª¥“á«®¢¨¥ ®£à ­¨ç¥­­®á⨠­  ®â१ª¥®¤-­®à®¤­ëå ªà ¥¢ëå ãá«®¢¨© ¤«ï § ¤ ç¨ ˜âãଠ{‹¨ã¢¨««ï:(³´d (1 − ξ 2 ) dΘ(ξ) +− dξdξ|Θ(ξ)| < ∞, ξ ∈ [−1; 1].k02Θ(ξ) = λΘ(ξ), ξ ∈ [−1; 1],1 − ξ2k = 0:³´d (1 − ξ 2 ) dΘ(ξ) = λΘ(ξ),− dξdξ|Θ(ξ)| < ∞, ξ ∈ [−1; 1]. áᬮâਬ á­ ç «  § ¤ çã ¯à¨(61®«ã祭­®¥ ãà ¢­¥­¨¥ ­ §ë¢ ¥âáïãà ¢­¥­¨¥¬ ‹¥¦ ­¤à ,ξ ∈ [−1; 1] à¥λ = n(n + 1), n ∈ N ∪ {0}, ¨ ®­® ¢ëà -ª®â®àë© ¤®ª § «, çâ® ®£à ­¨ç¥­­®¥ ­  ®â१ª¥è¥­¨¥ áãé¥áâ¢ã¥â, ¥á«¨¦ ¥âáï ä®à¬ã«®©Θn (ξ) = Pn (ξ) =£¤¥1 dn 2(ξ − 1)n ,2n n! dξ nPn (ξ) ­ §ë¢ ¥âáï ¯®«¨­®¬®¬n ∈ N ∪ {0},‹¥¦ ­¤à  (¤à㣮£®, «¨­¥©­®­¥§ ¢¨á¨¬®£® á í⨬ ¨ ®£à ­¨ç¥­­®£® ¢¬¥áâ¥ á ¯¥à¢®© ¯à®¨§¢®¤­®© à¥è¥­¨ï ­  ®â१ª¥ ­¥â).

®«¨­®¬ë ‹¥¦ ­¤à  ®àâ®-R1[−1; 1]: −1 Θn (ξ)Θm (ξ) dξ = 0, n 6= m,¯®«­ãî ¢ L2 [−1; 1] á¨á⥬ã.£®­ «ì­ë ­  ®â१ª¥¯à¥¤áâ ¢«ïîâˆâ ª,(³´d (1 − ξ 2 ) dΘ(ξ) = n(n + 1)Θ(ξ),− dξdξ⇐⇒|Θ(ξ)| < ∞, ξ ∈ [−1; 1],1 dn 2⇐⇒ Θn (ξ) = Pn (ξ) = n(ξ − 1)n ,2 n! dξ n¨n ∈ N ∪ {0}.’®£¤  à¥è¥­¨¥¬ § ¤ ç¨(³´d (1 − ξ 2 ) dΘ(ξ) +− dξdξ|Θ(ξ)| < ∞,ξ ∈ [−1; 1]k02Θ(ξ) = n(n + 1)Θ(ξ),1 − ξ2¯à¨ ª ¦¤®¬ 䨪á¨à®¢ ­­®¬ §­ ç¥­¨¨k0ïîâáï â ª ­ §ë-¢ ¥¬ë¥ ¯à¨á®¥¤¨­ñ­­ë¥ ¯®«¨­®¬ë ‹¥¦ ­¤à :Θkn0 (θ) = Pnk0 (ξ) = (1 − ξ 2 )k02d k0Pn (ξ) ⇐⇒dξ k⇐⇒ Pnk0 (ξ) = sink0 θ · Pn(k0 ) (cos θ),‘¨á⥬ â®£®­ «ì­ n ∈ N ∪ {0},k0 6 n.{Pnk0 (θ)}, n > k0 , nR = k0 , k0 + 1, k0 + 2, . . . ®à1k0 (ξ) dξ = 0, n 6= m­  ®â१ª¥ [−1; 1]: −1 Pnk0 (ξ)Pm(¯à®¨§¢®¤­ë¥ ®¤­®£® ¯®à浪  ã ¯®«¨­®¬®¢ à §­ëå á⥯¥­¥©) ¨ï¢«ï¥âáï ¯®«­®© ¢L2 [−1; 1].‚¨¤­®, çâ® ª ¦¤®¬ã+1䨪á¨à®¢ ­­®¬ã n ᮮ⢥âáâ¢ã¥â 2k +ᮡá⢥­­ëå ä㭪権 ®¯¥à â®à  ¥«ìâà ¬¨:Yn0 (ϕ, θ) = Pn(0) (cos θ) ≡ Pn (cos θ),62Ynk (ϕ, θ)="sink θPn(k) (cos θ) cos kϕ, k = 0, 1, 2, .

