The Asymptotic Solutions of an Ordinary Differential Equation in which the Coefficient of the Parameter is Singular |
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analytic function bounded arc length asymptotic forms asymptotic representations asymptotic solutions becomes infinite behaviors of y₁(z boundary of R₂ bounded from zero bounded in length boundedness coefficients complex plane considered by Langer converges uniformly curves of integration defined denote Doctor of Philosophy domain dz₁ Edmond Darrell Cashwell elementary functions ential equation exponents find the asymptotic follows forms Z u₁(2 formula 16 Idzı independent solutions induction infinite series integration in 16 J₁(z J₂(z K₂ kernel Mn+1 monotone decreasing ORDINARY DIFFERENTIAL EQUATION P₁ PARAMETER IS SINGULAR path of integration pole of second quantity R(F range region Rw regular singular point related equation right of 20 schlicht Second Solution set of curves solution of equation Stokes subregions suitably large constant the-curves corresponding transformation 13 U₁ u₁(z u₂(z University of Wisconsin vanishes or becomes Wisconsin in partial Y₁ Y₂ Z₁ гр