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analytic applied approximation axis boundary conditions boundary value calculation Calculus of Variations Cauchy Chapter coefficients Complex Variables compute consider constant continuous contour convergence coordinates curve defined denote Derive discussed diverges domain eigenvalue problem eigenvectors Euler evaluate Example Exercise exist expansion expressed fact Figure finite fluid follows formula Fourier series Green's function hence Hint infinite inner product instance integral interval inverse Laplace transform limit Linear Analysis Mathematical matrix method nonlinear obtain operator Ordinary Differential Equations orthogonal Partial Differential Equations perturbation phase plane plane Real Variable Theory recall region Residue Theorem result satisfies scalar second-order Section self-adjoint Separation of Variables sequence shown in Fig simply singular point Sketch so-called solve Suppose surface Taylor series theorem tion value problem Vector Field velocity whereas York zero