(20)按相应齐次问题的原征函.解 (r,0)=∑(r)P(cos) r R(0)有界R(a)=0 的l=1时,R1() R1()有界→B1=0 的l≠1时,R1(r)=A1r1+Bar-1-1 R(0)有界→B=0 R(r)=0,l≠ (a)=0 Al=0 学表述正确 数法正确 R(r)满足的微数函 R(r)满足的边界件 =1时的解 ≠1时的解 (3) 解式 (2)❋✱ (20 ✲) ●❍■❏❑❁❂✷ ✾✿▲❉▼◆ u(r, θ) = X∞ l=0 Rl(r)Pl(cos θ) 1 r 2 d dr  r 2 dRl dr  − l(l + 1) r 2 Rl = −4πrδl1 Rl(0)❖ ✻ Rl(a) = 0 P l = 1 ◗❘ R1(r) = A1r + B1r −2 − 2 5 πr3 ❘ R1(0)❖ ✻ ⇒ B1 = 0 R1(a) = 0 ⇒ A1 = 2 5 πa2 R1(r) = 2 5 π ￾ a 2 − r 2
P l 6= 1 ◗❘ Rl(r) = Alr l + Blr −l−1 ❘ Rl(0)❖ ✻ ⇒ Bl = 0 Rl(a) = 0 ⇒ Al = 0 Rl(r) = 0, l 6= 1 ❙❚ u(r, θ) = 2 5 π ￾ a 2 − r 2
r cos θ ❉❯❱❲❳❨ (2 ✲) ❅❂✸❩❳❨ (2 ✲) ❬❭❪ φ ❫❴ (2 ✲) ✾✿▲❉ (2 ✲) Rl(r) ❵❛✷❜✲✸✹ (2 ✲) Rl(r) ❵❛✷✺✻✼✽ (2 ✲) l = 1 ◗✷❅ (3 ✲) l 6= 1 ◗✷❅ (3 ✲) ❅ ❊ (2 ✲) 14