. . , n;sin|k| θPn(|k|) (cos θ) sin |k|ϕ, k = −1, −2, . . . , −n,ª ¦¤ ï ¨§ ª®â®àëå ­ §ë¢ ¥âáï áä¥à¨ç¥áª®© ä㭪樥©. ˆå  «£¥¡à ¨ç¥áª ï á㬬 nPk=−nank Ynk (ϕ, θ)ª®© ä㭪樥©. Ž­  ®¡®§­ ç ¥âáïä㭪樥© ¯®à浪 n,⮦¥ ï¥âáï áä¥à¨ç¥á-Yn ,­ §ë¢ ¥âáï áä¥à¨ç¥áª®©¨, ª ª ᮡá⢥­­ ï äã­ªæ¨ï ®¯¥à â®à ¥«ìâà ¬¨, 㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î−µ1 ∂sin θ ∂赶1 ∂ 2 Yn (ϕ, θ)+=∂ϕ2sin2 θ= n(n + 1)Yn (ϕ, θ), n ∈ N ∪ {0},nXank Ynk (ϕ, θ).Yn (ϕ, θ) =∂Yn (ϕ, θ)sin θ∂θ¶k=−n‡ ¬¥â¨¬, çâ® ¯®à冷ª ä㭪樨 ®¯à¥¤¥«ï¥âáï á⥯¥­ìî ¯®«¨­®¬  ‹¥¦ ­¤à  (­¥§ ¢¨á¨¬® ®â ¯®à浪  ¯à®¨§¢®¤­®© ¢ ¯à¨-ᮥ¤¨­ñ­­®¬ ¯®«¨­®¬¥).‘¨á⥬ {Ynk (ϕ, θ)}®à⮣®­ «ì­  ­  ¯®¢¥àå­®á⨠¥¤¨­¨ç-­®© áä¥àë ¨ ¯®«­  ­  ­¥© ¢L2 (S1 ).’ ª ª ª ¤¥«¥­¨¥ ¯¥à¥¬¥­­ëå ¯à®¨á室¨«® ¢ ®¤­®à®¤­®¬ãà ¢­¥­¨¨, â® ¯à¨¤ñâáï ­ ©â¨ ¨R(r):µ¶1 ddR(r)r2= λ = n(n + 1) ⇐⇒R(r) drdr⇔ r2 R′′ (r) + 2rR′ (r) − n(n + 1)R(r) = 0 ⇒ Rn (r) = Cn rn +Dn.rn+1”ã­ªæ¨ïµ¶bnnan r + n+1 sink θPn(k) (cos θ)(ck cos kϕ+dk sin kϕ), n ∈ N∪{0},rk6n㤮¢«¥â¢®àï¥â ãà ¢­¥­¨î ‹ ¯« á  ¨ ­ §ë¢ ¥âáï è à®¢®©ä㭪樥©.III.’¥¯¥àì ¨é¥¬ à¥è¥­¨¥ § ¤ ç¨ ¢ ¢¨¤¥ ä®à¬ «ì­ëå à冷¢,£¤¥ ª ¦¤®¥ á« £ ¥¬®¥ ï¥âáï à¥è¥­¨¥¬ ãà ¢­¥­¨ï ‹ ¯« á :63 ) ¢­ãâਠè à  ¢ ¢¨¤¥ à鸞u(r, ϕ, θ) =∞Xαn rn Yn (ϕ, θ) =∞Xrnn=0n=0nXank Ynk (ϕ, θ),r < R,k=−n¡) ¢­¥ è à  ¢ ¢¨¤¥u(r, ϕ, θ) =n∞∞XX1 Xαnank Ynk (ϕ, θ), r > RY(ϕ,θ)=nrn+1rn+1n=0n=0¨k=−n¢) ¢­ãâਠè à®¢®£® á«®ï ¢ ¢¨¤¥u(r, ϕ, θ) =¶ n∞ µXBn X knAn r + n+1=sin θPn(k) (cos θ)(Ck cos kϕ+Dk sin kϕ),rn=0k=0R1 < r < R2 .