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Differential Equations Lecture Work Solutions 128

Differential Equations Lecture Work Solutions 128 - 1 u(r =...

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1. u ( r, θ, ϕ ) = n =0 A n 0 r n P n (cos ϕ ) + n =0 m =1 r n P m n (cos ϕ ) ( A nm cos + B nm sin m ϕ ) (7.7.37) u r ( a, θ, ϕ ) = f ( θ ) = = n =0 n A n 0 a n 1 P n (cos θ ) + n =0 m =1 n a n 1 P m n (cos θ ) ( A nm cos + B nm sin m ϕ ) A n 0 n a n 1 = π 0 f ( θ ) P n (cos θ ) sin θ dθ π 0 P 2 n (cos θ ) sin θ dθ n a n 1 A nm = π 0 2 π 0 f ( θ ) P m n (cos θ ) cos m ϕ sin θ dϕ dθ π 0 2 π 0 [ P m n (cos θ ) cos m ϕ ] 2 sin θ dϕ dθ n a n 1 B nm = π 0 2 π 0 f ( θ ) P m n (cos θ ) sin m ϕ sin θ dϕ dθ π 0 2 π 0 [ P m n (cos θ ) sin m ϕ ] 2 sin θ dϕ dθ A nm = π 0 2 π 0 f ( θ ) P m n (cos θ ) cos m ϕ sin
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