37 hyperboloid 40 initial value problem 85 98 103 124

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Unformatted text preview: neous initial condition u(r, θ, 0) = h(r, θ) is also satisfied. If we set t = 0 in (5.33), this condition becomes ∞ ∞ ∞ a0,k f0,k + k=1 [an,k fn,k + bn,k gn,k ] = h. (5.34) n=1 k=1 Thus we need to express an arbitrary initial displacement h(r, θ) as a superposition of these eigenfunctions. It is here that the inner product , on V comes to the rescue. The lemma implies that eigenfunctions corresponding to distinct eigenvalues are perpendicular. This implies, for example, that f0,j , f0,k = 0, 152 unless j = k. Similarly, for arbitrary n ≥ 1, fn,j , fn,k = fn,j , gn,k = gn,j , gn,k = 0, unless j = k. The lemma does not immediately imply that fn,j , gn,j = 0. To obtain this relation, recall that in terms of polar coordinates, the inner product , on V takes the form 1 2π f, g = f (r, θ)g (r, θ)rdθ dr. 0 0 If we perform integration with respect to θ first, we can conclude from the familiar integral formula 2π sin nθ cos nθdθ = 0 0 that fn,j , gn,j = 0, as...
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This document was uploaded on 01/12/2014.

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