We can construct a table of values for the

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Unformatted text preview: gives a decomposition into eigenfunctions which solve the eigenvalue problem. Such a theory can be developed for an arbitrary bounded region D in the (x, y )-plane bounded by a smooth closed curve ∂D. Let V denote the space of smooth functions f : D → R whose restrictions to ∂D vanish. It is important to observe that V is a “vector space”: the sum of two elements of V again lies in V and the product of an element of V with a constant again lies in V . We deﬁne an inner product , on V by f, g = f (x, y )g (x, y )dxdy. D Lemma. With respect to this inner product, eigenfunctions corresponding to distinct eigenvalues are perpendicular; if f and g are smooth functions vanishing on ∂D such that ∆f = λf, ∆g = µg, (5.32) then either λ = µ or f , g = 0. The proof of this lemma is a nice application of Green’s theorem. Indeed, it follows from Green’s theorem that −f ∂D ∂g ∂g dx + f dy = ∂y ∂x = D D ∂ ∂ [f (∂g/∂x)] − [−f (∂g/∂y )] dxdy ∂x ∂y ∂f ∂g ∂ f ∂g + dxdy + ∂x ∂x ∂y ∂y f D ∂2g ∂...
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This document was uploaded on 01/12/2014.

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