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1994 International Press Ac(e Anal analytic Appl Applications of Analysis assume asymptotic analysis asymptotic approximation asymptotic expansion asymptotic solutions axis behavior Bessel functions blow-up boundary conditions boundary layer boundary value bounded C. G. Lange circular cylindrical domain class C1 coefficients consider constant contour convergence corresponding cubic fold point curve defined denote derive determined differential equation eigenvalue entire functions evaluated exists exponentially small Figure fluid follows formula given Hence integral equations jump Lemma limit linear Math Mathematics Subject Classification Mathieu functions Methods and Applications nearly homoclinic orbit nonlinear numerical obtain orthogonal orthogonal polynomials paper parameter polynomials positive power series proof Proposition saddle point saddle region satisfies Section SIAM singular perturbation solitary waves steepest descent tanh Theorem theory trajectory transformation turning point uniform asymptotic uniformly upper half-plane value problem variable vector zero