Žáâ «®áì 㤮¢«¥â¢®à¨âì ªà ¥¢ë¬ ãá«®¢¨ï¬.„«ï í⮣®¯à¨¤ñâáï à §«®¦¨âì ¨å ¢ àï¤ë ”ãàì¥ ¯® áä¥à¨ç¥áª¨¬ äã­ªæ¨ï¬.à¨¬¥à 20.‡ ¤ ç  „¨à¨å«¥ ¢ è à¥: ∆u = 0, r < r0 ,u|r=r0 = u0 (ϕ, θ),u(r, ϕ + 2π, θ) = u(r, ϕ, θ).ãáâì ªà ¥¢®¥ ãá«®¢¨¥ à §«®¦¥­® ¢ àï¤ ¯® áä¥à¨ç¥áª¨¬◮äã­ªæ¨ï¬:u0 (ϕ, θ) =∞Pβn Yn (ϕ, θ).n=0’®£¤ ∞Xαn r0n Yn (ϕ, θ) =∞Xn=0n=0βn Yn (ϕ, θ) ⇐⇒⇔ αn r0n = βn ⇔ αn =Žâ¢¥â.∞Pn=064³βn rr0∞Xβnβn⇒ u(r, ϕ, θ) =nr0´nn=0Yn (ϕ, θ).µrr0¶nYn (ϕ, θ).◭‚ ®¬ ¢¨¤¥ ¬ë ­¨ç¥£® ­¥ ¯®«ã稬, ªà®¬¥ ä®à¬ã« (ª ª ¨¤«ï ä㭪権 ¥áᥫï), â ª ª ª ¯à¨ à §«®¦¥­¨¨ ¯® áä¥à¨ç¥áª¨¬ äã­ªæ¨ï¬ ¯à¨å®¤¨âáï à áª« ¤ë¢ âì ¯® ¯à¨á®¥¤¨­ñ­­ë¬¯®«¨­®¬ ¬.‚ ¯à¨­æ¨¯¥ íâ® ¬®¦­® ᤥ« âì â ª.‘­ ç «  ¬®¦­® à §«®¦¨âì, ­ ¯à¨¬¥à,®¡ëç­®© âਣ®­®¬¥âà¨ç¥áª®© á¨á⥬¥:+∞Pu0 (ϕ, θ)¢ àï¤ ¯®u0 (ϕ, θ) =a0 (θ)2 +(ak (θ) cos kϕ+bk (θ) sin kϕ),   § â¥¬ ª®íää¨æ¨¥­âë ¯® á¨á-k=1⥬¥ ¯à¨á®¥¤¨­¥­­ëå ¯®«¨­®¬®¢:ak (θ) =∞Xckn Pnk (θ)=ckn sink θPn(k) (cos θ),n=kn=k£¤¥∞X(k)Rπak (θ) sink θPm (cos θ)dθckm = R0 ³,´2(k)πkdθsinθP(cosθ)m0€­ «®£¨ç­®,bk (r, θ) =∞Xdkn Pnk (θ)=∞Xdkn P sink θPn(k) (cos θ).n=kn=k®í⮬ã, çâ®¡ë ¨§¡¥¦ âì í⮩ £à®¬®§¤ª®áâ¨, ¢ ­ è¨å § ¤ ç å ªà ¥¢ë¥ ãá«®¢¨ï ïîâáï ª®­¥ç­ë¬¨ á㬬 ¬¨ áä¥à¨ç¥áª¨å ä㭪権.7.2.