Then a nk 2 nk a2 and rr jn nk ra will

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Unformatted text preview: h the theory of Fourier series, allows us to solve the Dirichlet problem for Laplace’s equation in a disk. Indeed, we can now formulate the Dirichlet problem as follows: Find u(r, θ), for 0 < r ≤ 1 and θ ∈ mathbbR, such that 1. u satisfies Laplace’s equation, 1∂ r ∂r r ∂u ∂r + 1 ∂2u = 0, r2 ∂θ2 (5.25) 2. u satisfies the periodicity condition u(r, θ + 2π ) = u(r, θ), 3. u is well-behaved near r = 0, 4. u satisfies the boundary condition u(1, θ) = h(θ), where h(θ) is a given well-behaved function satisfying the periodicity condition h(θ +2π ) = h(θ). The first three of these conditions are homogeneous linear. To treat these conditions via the method of separation of variables, we set u(r, θ) = R(r)Θ(θ), where Θ(θ + 2π ) = Θ(θ). Substitution into (5.25) yields 1d r dr R d2 Θ = 0. r2 dθ2 r dR dr Θ+ r dR dr Θ+R We multiply through by r2 , r d dr d2 Θ = 0, dθ2 and divide by −RΘ to obtain − rd R dr r dR dr = 1 d2 Θ . Θ dθ2 The left-hand side of this equation does not depend on θ while the right-hand side d...
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