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Asymptotic Expansion of Integrals
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Airy function assume asymp asymptotic expansion asymptotic methods asymptotic sequence behavior boundary conditions boundary layer branch cut branch point Cauchy's Integral Theorem coefficient complex conclude contour integration d2Yi defined determine the leading-order differential equations dimensionless distinguished limit Duffing's equation evaluate exact solution example Figure find the leading-order first-order problem flow Fourier integral Gamma function Hence initial conditions inner and outer inner expansion integrand Jordan's Lemma Laplace integral Laplace's Method leading-order term Mach number matching Method of Stationary Method of Steepest multiple-scale analysis numerical outer expansion perturbation methods perturbation solution real axis regular-perturbation problem Riemann-Lebesgue Lemma saddle point scale secular terms singular slender body slender-body theory small parameter solved subject Stationary Phase Steepest Descent steepest-descent contour subsection Substituting into Equation surface Taylor series theory transonic turbulence two-point boundary value value problem valid velocity Watson's Lemma wherefore Wilcox yields zeroth zeroth-order problem