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6 Local Analysis of Inhomogeneous Linear Equations
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Airy equation Airy function analytic approximation to y(x asymptotic analysis asymptotic expansion asymptotic matching asymptotic relation asymptotic series behavior of solutions behavior of y(x boundary conditions boundary layer boundary-layer theory boundary-value problem branch points Clue coefficients compute constant contour controlling factor critical point derive determined difference equation differential equation eigenvalue problem exact solution Example Find the leading finite first-order formula Frobenius series gives higher-order inhomogeneous initial conditions initial-value problem inner integral representation irregular singular point Laplace's method leading behavior leading-order linearly independent linearly independent solutions nonlinear obtain optimal asymptotic approximation outer solution Pade approximants Pade sequence parameter perturbation problem perturbation series perturbation theory plot poles polynomial Prob radius of convergence result saddle point satisfies second-order Show solution y(x solve steepest steepest-descent Stirling series Substituting Taylor series tion trajectories uniform approximation valid values vanishes verify WKB approximation WKB theory